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Photovoltaic phenomena in inhomogeneous semiconductors
ARTICLE IN PRESS
Progress in Quantum Electronics 27 (2003) 295–365
Review
Photovoltaic phenomena in inhomogeneous semiconductors S. Sikorski, T. Piotrowski* Institute of Electron Technology, Al. Lotnikow 32/46, Warsaw 02-668, Poland
Abstract The photovoltaic phenomena in inhomogeneous semiconductors are discussed here. (i) The Rooesbroeck potential is generalised and the formulae defining the Tauc bulk photovoltaic effect (BPV), in semiconductors with position-dependent doping, are derived. (ii) The domain of the inhomogeneities is expanded, and besides doping the cases of position-dependent energy gap EG and mobilities in the presence of temperature gradient are described. Thus a new thermo-photovoltaic effect is revealed. (iii) The equations describing the BPV potential distribution in the case of point illumination are derived. Using the Green function formalism the equations are obtained, which give a base to examine the inhomogeneities especially of resistivity distribution. (iv) The method of images is used to calculate the BPV potential distribution. The potential distributions in rectangular plates and circular slices are presented. (v) The results of photovoltaic measurements carried out on a sample with temperature gradient are presented, giving an experimental confirmation of the thermophotovoltaic effect. Application of BPV to determine experimentally resistivity distribution is demonstrated. (vi) The theory and measurements of the bulk photoelectromagnetic effect (BPEM) taking place in an inhomogeneous semiconductor exposed to magnetic field are presented. r 2003 Elsevier Ltd. All rights reserved.
Contents 1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
299
2.
Photovoltaic effects. Transport equations . . . . . . . . . . . . . . 2.1. Transport of excess carriers in a homogeneous semiconductor 2.2. Transport of excess carriers in a semiconductor with position-dependent doping . . . . . . . . . . . . . . . . . . .
300 300
*Corresponding author. Tel.: +48-22-54-87-942; fax: +48-22-847-0631. E-mail address: [email protected] (T. Piotrowski). 0079-6727/03/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0079-6727(03)00004-1
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Nomenclature B magnetic induction b unit vector parallel to the magnetic induction BPEM bulk photoelectromagnetic effect BPV bulk photovoltaic effect D ambipolar diffusion coefficient (2.45) D dipole moment # ¼ Dx þ iDy complex dipole moment D d slice thickness Dp ðDn Þ hole (electron) diffusion coefficient Dx ; Dy ; Dz components of dipole moment (EMF) electromotive force EBPEM vector of BPEM (6.20) EBPV ; EMPV ; ETPV ; ETEV vectors defined by (3.19)–(3.22) VC T EN BPV ; EBPV ; EBPV vectors defined by (3.28)–(3.31) Efi intrinsic Fermi level Efp ðEfn Þ hole (electron) Fermi level EG energy band gap Eth thermodynamic force defined by (2.11) and (2.17) EV ðEC Þ valence (conduction) band edge g electron–hole pairs generation rate gs electron–hole pairs surface generation GðA; BÞ; G0 ðA; BÞ 3D Green functions I current intensity i; j; k unit vectors of axes x; y; z; respectively J total current density J p ðJ n Þ hole (electron) current density J 0p ðJ 0n Þ hole (electron) current density in the absence of magnetic field J internal current density (2.40) J0 total current density in the absence of magnetic field kB Boltzmann constant LD excess carrier diffusion length Lp ðLn Þ hole (electron) thermoelectric transport coefficient (3.3) and (3.4) M; M N ; M E ; M T ; M M vectors defined by (4.2)–(4.5) mv ðmc Þ hole (electron) effective mass n non-equilibrium electron concentration n0 equilibrium electron concentration þ þ N ¼ ND NA ND ðNA Þ donor (acceptor) concentration n normal unit vector ni intrinsic concentration NV ðNC Þ effective density of states of valence (conduction) band p non-equilibrium hole concentration
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P total number of hole–electron pairs p0 equilibrium hole concentration q elementary charge R radius Rint internal resistance (2.29) s surface recombination velocity s cross-sectional area T temperature TB bulk of the sample tg Y quantity defined by (5.47) U voltage V shockley potential (3.7) V0 shockley potential in equilibrium w ¼ uþiv complex variable x; y; z co-ordinates z ¼ xþiy complex variable CðzÞ function defined by (4.46) Dp excess hole–electron pairs concentration Ds excess conductivity (2.37) dðA BÞ 3D Dirac delta function d2D ðA BÞ 2D Dirac delta function dG Green function defined by (4.25) dC potential defined by (4.34) ep ðen Þ hole (electron) thermoelectric coefficient (3.3) and (3.4) Y angle defined by (6.14) YH angle defined by (6.13) Yp ðYn Þ Hall angle for holes (electrons) (6.3) and (6.4) m ambipolar mobility with illumination m group mobility (2.49) m0 ambipolar mobility in equilibrium mp ðmn Þ hole (electron) mobility mpH mnH Hall mobility of holes (electrons) r non-equilibrium resistivity r0 equilibrium resistivity s non-equilibrium conductivity s0 equilibrium conductivity sp ðsn Þ non-equilibrium hole (electron) conductivity sp0 ðsn0 Þ equilibrium hole (electron) conductivity j ¼ 4a angular width (5.57)–(5.50) j0 equilibrium Fermi potential jp ðjn Þ hole (electron) Fermi potential ðnÞ ðnÞ jðnÞ x ; jy components of potential C C potential defined by (2.14)
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2.2.1. 2.2.2. 2.2.3. 2.2.4.
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303 307 309 311
Transport of excess charge carriers in a semiconductor with extended inhomogeneity in the presence of temperature gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Theoretical foundation . . . . . . . . . . . . . . . . . . 3.2. Numerical example . . . . . . . . . . . . . . . . . . . . 3.3. Experimental investigation of thermophotovoltaic effect .
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312 312 317 318
Photovoltaic electrical fields . . . . . . . . . . . . . . . . . . . . 4.1. Basic equations . . . . . . . . . . . . . . . . . . . . . . . . 4.2. General solution for an open-circuit sample . . . . . . . . . 4.3. Fundamentals of the measurement of the bulk photovoltaic effect (BPV) and the relative gradient of resistivity . . . . . 4.4. Measurement of the resistivity distribution . . . . . . . . . . 4.5. The Green function and the potential distribution in thin samples . . . . . . . . . . . . . . . . . . . . . . . . .
. . .
320 320 322
. .
325 327
.
329
The BPV—potential distributions . . . . . . . . . . . . . . 5.1. The method of images . . . . . . . . . . . . . . . . . 5.2. Half-space and space between two parallel half-planes . 5.3. Infinitely long plate . . . . . . . . . . . . . . . . . . . 5.4. Cuboid with electrodes at two opposite faces . . . . . 5.5. Slices . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1. Half-plane and infinite strip . . . . . . . . . . 5.5.2. Circular slice . . . . . . . . . . . . . . . . . . 5.5.3. Absolute measurements of grad r in a circular slice . . . . . . . . . . . . . . . . . . . . . . 5.5.4. Circular slice with extended electrodes . . . . .
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331 331 332 335 337 339 339 342
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343 347
Bulk photoelectromagnetic effect . . . . . . . . . . . . . . . . . . . 6.1. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . .
350 350 357
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
358
A. Derivation of Eq. (2.17) . . . . . . . . . . . . . . . . . . . . . . .
358
B.1. B.2.
Derivation of (2.22) . . . . . . . . . . . . . . . . . . . . . . . . Derivation of (2.23) . . . . . . . . . . . . . . . . . . . . . . . .
360 361
C.1. Calculation of grad n0 =ni . . . . . . . . . . . . . . . . . . . . . . C.2. Calculation of kB T=q grad ln ni . . . . . . . . . . . . . . . . . .
362 363
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
364
3.
4.
5.
6.
Bulk photovoltaic effect (BPV) . . . . The hole and electron current densities Continuity equations . . . . . . . . . . Stationary transport equation system .
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1. Introduction The bulk photovoltaic effects, resulting from the existence of an inhomogeneity of continuous (non-abrupt) character, have long been ignored and, what is more, no advantage was gained from their occurrence. Initially, studies on the photovoltaic effect due to a continuous inhomogeneity were inspired by its disadvantageous influence upon the results of other electrical measurements, such as e.g., the measurement of the Hall mobility or resistivity. Tauc [1] in 1955 and Trousil [2] in 1956, who discovered the bulk photovoltaic phenomenon (BPV) and formulated its strict theory, explained the mechanism of this influence. The paper of Lashkarev and Romanov [3] devoted to BPV too was published in the next year. These authors discovered BPV independent of Tauc and Trousil, but their results could be published in USSR only after a considerable delay. These studies initiated investigations of the photovoltaic phenomena in inhomogeneous semiconductors, where the relative change of certain material parameters along the diffusion length is well below unity. These parameters include in the first place the concentration of impurities and the chemical composition. The carrier mobility and the width of the band gap, as well as their variability, are associated with these parameters and with the temperature. The inhomogeneous semiconductors have also other features that distinguish them from junction structures, namely, the effect of rectification and excess carrier injection are small and can be neglected. The BPV phenomenon has found its application in the measurements of the resistivity gradient and has become a widely used diagnostic method to which many papers have been devoted. At present, interest in the photovoltaic effects in inhomogeneous semiconductors is inspired by the development of new technologies designed for producing regions with varying band gap, which, when formed in semiconductor structures, greatly improve their performance parameters. The generalised theory of the photovoltaic effects permits modelling the operation of semiconductor devices in a more precise way, and enables us to take advantage of the effects, induced by built-in inhomogeneities, or produced purposely for diagnostic examinations. The present study synthesises the achievements of earlier and more recent studies devoted to the photovoltaic effects active in inhomogeneous semiconductors. Section 2 discusses in detail the classical BPV phenomenon and introduces a new formalism that permits us to interpret it in clear physical terms. Section 2.2 is devoted to the derivation of the continuity equations, which constitute a system of transport equations. Although these equations are confined to the description of the BPV effect here, they have a general significance. Section 3 reports on certain new results based on an extended concept of inhomogeneity in the presence of a temperature gradient. The equations derived in this chapter classify and describe the photovoltaic effects that can occur in materials with such inhomogeneity. Based on the equations obtained in Section 2, Section 4 contains the equations that define the potential distribution created by point illumination of the semiconductor surface. These equations, obtained using the Green function
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formalism, can be used for determining the potential distribution in samples of various shapes. The generality of their mathematical form justifies the application of these equations to the measurement of resistivity. Section 4 also includes the general mathematical foundations of the procedure used by Blackbourn and Larrabee for determining the absolute value of the resistivity gradient. Section 5 provides the general principles and examples of calculating the BPV potential for various sample geometries, based on the method of images and using again the Green function formalism. The potential distributions for 3D cases (cuboidal sample) and 2D cases (circular slices) are also shown. Section 6 describes the bulk photoelectromagnetic (BPEM) effect that was discovered during theoretical and experimental studies on the photovoltaic phenomena in inhomogeneous semiconductors. The theory of BPEM and its experimental confirmation are discussed here in detail. 2. Photovoltaic effects. Transport equations 2.1. Transport of excess carriers in a homogeneous semiconductor Equations describing the carrier transport in a homogeneous semiconductor were derived in a strict form by van Roosbroeck [4,5]. His results are still valid and are utilised by many scientists. The starting point of the Roosbroeck theory are equations, used by Shockley [6] in 1949 that describe the densities of the hole and electron currents J p and J n J p ¼ sp grad V qDp grad p ¼ sp grad V kB Tmp grad p;
ð2:1Þ
J n ¼ sn grad V þ qDn grad n ¼ sn grad V þ kB Tmn grad n;
ð2:2Þ
where sn and sp are the electron and the hole conductivities, q is the elementary charge, Dn and Dp are the electron and the hole diffusion coefficients, mn and mp are the electron and the hole mobilities, kB is the Boltzmann constant and T is the temperature. The hole and electron concentrations (p and n, respectively) include the concentration of the excess carriers (Dp) generated as a result of the semiconductor being illuminated. The potential V corresponds to the intrinsic Fermi level Efi : It is normally found near the centre of the band gap. The potential Efi is the Fermi level in the case when the equilibrium concentration of holes is equal to the concentration of electrons. This is why the distance of Efi to the valence band edge and the distance to the conduction band are approximately the same. The small deviation of Efi from the band gap centre results from the difference between effective masses of the holes and electrons and is of the order of kB T. Eqs. (2.1) and (2.2) have been playing a very important role in semiconductor electronics. Based on the monograph written by Chandrasekhar [7], Roosbroeck considers in Ref. [4] the equations for Jp and Jn to be the Smoluchowski equations adapted to describe the flow of holes and electrons. Both Eqs. (2.1) and (2.2) defining the hole and electron currents contain the ohmic and diffusion terms. The diffusion term can be expressed in two forms, which involve
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the diffusion coefficient or the mobility. This possibility results from the relationship discovered by Eintstein and independently by Smoluchowski. The diffusion coefficient is chosen according to the D1 formulation, which is simpler especially as regards the Einstein–Smoluchowski relationship and the current densities, as pointed out by Landsberg [8]. In the theory of the p–n junction, still another form of Eqs. (2.1) and (2.2) played an important role. Shockley noticed [6] that the mobilities and concentrations of charge carriers were involved in both the conductivities sp and sn and in the term describing diffusion. Hence, if we introduce two Fermi potentials jp and jn ; which correspond to the hole and electron Fermi levels we can express the two currents J p and J n in the form of a single term. Then we obtain the equations: J p ¼ sp grad jp ;
ð2:3Þ
J n ¼ sn grad jn :
ð2:4Þ
These equations are also valid for degenerate semiconductors, and thus are more general than Eqs. (2.1) and (2.2). Eqs. (2.3) and (2.4) are also known in the thermodynamics of irreversible processes [9], where grad jp and grad jn are referred to as thermodynamic forces. When describing photovoltaic phenomena, it is of crucial importance to formulate the equation that defines the density of the total current J: By adding Eqs. (2.3) and (2.4) we obtain J ¼ sEth ;
ð2:5Þ
where s is the total conductivity, and Eth is a thermodynamic force defined by sp sn ð2:6Þ Eth ¼ grad jn grad jn : s s Unlike the potential forces grad jp and grad jn their linear combination Eth may contain generally a non-potential component, and this is the origin of the electromotive force responsible for the occurrence of the photovoltaic effect. The present paper is devoted to the photovoltaic effects in inhomogeneous semiconductors with no abrupt changes of the material parameters (as would be the case in e.g., p–n and l–h junctions). Therefore, the samples examined obey the principle of quasi-neutrality, which says that the concentration of excess holes is equal to the concentration of excess electrons. This is written as p p0 ¼ n n0 ¼ Dp;
ð2:7Þ
where p0 and n0 are the equilibrium concentrations of the holes and electrons, and Dp is the concentration of the excess carriers. The equations presented thus far are valid for any inhomogeneous semiconductor with p0 and n0 dependent on position. However in this section, we will confine our attention to a homogeneous semiconductor, where p0 and n0 are independent of position. Using the grad operator in Eq. (2.7), we obtain grad p ¼ grad n ¼ grad Dp:
ð2:8Þ
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Since this result cannot be directly substituted in Eqs. (2.3)–(2.5), it is necessary to use (2.1) and (2.2) where p and n occur in explicit forms. Adding these equations gives J ¼ s grad V þ kB Tmn grad n kB Tmp grad p:
ð2:9Þ
Substituting (2.8) into (2.9) we obtain J ¼ s grad V þ kB Tðmn mp Þgrad Dp:
ð2:10Þ
In this equation, the carrier mobility, temperature and equilibrium concentrations n0 and p0 are independent of position, and, thus, we can derive from it the equation for the thermodynamic force Eth (as defined by (2.6)): ! J k B T m n mp s ln : ð2:11Þ Eth ¼ ¼ grad V þ grad s q m n þ mp s 0 This was done using the formula " # " # Ds s grad Dp ¼ grad ¼ grad qðmn þ mp Þ qðmn þ mp Þ " # ln s ¼ s grad qðmn þ mp Þ " # ln s=s0 ¼ s grad ; qðmn þ mp Þ
ð2:12Þ
and introducing the equilibrium conductivity s0 so as to avoid the use logarithms for a dimensional quantity. Since s0 is constant here, this substitution does not change the value of the gradient. As will be shown later, even in the case s0 aconst; the ln s=s0 term remains unchanged. Then, Eq. (2.11) can be written in the form J ð2:13Þ Eth ¼ ¼ grad Chom ; s where Chom ¼ V
kB T mn mp s ln q mn þ mp s0
ð2:14Þ
is the potential introduced by Roosbroeck [5, Eq. (19)]. Eq. (2.11) describes the thermodynamic force Eth which occurs in inhomogeneous semiconductors. We can see that Eth has two components. One is the gradient of the potential V and the other is the gradient of a certain function which contains the difference between the carrier mobilities mn mp : The latter component is known as the Dember field and, thus, it is also a potential component. Therefore, the thermodynamic field Eth is defined by the gradient of the single potential Chom ; determined by Eq. (2.13). Hence, we can infer that the photoconductivity in an inhomogeneous semiconductor is not accompanied by the bulk photovoltaic effect which would have acted as origin of the electromotive force.
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2.2. Transport of excess carriers in a semiconductor with position-dependent doping 2.2.1. Bulk photovoltaic effect (BPV) If a semiconductor is doped so that the concentrations of donors ND and acceptors NA are position-dependent and the concentration of excess charge carriers Dp is equal to zero, the total equilibrium concentrations of holes and electrons are also position-dependent and equal to p0 ðx; y; zÞ and n0 ðx; y; zÞ; respectively. Under these conditions, the total current J is zero. Then by substituting s ¼ s0 ; V ¼ V0 ; p ¼ p0 ; n ¼ n0 in the general Eq. (2.9), we obtain 0 ¼ s0 grad V0 þ kB Tmn grad n0 kB Tmp grad p0 :
ð2:15Þ
This equation can be understood as the flow of the drift current s0 grad V0 ; which is reduced by the hole and electron diffusion currents, so that the total current is equal to zero. The term grad V0 represents an electric field called the ‘built-in field’ which is an important parameter that characterises semiconductors doped in a position-dependent manner. By subtracting (2.15) from (2.9) and dividing the result by s we obtain J s0 kB T ðmn mp Þgrad Dp: Eth ¼ ¼ grad V þ grad V0 þ ð2:16Þ s s s This can be done because Eq. (2.9) is also valid for inhomogeneous semiconductor. After several transformations, given in Appendix A, we obtain ! J kB T mn mp s Eth ¼ ¼ gradðV V0 Þ þ grad ln s q mn þ mp s0
2qm0 Dp grad V0 ; s
ð2:17Þ
where m0 ¼
sp0 sn0 mn þ m s0 s0 p
ð2:18Þ
is the equilibrium ambipolar mobility. The concept of the ambipolar diffusion coefficient was introduced by Roosbroeck [5] (see Eq. (2.45)). This coefficient is related to the ambipolar mobility through the Einstein–Smoluchowski relationship. These parameters appear in the equations that describe the transport of excess carriers, i.e. holes and electrons that occur simultaneously and have the same concentrations (Dp ¼ Dn). We therefore deal with the ambipolar transport. It can easily be seen from Eq. (2.18) that, in an n-type semiconductor where sn bsp ; the ambipolar mobility m0 ¼ mn : Similarly m0 ¼ mp for the p-type. This means that the minority carriers ‘impose’ the mobility on the majority carriers. This apparent paradox has led to the use of the erroneous term ‘minority carriers’ instead of ‘the excess carriers’. It is interesting to note that Roosbroeck [5] used the term ‘added current carriers’. Continuing the reasoning that led to Eq. (2.17), let us note that, after introducing the built-in field grad V0 (which results from the inhomogeneity of the semiconductor), this equation has become a generalised form of Eq. (2.11). Eq. (2.17) can also be
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written in the form Eth ¼
J 2qm0 Dp ¼ grad C þ grad V0 ; s s
ð2:19Þ
where C V V0
k B T m n mp s ln þ j0 q m n þ mp s 0
ð2:20Þ
is a generalised form of the potential Chom defined by (2.14). The potential C is normalised by using the constant j0 ; which is the value of the single Fermi potential occurring in a given semiconductor when it is in the state of equilibrium. This means that J ¼ 0 and Dp ¼ 0 at any point. If this condition is not fulfilled, i.e., Ja0; Dp ¼ 0 and s ¼ s0 at certain points within the semiconductor, such as e.g. point r0 with the co-ordinates x0 ; y0 ; z0 ; then by Eq. (2.20) we have Cðr0 Þ ¼ V ðr0 Þ V0 ðr0 Þ þ j0 :
ð2:21Þ
An analysis of this equation, given in Appendix B, leads to the conclusion that Cðr0 Þ ¼ jðr0 Þ:
ð2:22Þ
Therefore, the normalised potential C at the point r0 ; where Dp ¼ 0; is equal to the Fermi potential jðro Þ and hence these two potentials are here equivalent from the point of view of the measurement of the potential C and of the photovoltage. According to the principles of non-equilibrium thermodynamics, the Fermi potential jM of an arbitrary galvanic measuring probe at these points will acquire a potential equal to the Fermi potential of the given semiconductor. The potential jM of the probe will thus represent the potential Cðr0 Þ: How the two sets of charge carriers within the contact region are brought to thermodynamic equilibrium and the physical sense of the Fermi level are explained in Ref. [10]. The above analysis of the situation when Dp ¼ 0 permits us to conclude that the Dember component of the thermodynamic force Eth is not involved here. This suggests that the Dember effect occurs only when contact phenomena are in question. In the regions where Dpa0 we have two quasi-Fermi potentials jp and jn : It would be interesting to know how the potential C is related to them. It is demonstrated in Appendix B that for small Dp we have sp0 sn0 C¼ j þ j : ð2:23Þ s0 p s0 n Hence we can see that the value of C lies somewhere between the values of jp and jn ; rather nearer to the Fermi potential of the majority carriers. The field Eth has a potential and non-potential components defined by Eq. (2.19). We should ask whether this decomposition of the field Eth into components is unique, or, in other words, whether this field can have another non-potential component. When we however assume that C cannot be a direct function of the coordinates since it is a function of state, i.e. a function of V0 and Dp; it appears that its non-potential component is well defined by Eq. (2.19).
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In order to illustrate the equations and results discussed above let us consider the simplest case of a cuboidal sample in which all the parameters depend on the coordinate x alone, as shown in Fig. 1. When adapting Eq. (2.19) to this onedimensional case we have Jx dC 2qm0 Dp dV0 Ethx ¼ ¼ ð2:24Þ dx s s dx and hence dC Jx 2qm0 Dp dV0 ¼ ; ð2:25Þ dx s s dx which permits us to calculate the voltage at the sample contacts as Z B Z B Z B dC dx 2qm0 Dp dV0 UBA ¼ dx ¼ CB CA ¼ Jx dx: ð2:26Þ dx s s dx A A A In order to interpret Eq. (2.26) let us use the concept of the electromotive force Z B 2qm0 Dp dV0 dx; ð2:27Þ ðEMFÞ ¼ s dx A the current outflowing from the sample I ¼ Jx s and the internal resistance of the sample Z 1 B dx ; Rint ¼ s A s
ð2:28Þ
ð2:29Þ
where s is the sample’s cross-section, we can write Eq. (2.26) in the form UBA ¼ ðEMFÞ Rint I: When the sample is in open circuit, the current I is zero and we have Z B 2qm0 Dp dV0 dx ¼ ðEMFÞ: UBA ¼ UOC ¼ s dx A
ð2:30Þ
ð2:31Þ
Fig. 1. Illuminated inhomogeneous sample with internal field grad V0 : A; B—supply electrodes. Point electrodes illustrate physical meaning of the potentials jp ; jn and C:
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We can see from this equation that, upon illumination of the semiconductor, the built-in field grad V0 becomes the origin of the electromotive force, which results in the occurrence of the photovoltaic phenomenon. The short-circuit current ISC in the sample may be calculated form Eq. (2.30) by assuming UBA ¼ 0: Then we obtain RB 2qm Dp=s ðEMFÞ ISC ¼ ¼ A R B0 : ð2:32Þ Rint 1=S dx=s A
The above reasoning is based on the fact that, in the non-illuminated region, the potential C is equal to the Fermi level j: Any metallic electrode or probe placed in this region will acquire a potential equal to the potential C: Within the illuminated region, the potential of the electrode depends on the structure formed in the contact area. If it is a p–n junction, the potential of the metallic contact will be equal to the quasi-Fermi potential of the minority carriers jmin : A point contact 1 (Fig. 1) of this type is for example formed when a point probe is used for measuring the so-called floating potential, which is just equal to the Fermi potential of the minority carriers as shown by Bardeen [11] and Henisch [12, Eq. (833.12)]. The point-probe measurements played an important role at the beginnings of the development of semiconductor electronics. An electrode of another type is the probe 2 shown in Fig. 1, which contains an l–h structure. It can be shown that it acquires a potential equal to the quasi-Fermi potential of the majority carriers jmaj : If the sample has been prepared so that it possesses a non-illuminated outgrowth 3, the probe of this type will acquire the potential equal to the potential C: This can be proved on the assumption that the length of the outgrowth exceeds the diffusion length LD of the excess carriers. This reasoning indicates that when measuring the parameters characteristic of the fields, potentials and currents in an illuminated semiconductor that contains excess carriers, it is very important to be aware of the method employed for the measurements. When measuring the voltage UBA under open-circuit conditions, i.e., measuring the electromotive force, Eq. (2.31) gives us the built-in field dV0 =dx: In practice however we are rather interested in the gradient of the equilibrium resistance r0 : To analyze this parameter, we should first determine the derivative of the carrier concentration as a function of dV0 =dx: In an inhomogeneous non-degenerate semiconductor, the concentrations n0 and p0 are given by n0 ¼ ni exp
q ðV0 j0 Þ; kB T
p0 ¼ ni exp
q ðj V0 Þ: kB T 0
ð2:33Þ
Hence, if n0 and p0 are functions of the co-ordinate x alone, we have dn0 q dV0 ¼ n0 ; kB T dx dx
dp0 q dV0 ¼ p0 kB T dx dx
ð2:34Þ
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since j0 ¼ const: By multiplying the first of Eqs. (2.34) by qmn; the other equation by qmp and then adding we obtain d ds0 q dV0 ðsp0 þ sn0 Þ ¼ ¼ ðsn0 sp0 Þ ð2:35Þ dx kB T dx dx and hence dV0 kB T 1 ds0 kB T 1 1 dr0 ¼ ¼ q sn0 sp0 dx q sn0 sp0 r20 dx dx kB T s20 dr0 : ¼ q sn0 sp0 dx
ð2:36Þ
Then substituting this result into Eq. (2.27), taking sEs0 ; and denoting Ds ¼ qðmn þ mp ÞDp; we obtain the electromotive force EMF as Z B 2qm0 Dp dV0 dx ðEMFÞ ¼ s dx A Z B kB T 2qm0 Dp s20 dr0 dx ¼ q A s sn0 sp0 dx Z kB T B 2m0 Ds s0 dr0 E dx: ð2:37Þ q A mn þ mp sn0 sp0 dx (1) In an n-type semiconductor m0 Emp according to (2.18) and thus we have m0 =ðmn þ mp ÞEmp =ðmn þ mp Þ ¼ 1=ðmn =mp þ 1Þ ¼ 1=ðb þ 1Þ; where b is the ratio of mobilities. Moreover, since here sn bsp ; we have sn sp Es0 and then Z B kB T 2 dr ð2:38Þ ðEMFÞn ¼ Ds 0 dx: q bþ1 A dx (2) In a p-type semiconductor, the relationship between the mobilities is m0 =ðmn þ mp ÞEmn =ðmn þ mp Þ ¼ ðmn =mp Þ=ðmn =mp þ 1Þ ¼ b=ðb þ 1Þ and sp bsn: and sn sp E s0 : Hence we obtain Z B kB T 2b dr ð2:39Þ ðEMFÞp ¼ Ds 0 dx: q bþ1 A dx These results are fully consistent with Eqs. (3.25) and (3.24) in Ref. [13] and with Eqs. (7) in Ref. [3]. Eqs. (2.38) and (2.39) are useful in determining experimentally dr0 =dx; and also in determining the precise course of the r0 ðxÞ distribution based on the bulk photovoltaic effect. However, Eq. (2.27) is more general since it describes both n-type and p-type semiconductors and also semiconductors with a continuous (without the barrier) transition from n-type to p-type. 2.2.2. The hole and electron current densities In order to formulate the equations that describe the transport of excess carriers in a complete form, it is necessary to transform the equations for J p and J n : Let the starting point be a current which we will call the internal current. The density of this
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current J is defined as sn sp J ¼ J n J p: s s
ð2:40Þ
The densities of the electron and hole currents calculated from this equation are sp J p ¼ J þ J ; ð2:41Þ s Jn ¼
sn J J : s
ð2:42Þ
Now we add these two equations and see that the current J ; which is a component of the current due to the holes, and the current J ; which is a component of the electron current, cancel each other and the resultant is the total current J: This is the explanation why the current J has been called ‘internal’: it does not contribute to the total current J: In physical terms, these two current components cancel each other, since the excess holes in the current J move in the same direction as the excess electrons in the current J : It can be interpreted as an electrically neutral flux of electron–hole pairs. Substituting Eqs. (2.1) and (2.2) into Eq. (2.40) and ordering the terms we obtain s sp n mp grad p þ mn grad n : J ¼ kB T ð2:43Þ s s As can be seen, J is determined by the diffusion components, whereas the terms that contain grad V0 cancel each other. It follows from (2.7) that p ¼ p0 þ Dp and n ¼ n0þ Dp. Using these relationships in Eq. (2.43) gives s kB T sp n ðsn mp grad p0 þ sp mn grad n0 Þ kB T J ¼ mn þ mp grad Dp s s s kB T ðsn mp grad p0 þ sp mn grad n0 Þ kB Tm grad Dp; ð2:44Þ ¼ s where m is the ambipolar mobility in an illuminated semiconductor, and also fulfils the equation for the ambipolar diffusion coefficient D: D ¼ kB Tm;
m¼
sn sp mp þ mn : s s
ð2:45Þ
To proceed further, let us differentiate Eqs. (2.33) and (2.34). When this is done we have grad p0 ¼
q p0 grad V0 ; kB T
grad n0 ¼
q n0 grad V0 : kB T
Substituting these results into Eq. (2.44) gives s sp n J ¼ q mp p0 mn n0 grad V0 kB Tm grad Dp: s s
ð2:46Þ
ð2:47Þ
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Now, if we adopt the quasi-neutrality principle (2.7), the bracketed expression that appears in the grad V0 term can be transformed to take the form sn sp 1 m p0 mn n0 ¼ ½sn mp ðp DpÞ sp mn ðn DpÞ s s p s
1 1 1 sn spn sn sp sn snp mp mnp Dp ¼ s q q s s ¼ mn Dp;
ð2:48Þ
where sn sp m m s p s n is known as the group mobility. Using (2.48) in Eq. (2.47)we obtain m ¼
ð2:49Þ
J ¼ qm Dp grad V0 kB Tm grad Dp:
ð2:50Þ
We can infer from this equation that, in an illuminated inhomogeneous semiconductor, the internal current is composed not only of the ambipolar diffusion current, which is proportional to the gradient of the excess carrier concentration Dp; but also contains the drift current of the excess carriers induced by the built-in field grad V0 : This is thus another type of the neutral flux of the electron–hole pairs, which is a result of the inhomogeneously distributed doping in the semiconductor. It is worthwhile to analyse the behaviour of this neutral flux under the action of a magnetic field. Thanks to the Lorentz force, the magnetic field tends to split the flux along the direction perpendicular to itself and to the flux axis, thereby generating an electromotive force. The result of the action of the magnetic field on the component kB Tm grad Dp (Eq. (2.50)) is known as the photoelectromagnetic effect [14,15]. The action of magnetic field B upon the component qm grad V0 is the bulk photomagnetoelectric effect, which will be discussed later in Section 6. Utilising Eqs. (2.19), (2.41), (2.42) and (2.50) we can write equations for the densities of the hole and electron currents in an inhomogeneous semiconductor that contains excess carriers. These equations are J p ¼ sp grad C qD grad Dp
2qm0 sp Dp grad V0 qm Dp grad V0 ; s
ð2:51Þ
2qm0 sn Dp grad V0 þ qm Dp grad V0 : ð2:52Þ s By putting grad V0=0 in these equations we obtain the equations derived by Roosbroeck [5] for homogeneous semiconductors. J n ¼ sn grad C þ qD grad Dp
2.2.3. Continuity equations To describe the transport of charge carriers with allowance made for the excess carriers, it is necessary to start with two continuity equations: first equation for the space charge and the second equation for the concentration of the excess carriers. When adopting the quasi-neutrality principle, the space charge and its time derivatives are equal to zero, from which it follows that the field of the total current
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density vector is solenoidal. Hence, the first continuity equation is div J ¼ 0:
ð2:53Þ
This equation may also be used for time-dependent processes, but we must remember that, there, it is an approximation valid only when the displacement currents are negligible. For stationary processes, Eq. (2.53) is strictly fulfilled even at high concentration Dp; irrespective of whether or not the quasi-neutrality principle is satisfied. Since J ¼ J p þ J n ; we can easily see that div J p ¼ div J n :
ð2:54Þ
The second continuity equation contains the excess carrier concentration Dp: qDp 1 1 ¼ div J p u þ g ¼ div J n u þ g; qt q q
ð2:55Þ
where u is the recombination rate and g is the generation rate of the electron–hole pairs. In view of the fact that the equilibrium carrier concentrations are independent of time, it is only the time derivative of the excess carrier concentration placed on the left-hand side of Eq. (2.55), which should be taken into account. An important assumption is that the carrier generation rate g due to an external factor, such as light, is the same for both the holes and the electrons. If we assume that the recombination mechanism is of the RSH (Shockley–Read–Hall) type, we obtain the well-known form [16] u¼
pn n2i ; tp ðn þ n1 Þ þ tn ðp þ p1 Þ
ð2:56Þ
where ni is the intrinsic concentration, whereas tn ; tp ; n1 and p1 are the time and the concentration parameters of the recombination centres, respectively. The RSH type is the basic mechanism of recombination in Ge and Si. To put Eq. (2.55) into the form that meets the practical application requirements, we should calculate div J p (or div J n ) using Eq. (2.41) (or (2.42)) and basing on the first continuity Eq. (2.53). When this is done we have s s sp p p div J p ¼ div J þ J ¼ div J þ J grad s s s sp ð2:57Þ þ div J ¼ 0 þ J grad þ div J ; s which when substituted into (2.55) gives qDp 1 sp ¼ div J þ J grad u þ g: qt q s
ð2:58Þ
In order to obtain a more detailed form of this equation, we should rearrange and transform this equation so as to calculate: grad sp =s: grad
sp sn0 sp þ sp0 sn q qm ¼ grad V grad Dp: þ 0 kB T s s2 s
ð2:59Þ
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2.2.4. Stationary transport equation system The results obtained thus far permit us to write a system of three equations that are fulfilled by the three fundamental quantities: (i) The total current density J; (ii) the potential C and (iii) excess carrier concentration Dp: These equations are J ¼ s grad C 2qm0 Dp grad V0 ;
ð2:60Þ
div J ¼ 0;
kB T m grad Dp þ m Dp grad V0 div q
sn0 sp þ sp0 sn m kB T grad V0 grad Dp J þ s2 s Dpðp0 þ n0 þ DpÞ þ g ¼ 0: tp ðn þ n1 Þ þ tn ðp þ p1 Þ
ð2:61Þ
ð2:62Þ
Eq. (2.60) follows immediately from Eq. (2.17). Eq. (2.61) is the continuity Eq. (2.53), and Eq. (2.62) is a detailed form of the continuity Eq. (2.58) and is composed of four members. The first member can be obtained by acting upon J ; as given by Eq. (2.50), with the operator div. The coefficient accompanying J in the second member of this equation is gradðsp =sÞ given by (2.59). The third member determines the recombination rate u given by Eq. (2.56), and the fourth represents the externally induced generation. The continuity Eq. (2.62) is the time-independent generalised form of the widely known Eq. (12) (given by Roosbroeck in Ref. [5]). That this is so can be seen by putting grad V0 ¼ 0 in Eq. (2.62). Compared to the equation which describes the behaviour of a homogeneous semiconductor, Eq. (2.62) has a very characteristic feature. If we assume g ¼ 0 (no illumination) and Ja0 then substituting Dp ¼ 0 gives the left-hand side of Eq. (2.62) proportional to (grad V0 ) J which cannot be zero. Hence the solution of Eq. (2.62) should be Dpa0: The obtained result shows that in an inhomogeneous semiconductor without illumination the current injection of excess carriers takes place, just as in the case in the neighbourhood of a p–n junction. In order to obtain a unique solution, we should specify the boundary conditions. Let us consider the case when the semiconductor sample has two metallic contacts (Fig. 2). On the surface S3 ; the boundary conditions for the hole and electron currents are J p n ¼ qsDp qgs ;
ð2:63Þ
J n n ¼ qsDp þ qgs
ð2:64Þ
where n is the normal unit vector, s is the surface recombination velocity and gs is the hole-electron pair generation rate per unit surface area. The normal components J p n and J n n of vector J p and J n define the charge of the holes and electrons that flow into the unit surface area. By adding these equations two-side, we obtain the first
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Fig. 2. Sample with arbitrary shape. S1 ; S2 —electrodes, S3 —the remaining area of the surface, Dp— concentration of excess carriers, TB—bulk of the sample.
boundary condition Jn ¼ 0;
ð2:65Þ
which means that the normal component of the total current J is equal to zero, thanks to which no resultant electric charge accumulates there. Now, multiplying the first of the equations by sn =s; the second equation by sp =s; and then adding, we obtain the second boundary condition J n ¼ qsDp qg:
ð2:66Þ
These boundary conditions for an inhomogeneous semiconductor may be formulated in a more detailed form by using Eqs. (2.50) and (2.60). Then, they become
qC 2qm0 Dp qV0 ¼ ; qn s qn
ð2:67Þ
qDp qV0 m Dp ¼ sDp gs : ð2:68Þ qn qn The operator q=qn means the derivative in the direction normal to the sample surface. D
3. Transport of excess charge carriers in a semiconductor with extended inhomogeneity in the presence of temperature gradient 3.1. Theoretical foundation The description of the excess carrier transport in an inhomogeneous semiconductor, with due allowance made for the position dependence of its band gap EG ; requires resorting to a sufficiently general theory. The most valuable contributions to such a theory were made by Tauc [13], Marshak [17–21], van Vliet [22,23], van Ruyven [24], Emtage [25] and Kishore [26]. Based on the results obtained by these
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investigators, the present author [27] formulated equations that described the transport of energy and of charge carriers with account of the presence of excess carriers. Then the concept of semiconductor inhomogeneity was extended [28] to include not only the position-dependent doping but also the position-dependent band gap EG : In later publications [29,30], the transport equations were generalised to include the position-dependence of temperature (grad Ta0). The generalised equations permit us to identify all the photovoltaic effects that occur under these conditions. As in Section 2.1, we begin the description of the carrier transport with formulating the general equations that describe the hole and electron current densities. According to Marshak [20], we have Efp J p ¼ sp grad Lp grad T; ð3:1Þ q J n ¼ sn grad
Efn Ln grad T; q
ð3:2Þ
where Efp and Efn are the Fermi levels for the holes and electrons, respectively, and Lp and Ln are the coefficients which define the thermoelectric currents. Eqs. (3.1) and (3.2) are analogous to Eqs. (2.3) and (2.4), except that the potentials jn and jp are replaced by the Fermi levels Efn and Efp divided by q; and that certain members proportional to grad T appear. In the paper of Marshak and Vliet [20] the coefficients Lp and Ln are defined as the ratios of the integrals of energy functions, Fermi functions and carrier relaxation constants. The integrals cover the first Brillouin zone. The authors mentioned above assumed that the relaxation constants depend on the energy alone and that the energy depends on the wave vector k in a spherical-parabolic manner. Based on these assumptions, the following results are obtained:
kB EV Efp ep L p ¼ sp ; ð3:3Þ q kB T
kB Efn EC en ; ð3:4Þ Ln ¼ sn q kB T where EV and EC are the edges of the valence and conduction bands, Efp and Efn are the hole and electron Fermi levels, and ep and en are the hole and electron thermoelectric coefficients, respectively. When the phonon scattering prevails, then for non-degenerate semiconductor with parabolic-spherical band, we have ep ¼ en ¼ 2: (Appendix 1 in Ref. [27]). By combining Eqs. (3.1) with (3.3) and Eqs. (3.2) with (3.4) we obtain
sp EV Efp grad Efp þ grad kB T ep grad kB T ; ð3:5Þ Jp ¼ q kB T
sn Efn EC grad Efn grad kB T þ en grad kB T : Jn ¼ ð3:6Þ q kB T
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Now we aim at transforming Eqs. (3.5) and (3.6) so as to give them a Shockley-like form. The Shockley potential V [6], which was already introduced in Section 2.1, is related to the intrinsic Fermi level Efi by the relationship V ¼
Efi : q
ð3:7Þ
We should also use certain equations from the Maxwell–Boltzmann statistics which involve the energy level Efi ; the effective densities of states NV in the valence band and NC in the conduction band, and the intrinsic concentration ni : These equations are: p Efi Efp ¼ kB T ln ; ni
EV Efp p ¼ ln ; NV kB T
ð3:8Þ
n Efn Efi ¼ kB T ln ; ni
Efn EC n ¼ ln : NC kB T
ð3:9Þ
Now, by using Eqs. (3.7)–(3.9) in Eqs. (3.5) and (3.6) and applying certain transformations (described in detail in Appendix 1 to our earlier paper [29]), we obtain the equations for Jp and Jn as J p ¼ sp grad V mp kB T grad p
kB T ni kB T grad ln ni sp ep ln ; þ sp grad q q NV J n ¼ sn grad V þ mn kB T grad n
kB T ni kB T grad ln ni þ sn en ln : sn grad q q NC
ð3:10Þ
ð3:11Þ
Putting grad ln ni ¼ 0 and grad T ¼ 0 in these equations gives Eqs. (2.1) and (2.2), a result which proves that our Eqs. (3.5) and (3.6) are Shockley-like equations. If we regard Eqs. (3.10) and (3.11) as belonging to the system of transport equations, we can consider the currents J p and J n ; the concentrations p and n and the potential V to be unknown quantities, with the remaining quantities as given. We assume that the distribution Tðx; y; zÞ is known. If the distribution of the chemical composition of the material is given (e.g. SiGe), then the intrinsic concentration ni ¼ ni ðT; x; y; zÞ; the mobilities mn and mp ; the effective densities of states NV and NC are all functions of temperature and co-ordinates. The introduction of the concentrations p and n into Eqs. (3.8) and (3.9) permits us to apply the neutrality principle, which is very useful for the photovoltaic phenomena that take place in inhomogeneous semiconductors. Now by adding Eqs. (3.10) and (3.11) we obtain the total current density J as kB T grad ln ni J ¼ s grad V þ ðmn grad n mp grad pÞkB T þ ðsp sn Þ q
ni ni kB T ; ð3:12Þ þ sn en ln sp ep ln grad q NC NV
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which, when assuming grad ln ni ¼ 0 and grad T ¼ 0; becomes Eq. (2.9). Hence, we can see that (3.12) is a generalised form of (2.9). The presence of the term grad ln ni results from the allowance made for the position-dependent chemical composition of the semiconductor, whereas the non-zero value of the grad T reflects the effect of varying temperature upon the flow of the current and upon the distribution of the electric field. For the sake of precision, it should be added that grad ln ni may be also related with the temperature gradient. In further transformations of Eq. (3.12), we will use certain relationships and notations consistent with the neutrality principle, such as p ¼ p0 þ Dp;
n ¼ n0 þ Dp;
sp ¼ sp0 þ Dsp ;
ð3:13Þ
sn ¼ sn0 þ Dsn ;
s ¼ s0 þ Ds;
ð3:14Þ
where Dsp ¼ qmp Dp;
Dsn ¼ qmn Dp;
Ds ¼ qðmn þ mp ÞDp:
ð3:15Þ
Moreover, it appeared necessary to utilize the relationship between the equilibrium concentrations p0 and n0 and the Fermi potential under conditions of equilibrium (no illumination). This relation is defined by the equation p0 q n0 ðj V0 Þ ¼ ln ; ð3:16Þ ln ¼ ni k B T 0 ni which results from the Maxwell–Boltzmann statistics adequate for the nondegenerate semiconductors. The presence of other members in Eq. (3.12) permits us to confirm that, apart from the bulk photovoltaic effect, other photovoltaic phenomena and, in addition, the Seebeck effect also take place in the semiconductor. To analyse these phenomena we need to transform Eq. (3.12) using Eqs. (3.13)–(3.16), which requires a series of algebraic operations to be done. These were described in detail in Ref. [29]. These transformations give the equation for the thermodynamic force Eth similar to Eq. (2.17): J ð3:17Þ Eth ¼ ¼ grad C þ EBVP þ EMPV þ ETPV þ ETEV ; s where ! mn mp kB T s grad C ¼ grad V V0 j0 ln ; ð3:18Þ s0 mn þ mp q 2qm0 Dp gradðV0 j0 Þ; s 1 s0 s qðmn mp Þðn0 p0 Þ ¼ 2 ln kB T s0 Ds s0 s ! mn mp m Dp grad ln n ;
mn þ mp mp
EBPV ¼ EMPV
ð3:19Þ
ð3:20Þ
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q en þ ep EG sn0 sp0 n0 s0 s ðmn mp Þ þ ln ln s 2 2kB T s0 ni Ds s0
en ep 3 m c þðmn þ mp Þ þ ln 2 mv 2 kB T ;
Dp grad q
1 n0 p0 kB T : ¼ sn0 en ln sp0 ep ln grad s q NC NV
ETPV ¼
ETEV
ð3:21Þ ð3:22Þ
These equations are valid for the inhomogeneous semiconductor, with positiondependent doping, chemical composition and temperature. The excess carriers generated in the material by illumination induce the photovoltaic effects described by Eqs. (3.17)–(3.22). The components of the thermodynamic force Eth are: grad C is defined by Eq. (3.18) which differs from Eq. (2.20) only in that j0 is position-dependent whereas in the former it is not; EBPV is the component representing the photovoltaic effect, analogous to the second term of Eq. (2.19), except that it does not include the varying potential j0 : An analysis of this component shows that, in spite of its apparent simplicity, it has a complex structure (to be discussed later in the text); EMPV is the photovoltaic component resulting from the position dependence of the electron-to-hole mobility ratio; ETPV is the component of the thermophotovoltaic effect, which occurs wherever the simultaneous presence of a photoconductivity and temperature gradient takes place; ETEV is the component of the Seebeck classical thermoelectric effect, which does not belong to the group of photovoltaic phenomena. In order to examine the internal structure of the generalised bulk photovoltaic effect (Eq. (3.19)), we should derive an equation which will define the built-in field gradðV0 j0 Þ; and reflect the role of the position-dependent doping, positiondependent temperature and position-dependent chemical composition in the occurrence of this built-in field. To do this we proceed as follows. We can see from (3.16) that
kB T n0 n0 kB T kB T n0 ln þ grad ln : ð3:23Þ gradðV0 j0 Þ ¼ grad ¼ ln grad q q q ni ni ni The calculations were performed based on certain assumptions and using additional notations. These are: (a) The resultant concentration of the ionised impurities N does not depend on temperature and is only a function of the co-ordinates x; y; z; which can be expressed as þ N ¼ Nðx; y; zÞ ¼ ND NA ;
ð3:24Þ
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þ where ND and NA are the concentrations of the ionized donors and acceptors, respectively. (b) The energy band gap depends explicitly on the co-ordinates, because of the position-dependent composition, and on temperature, which is written as
EG ¼ EG ðT; x; y; zÞ
ð3:25Þ
and (c) similarly the product of the effective densities of states in the conduction and valence bands is a function of the co-ordinates x; y; z and temperature so that NC NV ¼ NC NV ðT; x; y; zÞ:
ð3:26Þ
This product satisfies the equation NC NV ¼ 3=2 lnðmc mv Þ þ 3 ln T þ const;
ð3:27Þ
where mc is the effective mass of an electron and mv is the effective mass of a hole. The term gradðV0 j0 Þ which represents the built-in field (see Eq. (3.19)) is derived in Appendix C (Eq. (C.15)) using Eqs. (3.23)–(3.27). This permits us to write the components of the generalised bulk photovoltaic effect in the form VC T EBPV ¼ EN BPV þ EBPV þ EBPV ;
ð3:28Þ
where EN BPV
2qm0 Dp kB T grad N s q p0 þ n0
results from the position-dependent doping, 2qm0 Dp p0 n0 3 kB T EG grad lnðm m Þ grad EVC c v xyz xyz BPV s p0 þ n0 4 q 2q
ð3:29Þ
ð3:30Þ
results from the position-dependent composition of the material, and
2qm0 n0 p0 n0 3 q 3 1 qEG EG T lnðmc mv Þ þ ln þ þ EBPV ¼ 2 2kB qT s ni p0 þ n0 4 qT 2kB T kB T ð3:31Þ Dp grad q is the thermo-photovoltaic component, which can be combined with the component ETPV (3.21) if we wish to estimate the contribution of the temperature gradient to the photovoltaic effects. 3.2. Numerical example In order to illustrate the theoretical results presented in this chapter, let us consider an inhomogeneous rectangular semiconductor plate equipped with two contact electrodes, as shown in Fig. 1. All the material parameters are assumed to
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depend on the co-ordinate x alone. The excess carriers, whose concentration is constant and equal to Dp; are generated as a result of uniform illumination in such a way that their concentration in the vicinity of the electrodes is equal to zero. To determine the electromotive force of the voltaic effect, we will use Eqs. (3.17)– (3.22) adapted to the one-dimensional case. The calculations are performed for the two cases: (A) when the inhomogeneity is due to the position dependence of the þ chemical composition and doping (grad EG a0 and gradðND NA Þa0), and (B) when it is due to the position-dependent doping density (grad EG ¼ 0 and þ gradðND NA Þa0). The temperatures taken in the calculations are 300 and 400 K. In group A the germanium–silicon samples are considered provided the chemical composition changes from 100% Ge to 100% Si. The position dependence of the doping density is also taken into account. Certain material parameters, such as the þ energy band gap EG ; doping density ND ; effective masses and mobilities, are assumed to be linear versus co-ordinate x. So the values of jgrad EG j ¼ 0:5 eV/cm þ and jgrad ND j ¼ 3 1023 m–4 are adapted to perform calculations. The parameters of group B concern pure Ge with constant energy gap EG ; where þ grad ND ¼ 3 1023 m–4, and average concentration NDC ¼ 2 1021 m–3 (the same as that in group A). The calculated variation of the total photovoltage Upv as a function of the concentration of excess carriers is shown in Fig. 3. The thick solid and dashed curves show the values of Upv for the parameters of group A. The thin solid and dashed curves represent the variation of the B-group parameters. The solid curves were taken at a temperature of 300 K and the dashed curves, at a temperature of 400 K. Curves a and b are obtained when grad EG and grad ND have the same signs, whereas curves c and d, when their signs are opposite. In examining case B; we can þ see that when grad EG =0, the sign of Upv changes together with the sign of grad ND (curves e,f and g,h). These results show that grad EG can give rise to considerable photovoltage.
3.3. Experimental investigation of thermophotovoltaic effect The results reported in our earlier papers [29,30] show that the phenomena predicted theoretically, in particular the thermophotovoltaic effect, need to be verified experimentally. A schematic diagram of the set-up used for the measurements is shown in Fig. 4. The samples, made of silicon, were illuminated with a light spot of a diameter of 1 mm. The temperature gradient within the samples was established using an appropriate heater and Peltier-cooler. The temperature distribution was measurement by three Pt micro-resistors, spaced at a distance of 1 cm from each other. The temperature control system maintains constant temperature TC in the light spot place irrespective of the temperature gradient. Fig. 5 shows the measured magnitudes of the thermophotoelectric voltage as a function of 9grad T9 along the direction of grad ND and in the direction opposite to it, at various initial values of the BPV voltage. The thermophotoelectric unit voltages
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Fig. 3. Total photovoltages calculated from Eqs. (3.17) to (3.22) as a function of Dp for a Ge Si alloy sample with ND ¼ 1 1021 m3, grad ND ¼ 2 1023 m4 and gradwEG ¼ 0:5 eV/cm (lines a, b, e, f). Lines c, d, g, h grad ND ¼ 2 1023 m4, grad EG ¼ 0:5 eV/cm. Lines e, f grad ND ¼ 2 1023 m4, g, h grad ND ¼ 2 1023 m4 and grad EG ¼ 0 (Ge samples). Temperatures: 300 K—solid lines, 400 K— dashed lines.
Fig. 4. Schematic diagram of the set-up for measuring the photovoltaic effects.
obtained during the experiment were of the order of 1 108 Vm K1. Two n-type semiconductor silicon samples with rE4:5 O cm were used. Fig. 6 shows the results of the theoretical calculations. We can easily see from this figure that the magnitude of the thermophotoelectric voltage greatly depends on the þ sample resistivity. These results clearly show how the concentration ND within the þ light spot region, the gradient ND ; and temperature TC in the centre of the light spot influence the photovoltage of the sample with different grad T:
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Fig. 5. Measured values of the thermophotoelectric voltage as a function of jgrad Tj. Directions of grad T identical and opposite to grad ND : Different initial values (grad T ¼ 0) of the voltages differ because of the bulk photovoltaic effect in two different n-type Si samples with rE4:5 O cm.
4. Photovoltaic electrical fields 4.1. Basic equations Section 2.2.1 discussed the simplest one-dimensional case. The results are given by Eqs. (2.24) and further, under the assumption that the excess carrier concentration Dp is constant or that it depends on the co-ordinate x alone. In Section 2.2.3, we formulated the continuity Eq. (2.54) for the total current density, and the continuity Eq. (2.58) fulfilled by Dp: These two equations, together with adequate boundary conditions, permit us to determine the distribution of the potential Cðx; y; zÞ and the concentration Dpðx; y; zÞ: We must mention here that when dealing with a threedimensional case, we face serious mathematical problems. When measuring the resistivity gradient, we use a light spot, which can be treated as nearly point-type illumination. This is a 3D problem, but, in this case, it is sufficient to solve solely the continuity equation for the total current J; which determines the distribution of grad C. We may simplify the problem in this way, provided that the length of the illuminated region is comparable with the diffusion length LD and that the sample dimensions greatly exceed LD : If these conditions are satisfied, we may assume that the distribution Dpðx; y; zÞ in a given small region does not affect the distribution outside this region. If this is so, the total number of the electron–hole pairs is the decisive factor. Analysing the general form of the continuity equation for the total current J and applying several approximations, one obtains a simple result, suitable for practical applications. The mathematical reasoning permits specifying the conditions under
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Fig. 6. Photothermoelectric voltages obtained for various values of grad T and grad ND using (3.17)– (3.22) in turn. (EG depend only on T). (a) NDC ¼ 1 1022 m3, grad NDC ¼ 2 1023 m4, (b) NDC ¼ 1 1022 m3, grad NDC ¼ 2 1023 m4, (c) NDC ¼ 1 1022 m3, grad NDC ¼ 4 1023 m4, (d) NDC ¼ 6 1021 m3, grad NDC ¼ 2 1023 m4, (e) NDC ¼ 1:2 1022 m3, grad NDC ¼ 2 1023 m4, (f) NDC ¼ 1 1022 m3, grad NDC ¼ 5 1023 m4, (g) NDC ¼ 1 1022 m3, grad NDC ¼ 0 (h) NDC ¼ 5 1022 m3, grad NDC ¼ 0: TC ¼ 300 K for a, b, c, d, e, g, h. TC ¼ 350 K for f. grad T range 200 to +200 K/m. Index C indicates the centre of the light spot.
which this simple result can be considered to be a sufficiently good approximation. The theory presented below gives the solutions that determine the distributions of the potential C and the voltage generated between the sample electrodes in many practical cases. The basic method of measuring the photovoltaic effects uses chopped light, thanks to which the thermoelectric component ETEV (occurring in Eq. (3.22)), which does not follow the modulation, is eliminated from the measurement and does not affect the measurement result. Now we can seek the continuity equation for the current J PV ; due only to the photovoltaic effects. From Eqs. (3.17) and (3.28) with the neglect of the component ETEV and introducing a new vector M; we obtain J PV ¼ s grad C þ sDpM;
ð4:1Þ
where C is that portion of the potential which corresponds to the current J PV ; follows the modulation and thus can be measured. The vector M has four components: M ¼ M N þ M E þ M T þ M M;
ð4:2Þ
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which can be expressed as 1 2m0 kB T grad N M N ¼ EN ¼ Dp BPV sðp0 þ n0 Þ 1 VC 2qm0 p0 n0 3 kB T EG ME ¼ E gradxyz ln mc mv gradxyz ¼ ; Dp BPV s p0 þ n 4 q 2q
ð4:3Þ ð4:4Þ
1 T 1 EBPV þ ETPV Dp Dp
2qm0 n0 p0 n0 3 q 3 1 qEG EG lnðmc mv Þ þ ln þ þ ¼ 2 2kB qT s ni p0 þ n0 4 qT 2kB T
q en þ ep EG sn0 sp0 n0 s0 s þ ln þ ðmn mp Þ ln s 2 2kB T s0 ni Ds s0
en ep 3 mc kB T ; ð4:5Þ þ ðmn þ mp Þ þ ln grad 2 mv q 2
MT ¼
1 EMPV Dp mn mp 1 s0 s q m ¼ 2 ln ðmn mp Þðn0 þ p0 Þ kB T grad ln n ; s0 Ds s0 s mn þ mp mp
MM ¼
ð4:6Þ
where the vectors E are taken from Eqs. (3.29)–(3.31), (3.21) and (3.20), respectively. The vector M N is associated with the classical bulk photovoltaic effect, M E with the position dependence of the band gap EG and of the effective masses (when the variable composition case is in question), M T determines in total the thermophotoelectric effect and M M is proportional to the variation, due chiefly to the positiondependent composition, of the mobilities ratio. What we examine is the sum M of these vectors and our aim is to get information on the properties of the material, especially when the position-dependent doping is accompanied by a positiondependent composition. The continuity equation derived directly from Eq. (4.1) is div J PV ¼ sDC s grad C þ div sDpM ¼ 0;
ð4:7Þ
which, after rearranging, gives DC þ grad ln s grad C ¼
divðsDpMÞ ; s
ð4:8Þ
where D is the Laplace operator. 4.2. General solution for an open-circuit sample We will examine a sample of arbitrary shape (Fig. 7), equipped with two contact electrodes (of an arbitrary surface area or point-type), illuminated in such a way that in the vicinity of the electrodes the concentration Dp is zero (so as to avoid contact effects). We consider only the open-circuit arrangement.
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Let us denote S1 ; S2 are the surface area of the electrodes and S3 is thesurface area of the sample beyond the electrodes. In our case, the boundary conditions are: 1. On the surface S1 C ¼ 0:
ð4:9Þ
2. On the surface S3 C ¼ const: 3. The total current is equal to zero that is Z Jn dS2 ¼ 0:
ð4:10Þ
ð4:11Þ
S2
4. The normal component of grad C on surface S3 fulfils the equation ðgrad CÞn ¼ DpM;
ð4:12Þ
where n is the unit vector perpendicular to this surface. It follows from the theory of the elliptical partial differential equations that Eq. (4.8) with the boundary conditions (4.9)–(4.12) can be solved using the Green function that satisfies the equation DA GðA; BÞ þ grad ln sðAÞgradA ðA; BÞ ¼ dðA BÞ;
ð4:13Þ
where A and B are points of the bulk of the sample and dðA BÞ—is the threedimensional Dirac delta function. Function G must in addition satisfy the boundary conditions that follow from Eqs. (4.9)–(4.11) and the equation ½gradA GðABÞ n ¼ 0;
ð4:14Þ
which replaces the boundary condition (4.12).
Fig. 7. Inhomogeneous semiconductor sample of arbitrary shape. S1 ; S2 electrodes. S3 is the remaining area. The light beam generates the excess electron–hole pairs Dp: TB—bulk of the sample.
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That there exist the solutions of Eqs. (4.13) and (4.14), we can see by constructing an electric model of the function GðA; BÞ; shown in Fig. 8. In this model, the current I ¼ sðBÞ is supplied to the sample at point B. The potential distribution established in the sample is a function of points A and B and is equal to the Green function. The solution of Eq. (4.8) with the boundary conditions (4.9)–(4.12) is the function
C ðx; y; zÞ ¼ C ðAÞ ¼
Z GðA; BÞ TB
þ
Z
divðsDpMÞ dtB s
GðA; BÞDpMn dSB ;
ð4:15Þ
S
where TB is the bulk of sample, dtB is the volume element, S is the total surface area of the sample and dSB is a surface element. Eq. (4.15) may be transformed to the form
C ðAÞ ¼
Z TB
gradB GðA; BÞMðBÞDp dtB
Z GðA; BÞMðBÞgrad ln s dtB ; TB
ð4:16Þ which is easier to interpret and to transform further into an approximate equation. Eq. (4.16) indicates that the potential distribution C ðAÞ results from the dipole sources whose distribution is MðBÞDpðBÞ and single sources with the distribution MðBÞgrad ln sðBÞDpðBÞ: When the illumination is of the point type, we can seek an approximate equation, since Dpa0 only in a small region of the light spot. An analysis of the Green function GðA; BÞ using the Legendre polynomials shows that a first approximation of Eq. (4.16) is C ðAÞEgradB GðA; B0 ÞMðB0 ÞP GðA; B0 ÞMðB0 Þgrad ln sðB0 ÞP;
Fig. 8. Electrical model of Green function GðA; BÞ.
ð4:17Þ
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where B0 is the centre of light spot and P is the total number of excess carriers given by Z P¼ DpðBÞ dtB : ð4:18Þ TB
Both the terms MðBÞ and grad ln s appearing in Eq. (4.17) are proportional to the degree of inhomogeneity. If the inhomogeneity is not significant, MðBÞgrad ln s can be neglected as small compared to MðB0 Þ: Even with the proposed simplifications, we still face the problem of determining the function GðA; BÞ; since it must satisfy Eq. (4.13), containing function grad ln sðBÞ: When we assume that this function is known, we obtain the equation for G0 ðA; BÞ in the form of the infinite series [32] GðA; BÞ ¼ G0 ðA; BÞ þ G1 ðA; BÞ þ G2 ðA; BÞ þ ?;
ð4:19Þ
in which the function G0(A,B) satisfies both the boundary conditions and the equation DA G0 ðA; BÞ ¼ dðA BÞ:
ð4:20Þ
The function G1 and the subsequent functions are found in a recurrent manner to be Z G0 ðA; CÞgrad ln s grad Gn ðC; BÞ dtC : ð4:21Þ Gnþ1 ðA þ BÞ ¼ TB
The condition for the convergence of this series is Rjgrad ln sjmax o0:1;
ð4:22Þ
where R is the maximum diameter of the semiconductor sample. The latter inequality roughly defines the range of the series convergence, even though we only take into account the first term G0 ðA; BÞ of the series. This reasoning permits us to write Eq. (4.17) in the approximate form C ðAÞ ¼ Cðx; y; zÞEgradB G0 ðA; Bsr ÞMðB0 ÞP;
ð4:23Þ
which describes the potential C and its gradient as a dipole field with the moment MP: Based on the derivation of the approximate Eq. (4.23) and on the complete Eq. (4.17) we can estimate the errors incurred during the measurements and to assess the suitability of the experimental methods employed. 4.3. Fundamentals of the measurement of the bulk photovoltaic effect (BPV) and the relative gradient of resistivity In this section we confine our attention to the bulk photovoltaic effect alone, since in practical situations, it forms the basis for determining the relative distribution of the resistivity gradient in semiconductor samples. Fig. 7 shows schematically the measurement of the photovoltage between the electrodes of an illuminated semiconductor sample with a resistivity gradient established in it. According to Eq. (4.23), the voltage UBPV is equal to UBPV ðB0 Þ ¼ gradB dGðB0 ÞM N ðB0 ÞP;
ð4:24Þ
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where dGðB0 Þ G0 ðC; B0 Þ G0 ðD; B0 Þ
ð4:25Þ
and the dipole moment M N is defined by (4.3). In order to find a simple relationship between this moment and the resistivity gradient r, we consider n-type semiconductor where n0 bp0 ; NEn0 ; m0 ¼ mp : Substituting these into Eq. (4.3), we obtain MN ¼
2mp kB T grad ln n0 : s
ð4:26Þ
Then multiplying n0 under the logarithm by the coefficient qmn ; (in the case of an ntype semiconductor this product is equal to s) and using the equality sr0 Es0 r0 ¼ 1; we can write 2mp kB T 2mp kB T grad ln s0 ¼ grad ln r0 s s 2mp kB T grad s0 ¼ E2mp kB T grad r0 ; s r0
MN ¼
ð4:27Þ
Substituting (4.27) into Eq. (4.22) gives UBPV ðB0 Þ ¼ 2mp kB TP grad dGðB0 Þgrad r0 ðB0 Þ ¼ 2mp kB TPjgrad dGðB0 Þj grad r ðB0 Þcos a; 0
ð4:28Þ
where a is the angle between the vectors grad dG and grad r0 : Knowing the geometry of the sample and its electrodes determines grad dG; therefore the projection of the vector grad r0 in the direction of the known vector grad dG may be expressed as UBPV ðB0 Þ grad r cos a ¼ : ð4:29Þ 0 B0 2mp kB T jgrad dGðB0 ÞjP By illuminating the sample at various points and measuring UBPV there, we can find the relative distribution of 9grad r0 9cos a; provided that the sum of the pairs P of the excess carriers is the same at all points B0 ; which is true when the carrier lifetime t is the same at all these points. When the angle a ¼ 0; and we are able to confirm this, and the measurement is performed along a straight line taken to be the axis x; then we can find the relative distribution dr=dx: The absolute distribution rðxÞ can be determined by measuring r at two points of the x axis, e.g. by means of the 4-point probe. In sufficiently long samples grad G0 may be assumed to be constant and formulae (4.24) and (4.26) define the relative distribution of grad r0 ; which can be determined by the measurements of UBPV : An early example of application of the bulk photovoltaic effect is the work of Oroshnik and Many [31]. The method used there was photovoltaic scanning to evaluate the homogeneity of germanium single crystals. In course of time the ! interesting results are obtained by Swiderski. In the conference paper publication [32] the micro-inhomogeneities were observed in germanium which enabled to investigate the details of discontinuities taking place in the process of single crystals production.
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4.4. Measurement of the resistivity distribution A very convenient method for determining the resistivity distribution rðxÞ in rectangular samples consists of the following three steps: (1) Measurement of UBPV along the x-axis R x of the sample. (2) Calculation of the integral IðxÞ ¼ x0 UBPV ðx0 Þ dx0 (x0 is arbitrarily chosen) which is the linear function A IðxÞ þ B of IðxÞ; where the two constants A and B are determined by measuring, with a 4-point probe, the resistivity at the two points x1 ; x2 : It is desirable to choose x1 ; x2 where UBPV has an extremum. (3) Having determined these quantities, we can express the distribution rðxÞ in the form rðxÞ ¼ fr1 ½IðxÞ Iðx2 Þ r2 ½IðxÞ Iðx1 Þ gðI1 I2 Þ1 :
ð4:30Þ
The method just described has been tested practically at our Institute. Fig. 9 shows the results of our experiments. The diagram at the top of the figure is UBPV plotted as a function of x; and at the bottom is the distribution rBPV ðxÞ; where the solid line represents the distribution calculated from the integral IðxÞ and formula (4.30), and the squared line the distribution measured with the 4-point probe. We can see that the two resistivity distributions, measured and calculated, are in good agreement, except in a certain region near x ¼ 35 mm. This difference may result from the poor geometrical resolution of the 4-point probe compared to that achieved in the light spot measurement, and from the contribution of the concentration gradient directed
Fig. 9. Distribution of the photovoltage measured in a rectangular inhomogeneous Ge sample (solid line in top figure). The distribution of resistivity obtained from the BPV effect (thick solid line); resistivity distribution obtained from 4-point probe measurements (squares).
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perpendicularly to the sample juncture line of the contacts; the value of UBPV only depends on the grad ND component that is parallel to the juncture line of the contacts. The relative distribution of the resistivity gradient can only be determined when the carrier lifetime in the semiconductor can be assumed to be independent of position. To obviate this restriction, Blackburn et al. [33] proposed that the bulk photovoltage and the photoconductivity should be measured at the same illuminated point, which permits determining the absolute resistivity gradient dr=dx with the position dependence of the carrier lifetime taken into account. These authors carried out the measurements on circular semiconductor slices using a small-diameter light spot. The formula determining the correction due to the circular geometry of the sample was derived using the conformal transformation. An improved method of Blackburn et al. [33] is presented by Larrabee and Blackburn [34]. The method based on the Green function presented in Section 4.1 makes it possible to develop a more general theory which may also be applied to wafers with a geometry different from circular even in the 3D case. To enable the photoconductivity measurement, a constant current Ics is forced through the sample using a current source, and the sample is illuminated by modulated light. During the period without illumination the current density in the sample is J 0 ¼ s0 grad C0 :
ð4:31Þ
During the period where the sample is illuminated the current density is described by (2.60), which, in the case of an n-type semiconductor (m0 Emp ), takes the form J ¼ s grad C 2qmp Dp grad V0 :
ð4:32Þ
Subtracting (4.30) from (4.31) we obtain J J 0 ¼ s grad dC Ds grad C0 2qmp Dp grad V0
! 2mp ¼ s grad dC qðmn þ mp Þ grad C0 þ grad V0 Dp; m n þ mp
ð4:33Þ
where dC ¼ C C0
ð4:34Þ
while the increase in the conductivity was assumed to be Ds ¼ qðmn þ mp ÞDp. The form of (4.33) makes it possible to notice that the components determining the final solution are grad C0 and 2ðm0 =mn þ mp Þgrad V0 : If (4.31) is to be simplified, the following condition must be met jgrad C0 jbjgrad V0 j
ð4:35Þ
because 2m0 =ðmn þ mp Þ is of the order of 1. The above condition may be fulfilled in practice because V0 is a given quantity, while grad C0 is proportional to the current Ics flowing through the sample and thus may be adjusted. If it is assumed that (4.33) is applicable, then (4.31) takes the form J J 0 E s grad dC qðmn þ mp ÞDp grad C0 :
ð4:36Þ
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The difference J J 0 is solenoidal, therefore the divergence of the right-hand side of the above equation is equal to zero. Thus divðs grad dCÞ ¼ div½qðmn þ mp ÞDp grad C0 :
ð4:37Þ
Transforming the left-hand side and dividing the equation by s one obtains
qðmn þ mp Þgrad C0 1 div grad dC þ grad ln s grad dC ¼ div sDp : s s ð4:38Þ This equation has the same form as (4.8), where M should be replaced by q=sðmn þ mp Þgrad C0 : According to (4.21) the solution of such an equation is q ð4:39Þ dCðAÞ ¼ dCðx; y; zÞ ¼ grad G0 ðA; B0 Þ ðmn þ mp Þgrad C0 ðB0 ÞP: s It should be noted that G0 is the proper function corresponding to an open sample because the supply current Ics is the same regardless of whether the sample is illuminated or not. As a consequence, G0 fulfils the boundary condition at the sample electrodes. The increase of the potential dC induced by the illumination causes a negative increase of the voltage between the sample electrodes. Thus, the resistance appearing in Eqs. (38) and (39) is changed by DR: If the solution of (4.37) is known, the value dUR of the measured voltage may be found in a manner analogous to (4.28): q dUR ðB0 Þ ¼ ðmn þ mp ÞP grad dGðB0 Þgrad C0 s q ¼ ðmn þ mp ÞPjgrad dGðB0 Þjjgrad C0 jcos b; ð4:40Þ s where b is the angle between vectors grad dG and grad C0 : Similar to (4.28) the term grad dG appears also in the above formula. Another important fact is that P is the total number of excess carrier pairs, is the same in Eqs. (4.40) and (4.28) because it corresponds to the same point B0 : This means that the factor P will be eliminated if (4.40) is divided by (4.28) and we have UBPV 2kB Ts mp grad r0 cos a : ð4:41Þ ¼ q mn þ mp jgrad C0 j cos b dUR The above leads to m þ mp UBPV grad r cos a ¼ q r n ; jgrad C0 jcos b 0 kB T 0 2mp dUR
ð4:42Þ
where s has been replaced by 1=r0 ; which in practice is a well-founded assumption. If 7grad C0 7cos b is known, Eq. (4.42) makes it possible to determine the projection of the gradient of r0 onto the direction of vector grad dG0 : 4.5. The Green function and the potential distribution in thin samples If we assume that the properties of the material and the concentration Dp at a given point do not vary along the co-ordinate z perpendicular to the sample surface,
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the mathematical problem becomes two-dimensional. The Green function G2D fulfils the equation analogous to (4.19) DG2D ðA; BÞ ¼ d2D ðA BÞ; where D¼
q2 q2 þ ; qx2 qy2
ð4:43Þ
d2D is the 2D Dirac delta funciton:;
Points A and B have the co-ordinates x and y. As in Section 4.3, we only consider the BPV effect in an n-type semiconductor. The dipole moment at point B0 is defined by Eq. (4.27), which, as we remember, has the form (with index N omitted) M ¼ 2mp kB T grad r0 :
ð4:44Þ
At a given dipole moment the distribution of the potential C is given by Eq. (4.21). The 2D-form of this equation is P C2D ðAÞ ¼ Cðx; yÞ ¼ gradB G2D ðA; B0 ÞM : ð4:45Þ d This equation may be used for determining the distribution of the potential and then calculating the voltage established between the sample electrodes. Having determined these, we can find the distribution of the resistivity gradient within the sample. Points A and B0 are defined by the two co-ordinates x and y and, thus, their positions may be expressed as the complex number z ¼ x þ iy: This permits us to introduce a function of the complex variable that describes grad G2D : Then the field generated by a single dipole in space is described by the function 1 # ð4:46Þ GðzÞ ¼ 2pz the real part of which fulfils Eq. (4.43). If the vector MP appearing in Eq. (4.45) is described in terms of the vector # ¼ Dx þ iDy ¼ M x þ iM y P; D ð4:47Þ then the potential due to this dipole will be # D : ð4:48Þ Cðx; yÞ ¼ Re 2pdz # given by (4.47) into Eq. (4.48) and taking z ¼ x þ iy; we obtain Now, substituting D Cðx; yÞ ¼ Re
ðDx þ iDy Þz ðDx þ iDy Þðx iyÞ Dx x þ Dy y ¼ Re ¼ ; 2pdðzz Þ 2pdjzj2 2pdjzj2
ð4:49Þ
which is the known equation that describes the potential due to a dipole with the components Dx ; Dy ; positioned at the origin of the co-ordinate system. In the case when the dipole is positioned at z0 ¼ x0 þiy0 ; Eqs. (4.45) and (4.46) may be easily transformed to take the form Dx þ iDy Dx ðx x0 Þ þ Dy ðy y0 Þ ¼ Cðx; yÞ ¼ Re : ð4:50Þ 2pdðz z0 Þ 2pdjz z0 j2
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In the next chapter, we derive equations describing the distribution of the BPV potential in thin slices of selected shapes. The results will be given in the form of the equipotential line plots.
5. The BPV—potential distributions 5.1. The method of images The basic equations and relationships describing the distribution of the field induced by BPV under conditions of point illumination have been given in the previous chapter. The equation of fundamental significance is Eq. (4.23). Its righthand side developed into the components is C ¼ Dx
qG0 qG0 qG0 þ Dy þ Dz ; qxB qyB qzB
ð5:1Þ
where we have introduced the notations that relate the components of vector D with those of vector M Dx ¼ Mx P;
Dy ¼ My P;
Dz ¼ Mz P:
ð5:2Þ
The potential C is a function of A, i.e., of the co-ordinates x; y; z: The components of the dipole D are defined at point B0 in the centre of light spot. The co-ordinates of this point are x0 ; y0 ; z0 : Formula (5.1) is valid for any shape of the sample. To determine the potential C in the definite case it is necessary to know the Green function G0 ; which is determined by the shape of the sample under consideration. The fundamental and simplest case of the Green function is 14prAB0 ; where rAB0 is the distance between points A and B0 : This function fulfils Eq. (4.20), which can be verified by using an appropriate substitution, and satisfies the boundary conditions in the infinity, thereby vanishing when r-N: So the function 14prAB0 is valid for the boundless sample. Taking the above into account and in accord with Eq. (4.23), we can write the potential induced by a single dipole MP positioned at point B0 in the infinite three-dimensional space C ðAÞ ¼
1 ½Dx ðx x0 Þ þ Dy ðy y0 Þ þ Dz ðz z0 Þ : 4pr3B0
ð5:3Þ
The potential C given by this equation vanishes when rAB0 -N thereby satisfies the boundary condition in the infinity. Unlike the case concerning the infinite space, the potential of the dipole situated in the half-space must fulfil following boundary conditions on the boundary plane. (1) In a semiconductor surface coated with a metal layer, whose conductivity is several times as high as that of the semiconductor, the boundary conditions is C¼ const: The same problem is encountered in electrostatics, and, there, we create the field of a point charge and of its image which is its reflection in the boundary plane. The charge and its image have opposite signs, as shown in Fig. 10a. We treat dipoles as two-charge objects, so we can say that the components perpendicular to
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Fig. 10. Charge q with its image, and dipole D with its image D0 in the case of (a) Metal–semiconductor interface and (b) semiconductor–insulator interface.
Fig. 11. Dipole at the surface of half-space.
the boundary plane of both the dipoles have the same directions, whereas the parallel components have opposite directions (see Fig. 10a). (2) When the semiconductor surface is not covered with metal we deal with a semiconductor/insulator boundary and the boundary condition is that the perpendicular to the boundary plane component of the field should be equal to zero. As a consequence, the directions of the dipole components are opposite to those that these components had in the previous case. Fig. 10b shows that the perpendicular components have opposite directions, whereas the parallel components have the same directions. Further in this section, we discuss the potential distributions determined in samples of various shapes by the method of images. 5.2. Half-space and space between two parallel half-planes Fig. 11 shows a co-ordinate system with a dipole positioned at its origin on the surface of a half-space; the dipole has the components Dx and Dy (Dz ¼ 0). In this situation, the image of the dipole is identical to the dipole itself. Using Eqs. (5.2) and making appropriate substitutions, we obtain C¼
1 ðDx x þ Dy yÞ: 2pr3AB0
ð5:4Þ
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Fig. 12. Dipole at the surface of the region between two parallel half-planes.
Fig. 13. Sequence of images for the case shown in Fig. 33.
Fig. 14. Components of the dipole sequence of Fig. 13. (a) Dipoles parallel to both planes and (b) dipoles perpendicular to the planes.
The region between two parallel half-planes and the position of the dipole are shown in Fig. 12. Because of the presence of the two mutually parallel planes, we must form images given by these planes, then images of these images, and so on. As a result we obtain the infinite sequence of dipoles ‘‘hinted at’’ Fig. 13. This sequence can be formed of the two other simple sets of dipoles shown in Figs. 14a and b. The Green function G0 of the sequence of single dipole sources placed at points B2 ; B1 ; B0 ; B1 and B2 has the form G0 ¼
N 1 X 1 : 4p n¼N rA;Bn
ð5:5Þ
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The function, which describes the components of the dipole sources (Fig. 14a), is the derivative of G0 with respect to n, which will be denoted as qG0 ¼ jð3Þ x ðx; y; zÞ: qx
ð5:6Þ
The dipole sources in the direction y (Fig. 14b) are described by the derivative of G0 with respect to y; which is qG0 ¼ jð3Þ y ðx; y; zÞ: qy
ð5:7Þ
To determine the function G0 which describes the potential of the single sources we should only calculate the potential induced by a sequence of point charges distributed periodically on a straight line. This was done by Madelung [35] who gave a formula in the form of a rapidly converging series, easy to differentiate. The ð3Þ differentiation gives the equations, which define the functions jð3Þ x and jy as
N 2x X 2pnr 2pny x þ ðx; y; zÞ ¼ nK ð5:8Þ cos jð3Þ 1 x rd 2 n¼1 d d 2pdr2 as well as jð3Þ y ðx; y; zÞ
N 2 X 2pnr 2pny ; ¼ 2 nK0 sin d n¼1 d d
ð5:9Þ
where K0 i K1 are the modified Hankel functions of imaginary argument and r is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5:10Þ r ¼ ðx2 þ z2 Þ: The assumption that the dipole has only two components Dx and Dy means that we neglect the inhomogeneity in the direction z: An analysis of the relationship between the two sets of dipoles shown in Figs. 13 and 14 and yields the equation for the distribution of the potential C within the region considered: ð3Þ Cð3Þ ðx; y; zÞ ¼ 2Dx ½jð3Þ x ðx; y; zÞ þ jx ðx; y 2a; zÞ ð3Þ þ 2Dy ½jð3Þ y ðx; y; zÞ þ jy ðx; y 2a; zÞ :
ð5:11Þ
The coefficients 2 appearing in both the terms of this equation indicates that the dipole images were taken to be identical with the dipoles themselves, just as in the half-space case. Fig. 15 shows the distribution of the potential C3 ðx; y; 0Þ induced by a dipole placed on the surface of the sample in the middle of its width; the moment of this dipole is given by the relationship Dx =d 2 ¼ 1 V, Dy ¼ 0: We can see that, at a certain distance from the dipole, the potential depends on x alone, the equipotential lines are straight lines and the potential may be expressed as 2Dx x Cð3Þ D : ð5:12Þ pdr2
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Fig. 15. Potential distribution in the x y plane of the dipole directed along the axis of the sample and placed symmetrically in the region shown in Fig. 12.
Fig. 16. Dipole localised at the surface of an infinitely long sample with finite width and thickness.
The photovoltage measured does not depend on the position of the measuring electrodes, provided that they are far enough from the light spot. 5.3. Infinitely long plate Fig. 16 shows an infinitely long sample and a dipole D that has the two components Dx and Dy (Dz ¼ 0). The position of the dipole on the sample surface (z ¼ 0) is defined by the distance a; b=2 and c=2 are the width and the thickness of the sample, respectively. The solutions can be obtained based on the distributions of the potentials induced by the dipoles parallel to the x-axis and parallel to the y-axis. The dipoles are placed on the plane z ¼ 0 at the nodes of the infinite orthogonal flat lattice shown in Fig. 17. This primary lattice should be accompanied by an image of secondary lattice (not shown in Fig. 17) shifted by 2a in parallel to the y-axis. The coordinates of the nodes of the primary lattice are (0; bl; cn), where l and n are integers within the interval (N, +N). Using these quantities we can determine the distance
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Fig. 17. Dipole lattices for the case of Fig. 16. (a) Dipoles parallel to the sample edge and (b) dipoles perpendicular to the sample edge.
ren between the points on the sample surface and the nodes of the primary lattice. This distance can be calculated from ren ¼ ½x2 þ ðy blÞ2 þ ðcnÞ2 1=2 :
ð5:13Þ
By substituting y bl 2a in the place of y bl in this equation we obtain r0l;n concerning the secondary lattice r0l;n ¼ ½x2 þ ðy bl 2aÞ2 þ ðcnÞ2 1=2 : In the reasoning which follows, we will use the function N X ðx; y; 0Þ ¼ ½x2 þ ðy bÞ2 þ ðcnÞ2 3=2 : jð4Þ x
ð5:14Þ
ð5:15Þ
l;n¼N
According to the method of images, the potential on the sample surface can be found using Eq. (5.3), which defines the dipole field in a general way. This potential is given by Dx x ð4Þ Cð4Þ ðx; y; 0Þ ¼ ½j ðx; y; 0Þ þ jð4Þ x ðx; y 2a; 0Þ 4p x Dy y ð4Þ ½j ðx; y; 0Þ þ jð4Þ þ y ðx; y 2a; 0Þ 4p y Dx x þ Dy y ð4Þ Dx x Dy y ð4Þ j ðx; y; 0Þ þ j ðx; y 2a; 0Þ: ð5:16Þ ¼ 4p 4p As in the previous case (5.12), the potential Cð4Þ for large values of x depends neither on y nor on z and may be expressed as 2Dx : ð5:17Þ Cð4Þ ðN; y; zÞ ¼ bc The potential distribution at xo0 is anti-symmetrical to the distribution at x > 0 and, thus, Cð4Þ ðN;y;zÞ ¼ 2Dx =bc: Hence the voltage between the sample far ends is U¼
4Dx Dx ¼ ¼ g; bc s
ð5:18Þ
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Fig. 18. Potential distribution of the dipole in an infinitely long sample with the square cross-section. Dipole parallel to the edge placed symmetrically.
where s is the cross-sectional area of the sample. This voltage does not depend practically on the dipole position, but it only depends on the dipole moment per unit cross-sectional area g: This observation is of importance in the photovoltaic diagnostics of the material. An analysis of Eqs. (5.13) and (5.14) indicates that when measuring the potential at a point with the co-ordinate x; and estimating the moment Dx ; we make the relative error equal to ! xd px exp pffiffi ð5:19Þ s s from which we can evaluate how long the sample should be in order that Eq. (5.17) might be used. The distribution of the potential Cð4Þ ðx; y; 0Þ is shown in Fig. 18. A dipole with the components Dx =s ¼ 1 V, Dy ¼ 0 is placed on the sample surface at its centre. The sample cross-section is square (b ¼ c). 5.4. Cuboid with electrodes at two opposite faces The distribution of the potential in a cuboidal sample equipped with two contact electrodes at its two end faces, shown in Fig. 19, is determined using the components of Eq. (5.15) that define the potentials of the dipole iDx and jDy in an infinitely long sample. These two potentials may be written as ð4Þ Fx ðx; y; zÞ ¼ 2Dx ½jð4Þ x ðx; y; zÞ þ jx ðx; y 2a; zÞ ;
ð5:20Þ
ð4Þ Fy ðx; y; zÞ ¼ 2Dy ½jð4Þ y ðx; y; zÞ þ jy ðx; y 2a; zÞ :
ð5:21Þ
The potential may be written as ð5Þ Cð5Þ ¼ Cð5Þ x þ Cy
ð5:22Þ
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Fig. 19. Cuboidal sample with electrodes covering the whole width of the sample.
where the first component related with the dipole iDx is N X Cð5Þ ðx; y; zÞ ¼ Fx ðx 2nf d; y; zÞ þ Fx ðx 2nf þ d; y; zÞ x
ð5:23Þ
n¼N
and the second component related with the dipole jDy is N X Cð5Þ ðx; y; zÞ ¼ Fy ðx 2nf d; y; zÞ þ Fy ðx 2nf þ d; y; zÞ: y
ð5:24Þ
n¼N
Eqs. (5.23) and (5.24) together with Eqs. (5.13) and (5.14) permit calculating the ð5Þ potential at any arbitrary point of the sample. In the case of Cð5Þ x and Cy ; the result is a series with a double index. In practical applications, it is interesting to know the value of Cð5Þ at two points in the vicinity of the electrodes, namely Cð5Þ x ð0; y; zÞ and ð5Þ ð5Þ Cð5Þ x ðf ; y; zÞ; and also Cy ð0; y; zÞ and Cy ðf ; y; zÞ: In view of the symmetry of the function, i.e., Fx ðx; y; zÞ ¼ Fx ðx; y; zÞ; the algebraic calculation of these quantities with the use of Eq. (5.23) yields the very simple result Cð5Þ x ð0; y; zÞ ¼ 0:
ð5:25Þ jð4Þ x
In the other case, we should also utilise the value of calculated in the previous section. The result is 4Dx : ð5:26Þ Cð5Þ x ðf ; y; zÞ ¼ bc By performing similar calculations based on the equation ð4Þ jð4Þ y ðx; y; zÞ ¼ jy ðx; y; zÞ
ð5:27Þ
we obtain Cð5Þ y ð0; y; zÞ ¼ 0;
Cð5Þ y ðf ; y; zÞ ¼ 0:
ð5:28Þ
This result means that, under open circuit conditions, the component Dy does not contribute to the generation of the voltage at the electrodes of the sample and, thus, the voltage is given by 4Dx Dx ð5Þ ¼ ¼ g: ð5:29Þ U ð5Þ ¼ Cð5Þ x ð0; y; zÞ Cx ðf ; y; zÞ ¼ bc s
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This is the same result as that obtained in the previous section (5.17), it however goes further. In a long sample not equipped with appropriate electrodes, Dx cannot be determined when the light spot is incident near a small-size electrode. When dealing with a cuboidal sample where the electrodes cover the whole surfaces of its end walls, the voltage is always Dx =s; irrespective of the position of the point considered, which is of great practical significance and cannot be predicted intuitively. We should of course remember the condition, which is still valid, that the distance between the illuminated points and the electrodes must exceed the length of the diffusion length. 5.5. Slices 5.5.1. Half-plane and infinite strip #z ¼ Let us consider a half-plane x > 0 that borders on an insulator. The dipole D Dx þ iDy (Fig. 20) is positioned at the point z0 ¼ x0 þ iy0 : Its image in the insulator # z ¼ Dx þ iDy situated at z ¼ x0þ iy0 : According to Eq. (4.50), the is the dipole D 0 resultant potential is
Dx þ iDy Dx þ iDy 1 Re C1 ðx; yÞ ¼ þ : ð5:30Þ 2pd z z0 z þ z0 In order to find the potential distribution within the infinite strip 0oxoh where h is the strip width, we have to consider the infinite sequence of the pairs D# z ; D# z shown in Fig. 21. The spacing between the recurring pairs of dipoles is 2h; and z0 is assumed to be equal to x0 : The potential is expressed as an infinite sum of the fields given by Eq. (5.30), shifted by a multiple of 2h: This is written as "
# N X Dx þ iDy Dx þ iDy 1 Re lim Cðx; yÞ ¼ þ : ð5:31Þ N¼N 2pd z x0 2kh z þ x0 þ 2kh k¼N
Fig. 20. Dipole Dz in a half-plane x > 0 and its image D0z .
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Fig. 21. The infinite sequence of dipole pairs Dz ; D0z for a strip with finite width h:
In this equation, the infinite sum is composed of two sums, which may be replaced by cotangents of adequate arguments. When this is done, we have lim
N¼N
N X
1 p p ¼ ctg ðz7x0 Þ: z7x 2h 2h 72kh 0 k¼N
ð5:32Þ
Substituting these functions into Eq. (4.49) we obtain h 1 p Re ðDx þ iDy Þctg ðz x0 Þ Cstr ðx; yÞ ¼ 4dh 2h i p þðDx þ iDy Þctg ðz þ x0 Þ : ð5:33Þ 2h The equipotential lines of the field induced by the dipoles Dx ; iDy and Dx þ iDy ; determined from this equation are shown in Figs. 22a–c. Fig. 22d shows the surface plot of the potential of the dipole Dx : Dx =dh and Dy =dh are equal to 1 V. We can see from the potential distribution shown in Fig. 22a that, at a sufficiently long distance from the dipole, the potential is constant and is equal to 0.5 V at the upper end and 0.5 V at the lower end. Fig. 22b shows that, in distant regions, the perpendicular dipole generates zero potentials, so that no potential difference is established at the ends. These observations are confirmed quantitatively by Eq. (5.33) when calculating the voltage established at the ends of a sufficiently long sample. The potentials of the points distant from the dipole are 1 Re½ðDx þ iDy Þ þ ðDx þ iDy Þ ctgð7NÞ Cstr ðx; 7NÞ ¼ 4dh
72Dy Dy 1 1 Re 72iDy ¼7 : ð5:34Þ ¼ ¼ 4dh i 4dh 2s Hence the voltage at the ends of a long sample, with the thickness d; width h and cross-sectional area s ¼ d h is Dy Cstr ðx; NÞ Cstr ðx; NÞ ¼¼ ¼ g: ð5:35Þ s This result is identical to that obtained for the sample with a finite thickness. The respective equations given in Ref. [35] permit evaluating the relative error incurred in the measurement of the resistivity gradient in rectangular plates as not exceeding 3 expð2pDx=hÞ; provided that the distance Dx between the light spot and the edge
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Fig. 22. Potential distribution in an infinitely long thin sample with finite width. (a) Dipole Dz parallel to the sample length, (b) dipole perpendicular to the sample length, (c) dipole deviated at 45 , (d) surface plot of potential distribution for the deviated dipole.
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Fig. 23. Circular slice with measuring arrangement and the location of the dipole.
where the measuring electrode is installed is below 0.64h. This also applies to the point-type electrode. 5.5.2. Circular slice Fig. 23 shows a circular slice of radius R; a plane of complex numbers z ¼ x þ iy # z ¼ Dx þ iDy : The circle jzjpR is surrounded by an insulator. We and the dipole D # w that satisfies the boundary condition seek the potential distribution of the dipole D jzj ¼ R at the circle periphery. To do this we utilise the solution (with adequately changed notation of the co-ordinates), obtained for a half-plane given by Eq. (5.30) and perform a transformation according to Rþz ; ð5:36Þ w¼ Rz where the point w0 corresponds to point z0 ; w0 ¼
R þ z0 ; R z0
w0 ¼
R þ z0 : R z0
# w is related to dipole D # z in the following: The dipole D
#w ¼D # z dw # z 2R : D ¼D dz z¼x0 ðR x0 Þ2 Substituting (5.36) and (5.37) into Eq. (5.30) we obtain 1 R Cðx; yÞ ¼ pd ðR z0 Þ2 Dx þ iDy
Re ðR þ zÞ=ðR zÞ ðR þ z0 Þ=ðR z0 Þ
Dx þ iDy þ : ðR þ zÞ=ðR zÞ þ ðR þ z0 Þ=ðR z0 Þ
ð5:37Þ
ð5:38Þ
ð5:39Þ
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By adding, in an appropriate proportion, the potentials due to the perpendicular dipole Dy and to the parallel dipole Dx ; we can determine the distribution of the potential C for any dipole positioned within the slice. The potential and force lines can be found from the real and imaginary parts of equation, respectively, (5.39). Fig. 24 shows these distributions for the dipoles with the intensity Rd 1V, directed along (a) u-axis, (b) v-axis and (c) deviated at an angle of 45 . The dipoles Du and Dv are located on the u-axis. The position of the Du þ iDv dipole is arbitrary. Let us now consider the experimental results. The voltage UBPV was measured using the set-up shown in Fig. 4. The sample was a typical circular n-type silicon slice. The resistivity of the material was 4.5 O cm. The measurements are performed along the diameter of the slice under open-circuit conditions. The measured values of the UBPV versus the position of the light spot are plotted in Fig. 25 (thin solid line). The thick solid line represents the values corrected using the geometrical factor ðR x0 Þ2 found from Eq. (5.39). The absolute values of rðxÞ can be determined from the corrected BPV distribution, if we in addition measure the resistivity at two appropriately matched positions using a 4-point probe. 5.5.3. Absolute measurements of grad r in a circular slice Studies on the inhomogeneity in circular semiconductor samples using the BPV effect were reported by Blackburn et al. [33] and Larrabee and Blackburn [34]. The latter investigators used the most advanced methods for scanning the light spot over the diameter of a large circular slice. The slice is equipped, at its periphery, with eight electrodes installed at eight points 1–8, as shown in Fig. 23. The voltage may be measured between various pairs of the contacts. If the light is scanned along the diameter 5–1, we measure the photovoltaic voltages U51 and U73 between the contacts 5–1 and 7–3. The method described above is referred to by the authors as the X–Y Geometry variant. Another variant—the ‘45 geometry’—consists of the measurements of U62 þ U48 ; U62 U48 ; and dURð64Þ in the presence of the current flow I82 : Here we shall only describe the X–Y Geometry variant, considering it to be sufficiently illustrative of this very valuable measurement idea. In order to find the voltage U51 we need to calculate the potentials CðR; 0Þ and CðR; 0Þ. Substituting w ¼ R into Eq. (5.39) results in an infinity in the denominators, which leads to Y ðR; 0Þ ¼ 0:
ð5:40Þ
Then, from Eq. (5.39) we obtain
Dx þ iDy Dx þ iDy 1 R CðR; 0Þ ¼ þ 2 Re 2R 2pd R x0 RðR x0 Þ R x0 R 2 R Dy ¼ pd R2 x0 R
ð5:41Þ
and hence we obtain the voltage U51 ¼ CðR; 0Þ CðR; 0Þ ¼
2RDx : pdðR2 x20 Þ
ð5:42Þ
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Fig. 24. Potential distribution and force lines, in a circular sample with point contacts. (a) Dipole parallel to the sample diameter, (b) dipole perpendicular to the sample diameter, and (c) dipole beyond the diameter, deviated at 45 with respect to the diameter. Contour and surface plot.
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Fig. 25. Measured distribution of the BPV voltage (thin solid line) in a circular 300 Si wafer. The distribution of the BPV voltage calculated with account of the circular geometry and point contacts (thick solid line). The distribution of the resistivity along the sample diameter (dotted line) determined from the corrected distribution of BPV.
The voltage U73 is equal to the difference of the potentials Cð0; iRÞ Cð0; iRÞ: Thus, using Eq. (5.39) we obtain U73 ¼ Cð0; iRÞ Cð0; iRÞ
R R iR R þ iR Re ðDx þ iDy Þ ¼ 2pdðR x0 Þ RðiR x0 Þ RðiR x0 Þ
2RDy R iR R þ iR þ ðDx þ iDy Þ 2 2 : ¼ R iRx0 R þ iRx0 pdðR2 þ x20 Þ
ð5:43Þ
Eqs. (5.42) and (5.43) are consistent with Eqs. (16) and (17) given by Larrabee and Blackburn [34]. According to notation (5.2) and Eq. (4.27), we have for n-type material qr ð5:44Þ Dx ¼ Mx P ¼ 2mp kB T 0 P: qx Substituting this equation into Eq. (5.42) and performing appropriate transformations we obtain
qr0 pdðR2 x20 Þ U51 : ð5:45Þ ¼ 4R qx x0 mp kB TP Similarly, Eq. (5.43) gives
qr0 pdðR2 þ x20 Þ U73 : ¼ 4R qy x0 mp kB TP
ð5:46Þ
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It should be remembered that these equations are only valid when we deal with ntype semiconductors. In the case of p-type semiconductors mp should be replaced by mn : It follows from the equations derived above that the voltages U51 and U73 only permit determining the approximate relative distributions of the derivative of r0 ; since in practice we are unable to determine precisely the number of carriers, P; generated by the light spot. This number depends on the excess carrier lifetime at a given point, which is unknown. This spoils the accuracy of the estimate of the existing inhomogeneity. Having however known Eqs. (5.45) for both the components of the grad r0 ; we can find its precise direction (Fig. 23) as tgY ¼
qr0 qr0 R2 þ x20 U73 : ¼ 2 : qy qx R x20 U51
ð5:47Þ
Larrabee et al. [34] and Blackburn [33] proposed to measure additionally the photo-induced resistance change DR; which permitted them to eliminate the dependence of the results on the carrier number P: They measured this change by passing a constant current I51 through the sample. The light spot was focused at point x0 and modulated, thanks to which the change of the voltage dU51 (established between electrodes 1 and 5) due to the photoconductivity of the light spot region could easily be measured. Section 4 described the generalised fundamentals of this method and gave Eqs. (4.40) that determined the voltage dUR ; which now will be denoted dU51 : Another equation derived there was (4.42) which permitted determining the value of the individual components of the grad r as functions of grad C0 ; the UPBV voltage (now denoted as U51 ), and dUR ¼ dU51 : Therefore, the only quantity to be found is the potential that occurs in the presence of the current I51 : In the case of a single source positioned at the edge of a half-plane, the potential distribution is of the 1=z type, and we have a singularity at the point z ¼ 0: Then, transforming the half-plane into a circle with the radius R and centre at z ¼ 0; we obtain I51 r0 R þ z C0 ¼ ; ð5:48Þ ln Rz pd where the function has a singularity at z ¼ 7R: In terms of complex numbers, gradient C0 ðx; yÞ is determined by the derivative with respect to z: dC0 I51 r0 2R ¼ : dz pd R2 z2
ð5:49Þ
Let us consider an arbitrary point x0 positioned on diameter 1–5. Substituting x ¼ x0 into Eq. (5.49), we obtain a real number with no imaginary part, which leads to qC0 ðx0 ; 0Þ I51 r0 2R ¼ ; qx pd R2 x20
ð5:50Þ
qC0 ðx0 ; 0Þ ¼ 0: qy
ð5:51Þ
The absence of the component perpendicular to the 1–5 diameter means that, in Eq. (4.42), cos a ¼ 1 and cos b ¼ 1: In this equation, the value of jgrad C0 j is defined
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by Eq. (5.50). Hence, we can conclude that the component of gradient r0 parallel to 1–5 diameter can be determined from
qr0 q 2 mp þ mn R I51 r ¼ U51 : ð5:52Þ qx x0 kB T 0 mp R2 x20 dU51 It should be noted that, in this equation, 1=DR appears as I51 =dU51 : It follows from the general theory that the equation that determines qr0 =qy cannot be found in this way. We may however use Eq. (5.47). By multiplying by sides Eqs. (5.47) and (5.52) we obtain
qr0 q 2 mp þ mn RðR2 þ x20 Þ I51 r ¼ U73 : ð5:53Þ qy x0 kB T 0 mp ðR2 x20 Þ2 dU51 Eq. (5.52) is consistent with Eq. (28) given by Larrabee [34], Eq. (28), and Eq. (5.53) is also confirmed there [34, Table 1, column III]. It should be noted that Eqs. (5.52) and (5.53) are valid for n-type semiconductor. 5.5.4. Circular slice with extended electrodes The problem of extended electrodes will first be considered for the case of a halfstrip shown in Fig. 26. The initial dipole D# z ¼ Dx þ iDy placed at the point z0 and its image D# z are placed at the point z0 : In order to obtain the equation for the potential induced by the original dipole placed at z0 ; we should transform Eq. (5.33) by subtracting iy from z: This means that in the first term x0 is replaced by z0 and in the second term by z0 : When this is done we obtain n h i h io 1 p p Re D# z ctg ðz z0 Þ D# z ctg ðz þ z0 Þ : Cor ðx; yÞ ¼ ð5:54Þ 4dh 2h 2h
Fig. 26. Dipole Dz situated at point z0 of the half-strip with a metal electrode and its image Dz placed at point z0 :
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The potential of the image D# z one can obtain by substituting into the above equation D# z instead of D# z and D# z instead of D# z : The point z0 should be replaced by z0 and vice versa. Then we have n h i h p io 1 p Re D# z ctg ðz z0 Þ þ D# z ctg ðz þ z0 Þ : Cim ðx; yÞ ¼ ð5:55Þ 4dh 2h 2h The potential in the half-strip is the sum of Cor and Cim n h i 1 p p Re D# z ctg ðz z0 Þ þ ctg ðz þ z0 Þ Cst=2 ðx; yÞ ¼ 4dh h 2h 2h io p p D# z ctg ðz þ z0 Þ þ ctg ðz z0 Þ : ð5:56Þ 2h 2h This equation forms the basis for calculating the potential distribution in a circular slice with one extended electrode. The first step is to formulate a conformal transformation so that the electrode segment becomes the (tg a; tg a) segment on the real axis of the half-plane and the sides of the half-strip pass on the remaining parts of the real axis. The second transformation converts the half-plane into a circle with the radius R positioned in the complex plane w ¼ u þ iv. The point electrode should be placed at w ¼ R: The extended electrode lies on an arc of the circle between the points Re2ia and R2ia . Thus, the size of the electrode is defined by 4a: Performing these two transformations, using the Christoffel–Schwarz integral, and transforming the circle to the y-axis we obtain h h i Rw : ð5:57Þ z ¼ þ arcsin 2 p tg a R þ w To perform transformation (5.57) we should find an equation that defines the relation between the dipoles D# z and D# w (placed at the point w0 ). This equation is
dz 2hRi # # qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: Dz ¼ Dw ¼ D# w ð5:58Þ dw w0 pðR þ w0 Þ tg2 aðR þ w0 Þ2 þ ðR w0 Þ2 The system of Eqs. (5.55)–(5.57) enables calculating the potential distribution of BPV (under open-circuit conditions) in a circular slice with one extended electrode. The values of the slice radius R; the angle a; the dipole D# w and its position w0 should be assumed. The value of z0 occurring in Eq. (5.54) can be calculated from Eq. (5.57). Fig. 27 shows the potential distribution established by the dipoles of intensities R d 1 V directed along: (a) u-axis, (b) n-axis and (c) deviated at 45 . The resistivity distribution determined based on the measurement of BPV bears an error dependent on the geometry of the sample and its electrodes. Typically, the measurements are made on circular slices with point or extended electrodes. A circular slice with two point electrodes is shown in Fig. 23. The variation of the BPV voltage with the light spot position is described by Eq. (5.39). Using the correction divisor BðR x0 Þ2 we can find the distribution of grad r: The BPV distribution in a circular slice with one point electrode and one extended electrode is defined by Eqs. (5.56)–(5.58). The more advanced calculations in the case of a circular slice with two extended electrodes were made. Assuming a1 ¼ j1 =4 and a2 ¼ j2 =4; the dipole placed on the u-axis of the electrodes and directed parallel to this diameter
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Fig. 27. Potential distribution in a circular sample with one from two extended electrode. Dipoles directed (a) along the axis of the electrodes, (b) perpendicular to the axis, and (c) beyond the sample diameter, deviated at 45 with respect to the diameter. Contour and surface plot of potential distribution. Angular width of the electrode is 60 .
one obtains UðR;RÞ 4Du C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ¼ h ; ð5:59Þ h i h i i2 hu i2 R 2 2 u u u 0 0 0 0 1 þ þ1 1 þ þ 1 tg2 a2 tg2 a1 R R R R where u0 is the distance between the light spot and centre of slice and Z p p=2 db qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: C¼ 2 0 ð1 ½tga1 tga2 2 sin2 bÞ
ð5:60Þ
The results of the calculation are shown in Fig. 28 in the form of a family of curves for various angular widths of the electrodes. Each curve represents the dependence of
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Fig. 28. Normalized photovoltage as a function of the position of a unit dipole directed along the diameter of a circular sample. Angular widths of both extended electrodes (a) 1 , (b) 5 , (c) 10 , (d) 20 , (e) 30 and (f) 60 .
the unit dipole voltage between the electrodes on its position. Two extended electrodes are placed symmetrically on both sides of circular slice. The unit dipole is placed in the u-axis of the electrodes being a diameter of the circle. The direction of the dipole is parallel to the diameter. Every curve is calculated for a specified angular size j ¼ 4a; the same for both electrodes. The voltages presented by the curves in Fig. 28 are obtained assuming a constant intensity of the dipole, so the voltage value for a definite position may serve as a correction coefficient. It is interesting to note that at j ¼ 4a ¼ 30 ; the voltage is almost constant. This means that in this case the correction coefficient is constant and equal to unity.
6. Bulk photoelectromagnetic effect 6.1. Theory The action of the magnetic field on the charge carriers in metals and semiconductors leads to a variety of interesting phenomena from magnetoresistance and thermomagnetic effects to the Hall effect. In view of the fact that excess carriers may be present in semiconductors, an additional group of photoelectromagnetic effects has to be considered. The well-known phenomenon, namely the photoelectromagnetic effect was discovered by Kikoin and Noskov [14] and was investigated by Bulliard [36] in germanium and silicon. The rigorous theory of this effect was given by van Roosbroeck [15]. In the publications mentioned above the authors assume that the semiconductor is homogeneous. In the case of an inhomogeneous semiconductor exposed to a magnetic field, another effect takes place, which was discovered and discussed by the present autor in Refs. [37–39]. This effect was called bulk photoelectromagnetic effect (BPEM).
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This chapter is devoted to the derivation of the transport equations for the excess current carriers in an inhomogeneous semiconductor exposed to a magnetic field, and to the theory of the bulk photoelectromagnetic effect and its experimental confirmation. It should be noted that the considerations presented throughout this chapter do not take into account any magnetoresistance effects, since only linear (with respect to the magnetic field) theory is considered here. According to the laws governing the action of the magnetic field on charge carriers, the additional hole J Bp and electron J Bn current appear J Bp ¼ ðJ 0p bÞYp ;
ð6:1Þ
J Bn ¼ ðJ 0n bÞYn ;
ð6:2Þ
where J 0p ; J 0n are hole and electron current densities in the absence of magnetic field, b is unit vector parallel to the magnetic induction B and the Hall angles Yp ; Yn are defined as Yp ¼ mpH B;
ð6:3Þ
Yn ¼ mnH B;
ð6:4Þ
where B is the absolute value of magnetic induction, and mpH ; mnH are the Hall mobilities of holes and electrons, respectively. The hole and electron current densities, J p and J n ; in the presence of magnetic field can be obtained by adding current (6.1) and (6.2) to the J p0 and J n0 : J p ¼ J 0p þ ðJ 0p bÞYp ;
ð6:5Þ
J n ¼ J 0n þ ðJ 0n bÞYn :
ð6:6Þ
In order to transform further these equations it will be useful to introduce the internal current J defined by (2.40). Then we obtain the form analogous to (2.41) and (2.42): sp J 0p ¼ J 0 þ J ; ð6:7Þ s sn J 0n ¼ J 0 J ; ð6:8Þ s where J 0 ¼ J 0p þ J 0n :
ð6:9Þ
Substituting (6.7) and (6.8) into Eqs. (6.5) and (6.6) one obtains the formulae that describe the hole and electron currents: sp sp J p ¼ J 0 þ J þ Yp J 0 b þ Yp J b; ð6:10Þ s s sn sn J n ¼ J 0 þ J þ Yn J 0 b þ Yn J b: ð6:11Þ s s
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Adding these formulae and combining the corresponding components one obtains the formula for the total current J s sn p Yp þ Yn ðJ 0 bÞ þ ðYp Yn ÞJ b: ð6:12Þ J ¼ Jp þ Jn ¼ J0 þ s s The introduction of the Hall angle sp sn Yp þ Yn s s and the angle of the photomagnetoelectric effect YH ¼
ð6:13Þ
Y ¼ Yp Yn ¼ ðmpH þ mnH ÞB
ð6:14Þ
enables formula (6.12) for the total current to be expressed in the simple form J ¼ J 0 þ YH J 0 b þ YJ b: 0
ð6:15Þ
The currents J and J present in Eq. (6.15) may be replaced by the equation derived in Section 2.2.2, which defines the same quantities. J 0 is the same as the total current density in the absence of the magnetic field; and thus one may use (2.60). Similarly J is defined by (2.50). Substituting these expressions into Eq. (6.15) and rearranging the terms appropriately one obtains an important equation defining the total current density in an inhomogeneous semiconductor that contains excess carriers and is exposed to magnetic field: J ¼ s grad Cð1 þ YH bÞ qDY grad Dp b 2qm0 Dp grad V0 ð1 þ YH bÞ qm DpY grad V0 b:
ð6:16Þ
Multiplying both sides of this equation by 1 YH b and neglecting the terms that include Y2H and YH Y give Jð1 YH bÞ ¼ s grad C qDY grad Dp b 2qm0 Dp grad V0 qm DpY grad V0 b:
ð6:17Þ
After rearranging the terms and dividing the equation by s one obtains the equation for gradient of the potential C J YH J b YqD 2qm0 Dp þ grad Dp b þ grad V0 grad C ¼ s s s s qm DpY grad V0 b: þ ð6:18Þ s An analysis of Eq. (6.18) enables each component of grad C to be linked to a certain law or phenomenon: (1) (2) (3) (4) (5)
ðJ=sÞ grad V0 ohmic component, YH J b=s Hall effect, ðqDY=sÞ grad Dp b Kikoin–Noskov photoelectromagnetic effect, ð2qm0 Dp=sÞ grad V0 bulk photovoltaic effect and ðqm DpY=sÞ grad V0 b bulk photoelectromagnetic effect.
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Fig. 29. Illuminated rectangular sample in the presence of a magnetic field. The gradient of carrier concentration and built-in field are perpendicular to the longest edge of the sample.
In order to describe the bulk photoelectromagnetic effect undisturbed by other phenomena, the case of a thin rectangular plate placed in a perpendicular magnetic field B (shown in Fig. 29) will now be considered. The internal field grad V0 is directed perpendicularly to the length of the plate. Hence the conductivity and the remaining parameters of the material are functions of the co-ordinate y alone. The steady state and a homogeneous magnetic field are assumed. As the illumination is homogeneous and the thickness of the plate is small, the assumption of Dp¼ const: is justified within the entire illuminated region. The non-illuminated region is wide enough to prevent the excess carriers from reaching the electrodes. The foregoing assumptions concerning the geometry of the sample, and the direction of magnetic and internal fields yield b ¼ k;
grad V0 ¼ k
dV0 ; dy
J ¼ iJx :
ð6:19Þ
Substituting these formulae into Eq. (6.16) one obtains the equation system:
qC Jx qm DpY dV0 Jx ¼ þ ¼ þ EBPEM ; qx s s dy s
ð6:20Þ
qC YH Jx 2qm Dp dV0 YH Jx ¼ þ ¼ þ EBPV ; qy s s dy s
ð6:21Þ
qC ¼ 0; qz
ð6:22Þ
where EBPEM and EBPV are the EMF—fields of the bulk photoelectromagnetic and bulk photovoltaic effects, respectively. The derivative q=qxðqC=qyÞ taken from (6.21) vanishes because the variable parameters only depend there on y. Consequently, q=qyðqC=qxÞ calculated from (6.20) is also equal to zero and hence qC Jx ¼ þ EBPEM ¼ const: qx s
ð6:23Þ
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Solving this for Jx yields
qC þ EBPEM : Jx ¼ s qx
ð6:24Þ
The above equations show that, generally, Jx must depend on y; but detailed computation proves that the assumption Jx Econst:
ð6:25Þ
results in an insignificant error, provided that grad V0 is sufficiently small. Using formulae (6.20) and (6.21) we will explain (i) the short-circuit, (ii) compensation and (iii) open-circuit measuring methods. (i) If the sample electrodes are short circuited, the Isc current may be calculated by substituting qC=qz ¼ 0 into Eq. (6.20) to obtain Isc dV0 ¼ qm DpY ¼ sEBPEM ; s dy
ð6:26Þ
where s is the cross-section of the sample. The above equation indicates that the short-circuit current flowing in the external circuit is proportional to the excess carrier concentration Dp and to the magnetic field (through the angle Y). The group mobility m is an important parameter in this equation. This parameter is positive in an n-type semiconductor, negative in a p-type one and is equal to zero in an intrinsic semiconductor. Thus the direction of the short-circuit current flow depends on the semiconductor type. It is interesting to calculate the distribution of the electric field in the direction y: It follows from (6.26) that Jx is proportional to Y; and thus substituting its value into Eq. (6.21) we obtain a term that contains B squared and proportional to Y YH ; which may be neglected. Thus in the case of a short-circuited sample we obtain
qC 2qm0 Dp dV0 ¼ ¼ EBPV : qy s dy
ð6:27Þ
It follows from (6.27) that only the bulk photovoltaic effect acts in the direction y: The component in the direction z does not exist nor does it exist in the cases considered below. The component dC=dy can be measured if additional electrodes (shown in Fig. 30) are used, and the electrodes A and B are short circuited.
Fig. 30. Components of the bulk photomagnetoelectric effect and the measurement contacts.
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(ii) To realise the compensation method a modulated illumination is used and a constant current I ¼ Jx s is passed through the electrodes A, B. Two modulated photovoltages are then observed. The first one is due to the BPEM and the second is related to the modulated photoconductivity. The current I (and thus Jx ) must then be chosen so as to suppress the resultant of both signals to zero. During the illumination the current is given by qC dV0 qm DpY ð6:28Þ J x ¼ ½s0 þ qðmn þ mp ÞDp qx dy and without illumination, by
qC J xc ¼ s0 ; qx C
ð6:29Þ
where s0 is the equilibrium conductivity and qðmn þ mp ÞDp is the photoconductivity. Subtracting (6.29) from (6.28) and taking into acount that Jx ¼ Jxc and qC=qx ¼ ðqC=qxÞC ; one obtains
qC dV0 ð6:30Þ 0 ¼ qðmn þ mp ÞDp þqm DpY qx C dy and hence
Jxc qC m dV0 : ¼ ¼ Y qx C s0 mn þ mp dy
ð6:31Þ
If the parameters of the semiconductor are known and Jxc is measured, the presented theory of BPEM can be proved using this equation. Using definitions (6.5), (6.6) and (6.14) one can transform Eq. (6.31) to obtain mnH þ mpH dV0 Jxc dV0 ¼ BkH m ; ð6:32Þ ¼ B m s0 mp þ mn dy dy where kH is the ratio of the Hall mobility and the drift mobility, m is the group mobility defined by (2.49). This result indicates the possibility of measuring the minority carrier mobility mm in the n- or p-type material, where m ¼ 7mm ; respectively. (iii) Considering the open-circuit case shown in Figs. 30 and 31 one should assume Jx ¼ 0
ð6:33Þ
according to the result (6.25). Substituting this value into Eqs. (6.20) and (6.21) one obtains the component of grad C in the direction x as
qC qm DpY dV0 ¼ ¼ EBPEM qx s dy
and the component in the direction y as qC 2qm0 DpY dV0 ¼ ¼ EBPV : qy s dy
ð6:34Þ
ð6:35Þ
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Fig. 31. Distribution of the electric field in an open-circuit, n-type sample with finite dimensions.
Experimental studies of the BPEM may be based on the measurement of both fields EBPEM and EBPV and the determination of their ratio which is tga ¼
EBPEM Y m ¼ : 2 m0 EBPV
ð6:36Þ
The parameter a is the angle between the direction of grad C and qC=qy or it is the inclination of equipotential lines with respect to the plate edge (Fig. 31). To calculate tga given by (6.36) one should only know the magnetic field B; the two Hall mobilities (according to (6.5) and (6.6)) and the ratio m =m0 : This last value is equal to +1, for n-type and 1 for p-type semiconductor. Thus formula (6.36) is more convenient than (6.31). The agreement between the values of tg a determined experimentally and theoretically proves that the theory is correct. Eq. (6.36) may be used for determining the sum of Hall mobilities. According to (6.14), (2.49) and (2.18) one can write 1 m 1 tga ¼ BðmpH þ mnH Þ ¼ 7 BðmpH þ mnH Þ; 2 2 m0
ð6:37Þ
where plus for n-type and minus for p-type hold. The above equation enables us to determine the sum of Hall mobilities and then the minority carrier mobility if the majority carrier mobility is known from classical Hall measurements. Thus, the BPEM measurements give two possibilities to measure the minority carrier mobility. The first possibility is demonstrated by (6.32) and the second is represented by Eq. (6.36). To realize the measurement of tg a according to Eq. (6.37) it is necessary to determine EBPEM and EBPV : It is seen from (6.34), (6.35) and Fig. 30 that UBPEM ; ð6:38Þ EBPEM ¼ l EBPV ¼
UBPV ; a
ð6:39Þ
where l and a are the length and width of the illuminated region, respectively, UBPEM is the voltage between the electrodes A, B, and the voltage UBPV is measured between the point electrodes C, D.
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6.2. Experiment Investigations were carried out using germanium p- and n-type plates [37] made from single crystals with a sufficiently high, though uniform, resistivity gradient. The plates were cut in such a direction so as to obtain an internal field grad V0 directed perpendicularly to he length of the plate. As a result, the transverse built-in field in the samples was at least an order of magnitude higher than the lateral component of this field. The thickness of the investigated samples ranged from 0.2 to 0.5 mm, the width was between 6 and 7 mm and the length between 12 and 15 mm. The sample was illuminated using an optical set-up with a tungsten bulb as the source of light. The light beam was directed to the sample using a mirror. A rectangular light spot with approximate dimensions of 5 5 mm was created at the surface of the sample. Both the sample (held appropriately) and the mirror were placed between the poles of an electromagnet capable of providing magnetic induction up to approximately 0.7 T. The placements of the sample and mirror, as well as the path of the light are shown in Fig. 32. The measurements enabled the voltages UBPEM and UBPV and thus the fields EBPEM and EBPV to be determined. The field EBPEM was measured versus the magnetic field B and the intensity of illumination. Fig. 33 shows the shape of EBPEM ¼ f ðBÞ for n-type germanium sample 1 (0.9 O cm) and n-type sample 2 (5.8 O cm) at constant illumination. The dependence of EBPEM on the magnetic field B was linear within the experimental error in agreement with the theory of small angles yp ; yn : The dependence of EBPEM on the illumination was also found to be linear. The magnitudes of EBPEM at B ¼ 0:7 T (points K, L in Fig. 33) were 0.40 mV (sample 1) and 0.30 mV (sample 2) at full voltage on the lamp. Moreover EBPV across points C, D was measured. In the sample 1 EBPV ¼ 2:0 mV, whereas in sample 2 EBPV ¼ 1:12 mV. Thus the computation of tg a was possible (Eqs. (6.34)–(6.36)) yielding in sample (1Þtg a ¼ 0:15; in sample (2) tg a ¼ 0:27: A rough theoretical computation yielded the value tg a ¼ 0:26; which was accepted as satisfactory.
Fig. 32. Experimental arrangement for measuring the bulk photomagnetoelectric effect; (a) side view of the sample, (b) front view; 1—Ge sample, 2—mirror.
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Fig. 33. UBPEM versus the magnetic induction B for two (1, 2) n-type Ge samples with different resistivities.
Acknowledgements The authors like to acknowledge Prof. Dr. Jaroslaw. Swiderski for the valuable discussions.
Appendix A. Derivation of Eq. (2.17) The equations involved in the calculations are: Ds ¼ s s0 ¼ qðmn þ mp ÞDp; grad Ds ¼ s grad ln
s Ds þ grad s0 ; s0 s0
kB T grad s0 ¼ ðsn0 sp0 Þgrad V0 : q
ðA:1Þ ðA:2Þ ðA:3Þ
Eq. (A.1) follows from the definition of the conductivity. To prove that Eq. (A.2) is correct we should apply the grad operator to the right side of it. Eq. (A.3) may be derived based on the formulae for the equilibrium concentrations p0 and n0 : q ðj V0 Þ ; p0 ¼ ni exp ðA:4Þ kB T 0 q n0 ¼ ni exp ðV0 j0 Þ ; ðA:5Þ kB T
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where j0 is the equilibrium Fermi level, which is position independent. By calculating the gradient of the two concentrations we obtain q grad p0 ¼ p0 grad V0 ; ðA:6Þ kB T grad n0 ¼
q n0 grad V0 : kB T
ðA:7Þ
Then, based on the assumption that mn and mp are position independent, we multiply both sides of these equations by qmp and qmn to obtain kB T grad sp0 ¼ sp0 grad V0 ; q
ðA:8Þ
kB T grad sn0 ¼ sn0 grad V0 ; q
ðA:9Þ
where sn0 and sp0 are the equilibrium conductivities. Summing Eqs. (A.8) and (A.9) gives (A.3). Now, let us use Eqs. (A.1) and (A.2) to transform Eq. (2.16). J Ds kB T mn mp ¼ grad V þ grad V0 grad V0 þ grad Ds s s qs mn þ mp Ds grad V0 ¼ gradðV V0 Þ s
k B T m n mp s Ds þ s grad ln þ grad s0 qs mn þ mp s0 s0 kB T mn mp s ¼ gradðV V0 Þ þ grad ln q mn þ mp s0 ! 1 mn mp 1 kB T grad s0 grad V0 Ds: þ s mn þ mp s0 q
Eth ¼
ðA:10Þ
By combining the first and second members of this equation, inserting Ds as defined by (A.1), and using the assumption that T; mp and mn are position independent, we obtain ! kB T mn mp s Eth ¼ gradðV V0 Þ þ grad ln q mn þ mp s0 " # 1 m n mp 1 þ ðsn0 sp0 Þgrad V0 grad V0 Ds s mn þ mp s0 ! kB T mn mp s ln ¼ gradðV V0 Þ þ grad q mn þ mp s0
q sn0 sp0 þ ðmn mp Þ ðA:11Þ ðmn þ mp Þ Dp grad V0 : s s0 s0
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Transforming the coefficient enclosed in the square brackets gives
sn0 sp0 ðmn mp Þ ðmn þ mp Þ s0 s0
sn0 sp0 sn0 sp0 1 mp þ1 ¼ mn s0 s0 s0 s0 mp mn ¼ ðsn0 sp0 s0 Þ ðsn0 sp0 þ s0 Þ s0 s0 sp0 sn0 m 2 m ¼ 2m0 ; ¼ 2 s0 n s0 p
ðA:12Þ
which is consistent with definition (2.18). Using these results we may write ! J kB T mn mp s Eth ¼ ¼ gradðV V0 Þ þ grad ln s q mn þ mp s0 2qm0 Dp grad V0 : s This is exactly Eq. (2.17).
ðA:13Þ
Appendix B. Derivation of Eqs. (2.22) and (2.23) Eqs. (2.22) and (2.23) are derived based on the Boltzmann statistics. The reasoning is as follows. B.1. Derivation of (2.22) If Dpa0 and Ja0; the concentrations p and n are given by q p ¼ ni exp ðj V Þ; kB T p n ¼ ni exp
q ðV jn Þ: kB T
ðB:1Þ ðB:2Þ
If Dp ¼ 0; but Ja0; then jp ¼ jn ¼ j and p ¼ p0 ; n ¼ n0 : In this case, Eqs. (B.1) and (B.2) become q ðj V Þ; ðB:3Þ p0 ¼ ni exp kB T n0 ¼ ni exp
q ðV jÞ kB T
ðB:4Þ
kB T ni kB T ni ln ¼ ln : q q p0 n0
ðB:5Þ
and hence V j¼
In the particular case when Dp ¼ 0 and J ¼ 0; the term V j is replaced by the term V0 j0 ; which is equal to the former term since it depends on p0 and n0 in the same
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way. Thus we have V0 j0 ¼ V j:
ðB:6Þ
Eq. (2.21) can now be transformed so that Cðr0 Þ ¼ V ðr0 Þ V0 ðr0 Þ þ j0 ¼ ½V ðr0 Þ jðr0 Þ ½V0 ðr0 Þ j0 þ jðr0 Þ:
ðB:7Þ
The Eq. (B.6) for point r0 is V ðr0 Þ jðr0 Þ ¼ V0 ðr0 Þ j0 ;
ðB:8Þ
which means that the terms in square brackets cancel one another and we arrive at Eq. (2.22): Cðr0 Þ ¼ jðr0 Þ:
ðB:9Þ
B.2. Derivation of (2.23) Because of (B.6), (B.3) and (B.4) for the point r0 can be written as q p0 ¼ ni exp ðj V0 Þ; kB T 0 n0 ¼ ni exp
q ðV0 j0 Þ: kB T
Now, dividing (B.1) by (B.10) and (B.2) by (B.11) gives p q ðj V j0 þ V0 Þ; ¼ exp p0 kB T p n q ðV jn V0 þ j0 Þ: ¼ exp n0 kB T Hence, we can derive the equation kB T p kB T n ln ¼ jn þ ln : V V0 þ j0 ¼ jp q p0 q n0 For small values of Dp; we have kB T Dp kB T Dp V V0 þ j0 Djp ¼ jn þ : q p0 q n0
ðB:10Þ ðB:11Þ
ðB:12Þ ðB:13Þ
ðB:14Þ
ðB:15Þ
Using Eq. (A.1) and knowing that sp0 =s0 þ sn0 =s0 ¼ 1; we obtain from (B.15)
sp0 kB T Dp sn0 kB T Dp V V0 þ j0 ¼ jp jn þ q p0 q n0 s0 s0
sp0 sn0 kB T Dp sn0 sp0 ¼ jp þ jn þ q s0 n0 s0 s0 p0 sp0 sn0 Dp ¼ j þ j þ kB T ðmn mp Þ s0 s0 p s0 n ! sp0 sn0 kB T mn mp Ds ¼ j þ j þ ðB:16Þ q mn þ mp s0 s0 p s0 n
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and hence sp0 sn0 kB T mn mp Ds jp þ jn ¼ V V0 : q mn þ mp s0 s0 s0
ðB:17Þ
For small values of Ds; the right-hand side of this equation is, by (2.20), equal to the potential C: Thus, in this case sp0 sn0 CD jp þ j : ðB:18Þ s0 s0 n This is exactly Eq. (2.23).
Appendix C. Calculation of gradðV0 j0 Þ appearing in Eq. (3.23) and in equations following (3.28) C.1. Calculation of grad n0 =ni The expression to be found contains the term grad ln
n0 grad n0 grad ni ¼ : ni n0 ni
ðC:1Þ
The equilibrium concentrations n0 and p0 are related to each other through the neutrality principle and the mass-action law: n0 p0 ¼ N;
ðC:2Þ
p0 n0 ¼ n2i
ðC:3Þ
which results in ðp0 þ n0 Þ2 ¼ N 2 þ 4n2i :
ðC:4Þ
Applying grad to (C.4) and using (C.3) yields ðp0 þ n0 Þgradðn0 þ p0 Þ ¼ 2N grad N þ 4n0 p0 grad ln ni :
ðC:5Þ
where N was replaced by (p0 n0 ) in accordance with (C.2). The application of grad to (C.2) and subsequent multiplication by (p0 þ n0 ) yields ðp0 þ n0 Þgradðn0 p0 Þ ¼ ðn0 þ p0 Þgrad N:
ðC:6Þ
Adding (C.5) and (C.6) and dividing the result by n0 one obtains ðp0 þ n0 Þgrad ln n0 ¼ grad N þ 2p0 grad ln ni : Subtracting the term ðp0 þ n0 Þgrad ln ni from both sides of (C.7) yields n0 ðp0 þ n0 Þgrad ln ¼ grad N þ ðp0 n0 Þgrad ln ni ; ni
ðC:7Þ
ðC:8Þ
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which leads to grad ln
n0 grad N p0 n0 ¼ þ grad ln ni : ni p0 þ n0 p0 þ n0
ðC:9Þ
C.2. Calculation of kB T=q grad ln ni The formula defining intrinsic concentration
pffiffiffiffiffiffiffiffiffiffiffiffiffi EG ni ¼ NC NV exp 2kB T
ðC:10Þ
leads to kB T kB T T EG grad ln ni ¼ grad lnðNC NV Þ grad q 2q 2q T
kB T T EG EG grad lnðNC NV Þ grad 2 grad T ¼ 2q 2q T T kB T 1 EG kB T ¼ grad lnðNC NV Þ grad EG þ : grad 2q 2q q 2kB T
ðC:11Þ
In order to take into account the dependence of NC NV and EG on the temperature and position we proceed in accordance with (3.25) 3 grad lnðNC NV Þ ¼ grad lnðmC mV Þ þ 3 ln T þ const 2 3 q 3 3 grad T ¼ lnðmC mV Þgrad T þ gradxyz lnðmC mV Þ þ 2 qT 2 T
ðC:12Þ
and 1 1 qEG kB T 1 grad EG ¼ þ gradxyz EG ; grad 2q 2kB qT q 2q
ðC:13Þ
where the operator gradxyz only refers to the co-ordinates. Substituting results (C.12) and (C.13) into Eq. (C.11) we obtain kB T grad lnni q kB T 3 q 3 3 grad T lnðmC mV Þgrad T þ gradxyz lnðmC mV Þ þ ¼ 2q 2 qT 2 T 1 qEG kB T 1 EG kB T gradxyz EG þ : grad grad 2kB qT q 2q q 2kB T
ðC:14Þ
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Substituting (C.14) into Eq. (C.9), rearranging the terms and substituting the result into Eq. (3.23) we obtain n0 kB T kB T n0 þ grad ln grad q q ni ni
kB T grad N n0 p0 n0 3T q 3 1 qEG EG þ ¼ þ ln þ lnðmC mV Þ þ ni p0 þ n0 4 qT 2kB T q p0 þ n0 2 2kB qT kB T p0 n0 3kB T EG þ gradxyz lnðmC mV Þ gradxyz
grad ðC:15Þ q p0 þ n0 4q 2kB T
gradðV0 j0 Þ ¼ ln
which is used in formulae (3.29) and (3.31).
References [1] J. Tauc, Theory of the bulk photovoltaic effect in semiconductors, Czechoslovak Journal of Physics 5 (1955) 178. [2] Z. Trousil, Bulk photovoltaic effect, Czechoslovak Journal of Physics 6 (1956) 96. [3] V.E. Lashkarev, V.A. Romanov, Bulk photo-emf in semiconductors, Trudy Instituta Fiziki AN USSR 7 (1956) 50. [4] W. Van Roosbroeck, Theory of flow of electrons and holes in germanium and other semiconductors, Bell System Technical Journal 29 (1950) 560. [5] W. van Roosbroeck, The transport of added current carriers in a homogeneous semiconductor, Physical Review 91 (2) (1953) 282. [6] W. Shockley, The theory of p–n junction in semiconductors and p–n junction transistors, Bell System Technical Journal 28 (1949) 435. [7] S. Chandrasekhar, Stochastic problems in physics and astronomy, Reviews of Modern Physics 15 (1943) 1–81. [8] P.T. Landsberg, S.A. Hope, Two formulations of semiconductor transport equations, Solid-State Electronics 20 (1977) 421. [9] S.R. de Groot, P. Mazur, Non-Equilibrium Thermodynamics, North-Holland, Amsterdam, Vol. 3, 1962 (Chapter VI). [10] A.G. Jordan, R.W. Lade, D.L. Scharfetter, A note on Fermi-levels, quasi-Fermi-levels and terminal voltages in semiconductor devices, American Journal of Physics 31 (1963) 490. [11] J. Bardeen, Theory of relation between hole concentration and characteristic of germanium point contact, Bell System Technical Journal 29 (1950) 469. [12] H.K. Henisch, Rectifying Semi-Conductor Contacts, Clarendon Press, Oxford, 1957. [13] J. Tauc, Photo-and Thermoelectric Effects in Semiconductors, Pergamon Press, NewYork, 1962. [14] I.K. Kikoin, M.M. Noskov, A new photoelectric effect in Cu2O, Physikalishe Zeitschrift der Sovietunion 5/6 (1934) 478, 586. [15] W. van Roosbroeck, Theory of the photomagnetoelectric effect in semiconductors, Physical Review 101 (1956) 1713. [16] P.T. Landsberg, Recombination in Semiconductors, Cambridge University Press, Cambridge, 1991. [17] A.H. Marshak, K.M. van Vliet, Current in semiconductors with position-dependent band gap and electron affinity, Physica Status Solidi (a) 42 (1977) 279. [18] A.H. Marshak, K.M. van Vliet, Carrier densities and emitter efficiency in degenerate materials with position-dependent band structure, Solid-State Electronics 21 (1978) 429. [19] A.H. Marshak, K.M. van Vliet, Electrical current in solids with position-dependent band structure, Solid-State Electronics 21 (1978) 417. [20] A.H. Marshak, C.M. van Vliet, Electrical current and carrier density in degenerate materials with nonuniform band structure, Proceedings of the IEEE 72 (2) (1984) 156.
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[21] A.H. Marshak, C.M. van Vliet, Semiclassical electrical current expressions for materials with position-dependent band structure in the near equilibrium approximation, Journal of Applied Physics 61 (19) (1997) 6800. [22] K.M. van Vliet, A.H. Marshak, The Shockley-like equations for the carrier densities and the current flows in materials with a nonuniform composition, Solid-State Electronics 23 (1980) 49. [23] K.M. van Vliet, Entropy production and choice of heat flux for degenerate semiconductors and metals, Physica Status Solidi (b) 78 (1976) 667. [24] L.J. van Ruyven, F.E. Williams, Electronic transport in graded-band-gap semiconductors, American Journal of Physics 35 (1967) 705. [25] P.R. Emtage, Electrical conduction and the photovoltaic effect in semiconductors with positiondependent band gaps, Journal of Applied Physics 33 (1962) 6. [26] R. Kishore, Generalized equations for the steady-state analysis of inhomogeneous semiconductors devices, Solid-State Electronics 33 (8) (1990) 1049. [27] S. Sikorski, Thermodynamical approach to the laws of current carrier and thermal energy transport in semiconductors, Electron Technology 31 (2) (1998) 227. [28] S. Sikorski, The transport of excess current carriers in an inhomogeneous semiconductor with position-dependent band gap, Semiconductor Science and Technology 13 (1998) 18. [29] S. Sikorski, T. Piotrowski, Transport of excess charge carriers in an inhomogeneous semiconductor in the presence of temperature gradient, Electron Technology 33 (4) (2000) 494. [30] T. Piotrowski, S. Sikorski, Photovoltaic effects in an inhomogeneous semiconductor with positiondependent temperature, Semiconductors Science and Technology 16 (2001) 750. [31] J. Oroshnik, A. Many, Evaluation of the homogeneity of germanium single crystals by photovoltaic scanning, Journal of the Electrochemical Society 106 (4) (1959) 360. ! [32] J. Swiderski, New applications of bulk photovoltaic effect measurements on germanium, Proceedings of the International Conference on Semiconductor Physics, Praque, 1960. [33] D.L. Blackburn, H.A. Schafft, L.J. Swartzendruber, Nondestructive photovoltaic technique for the measurement of resistivity gradients in circular semiconductor wafers, Journal of the Electrochemical Society 119 (12) (1972) 1773. [34] R.D. Larrabee, D.L. Blackburn, Theory and application of a nondestructive photovoltaic technique for the measurement of resistivity variations in circular semiconductor slices, Solid State Electronics 23 (1980) 1059. [35] E. Madelung, Das elektrische feld in systemen von regelm.asig angeordneten Punktladungen, Physikalische Zeitschrift. XIX (1918) 5524. [36] H. Bulliard, Photomagnetoelectric effect in germanium and silicon, Physical Review 94 (1954) 1564. [37] Sikorski, Bulk photoelectromagnetic effect in a non-homogeneous semiconductor, Bulletin de L’Academie Polonaise des Sciences, Serie des Sciences Techniques 8 (1960) 371. [38] S. Sikorski, Bulk photoelectromagnetic effect in semiconductors, Proceedings of the International Conference on Semiconductor Physics, Praque, 1960, p. 182. [39] S. Sikorski, Anew galvanomagnetic effect in semiconductors, Nature 193 (4810) (1962) 32.
Progress in Quantum Electronics 27 (2003) 295–365
Review
Photovoltaic phenomena in inhomogeneous semiconductors S. Sikorski, T. Piotrowski* Institute of Electron Technology, Al. Lotnikow 32/46, Warsaw 02-668, Poland
Abstract The photovoltaic phenomena in inhomogeneous semiconductors are discussed here. (i) The Rooesbroeck potential is generalised and the formulae defining the Tauc bulk photovoltaic effect (BPV), in semiconductors with position-dependent doping, are derived. (ii) The domain of the inhomogeneities is expanded, and besides doping the cases of position-dependent energy gap EG and mobilities in the presence of temperature gradient are described. Thus a new thermo-photovoltaic effect is revealed. (iii) The equations describing the BPV potential distribution in the case of point illumination are derived. Using the Green function formalism the equations are obtained, which give a base to examine the inhomogeneities especially of resistivity distribution. (iv) The method of images is used to calculate the BPV potential distribution. The potential distributions in rectangular plates and circular slices are presented. (v) The results of photovoltaic measurements carried out on a sample with temperature gradient are presented, giving an experimental confirmation of the thermophotovoltaic effect. Application of BPV to determine experimentally resistivity distribution is demonstrated. (vi) The theory and measurements of the bulk photoelectromagnetic effect (BPEM) taking place in an inhomogeneous semiconductor exposed to magnetic field are presented. r 2003 Elsevier Ltd. All rights reserved.
Contents 1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
299
2.
Photovoltaic effects. Transport equations . . . . . . . . . . . . . . 2.1. Transport of excess carriers in a homogeneous semiconductor 2.2. Transport of excess carriers in a semiconductor with position-dependent doping . . . . . . . . . . . . . . . . . . .
300 300
*Corresponding author. Tel.: +48-22-54-87-942; fax: +48-22-847-0631. E-mail address: [email protected] (T. Piotrowski). 0079-6727/03/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0079-6727(03)00004-1
303
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Nomenclature B magnetic induction b unit vector parallel to the magnetic induction BPEM bulk photoelectromagnetic effect BPV bulk photovoltaic effect D ambipolar diffusion coefficient (2.45) D dipole moment # ¼ Dx þ iDy complex dipole moment D d slice thickness Dp ðDn Þ hole (electron) diffusion coefficient Dx ; Dy ; Dz components of dipole moment (EMF) electromotive force EBPEM vector of BPEM (6.20) EBPV ; EMPV ; ETPV ; ETEV vectors defined by (3.19)–(3.22) VC T EN BPV ; EBPV ; EBPV vectors defined by (3.28)–(3.31) Efi intrinsic Fermi level Efp ðEfn Þ hole (electron) Fermi level EG energy band gap Eth thermodynamic force defined by (2.11) and (2.17) EV ðEC Þ valence (conduction) band edge g electron–hole pairs generation rate gs electron–hole pairs surface generation GðA; BÞ; G0 ðA; BÞ 3D Green functions I current intensity i; j; k unit vectors of axes x; y; z; respectively J total current density J p ðJ n Þ hole (electron) current density J 0p ðJ 0n Þ hole (electron) current density in the absence of magnetic field J internal current density (2.40) J0 total current density in the absence of magnetic field kB Boltzmann constant LD excess carrier diffusion length Lp ðLn Þ hole (electron) thermoelectric transport coefficient (3.3) and (3.4) M; M N ; M E ; M T ; M M vectors defined by (4.2)–(4.5) mv ðmc Þ hole (electron) effective mass n non-equilibrium electron concentration n0 equilibrium electron concentration þ þ N ¼ ND NA ND ðNA Þ donor (acceptor) concentration n normal unit vector ni intrinsic concentration NV ðNC Þ effective density of states of valence (conduction) band p non-equilibrium hole concentration
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P total number of hole–electron pairs p0 equilibrium hole concentration q elementary charge R radius Rint internal resistance (2.29) s surface recombination velocity s cross-sectional area T temperature TB bulk of the sample tg Y quantity defined by (5.47) U voltage V shockley potential (3.7) V0 shockley potential in equilibrium w ¼ uþiv complex variable x; y; z co-ordinates z ¼ xþiy complex variable CðzÞ function defined by (4.46) Dp excess hole–electron pairs concentration Ds excess conductivity (2.37) dðA BÞ 3D Dirac delta function d2D ðA BÞ 2D Dirac delta function dG Green function defined by (4.25) dC potential defined by (4.34) ep ðen Þ hole (electron) thermoelectric coefficient (3.3) and (3.4) Y angle defined by (6.14) YH angle defined by (6.13) Yp ðYn Þ Hall angle for holes (electrons) (6.3) and (6.4) m ambipolar mobility with illumination m group mobility (2.49) m0 ambipolar mobility in equilibrium mp ðmn Þ hole (electron) mobility mpH mnH Hall mobility of holes (electrons) r non-equilibrium resistivity r0 equilibrium resistivity s non-equilibrium conductivity s0 equilibrium conductivity sp ðsn Þ non-equilibrium hole (electron) conductivity sp0 ðsn0 Þ equilibrium hole (electron) conductivity j ¼ 4a angular width (5.57)–(5.50) j0 equilibrium Fermi potential jp ðjn Þ hole (electron) Fermi potential ðnÞ ðnÞ jðnÞ x ; jy components of potential C C potential defined by (2.14)
297
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2.2.1. 2.2.2. 2.2.3. 2.2.4.
. . . .
. . . .
. . . .
. . . .
303 307 309 311
Transport of excess charge carriers in a semiconductor with extended inhomogeneity in the presence of temperature gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Theoretical foundation . . . . . . . . . . . . . . . . . . 3.2. Numerical example . . . . . . . . . . . . . . . . . . . . 3.3. Experimental investigation of thermophotovoltaic effect .
. . . .
. . . .
. . . .
312 312 317 318
Photovoltaic electrical fields . . . . . . . . . . . . . . . . . . . . 4.1. Basic equations . . . . . . . . . . . . . . . . . . . . . . . . 4.2. General solution for an open-circuit sample . . . . . . . . . 4.3. Fundamentals of the measurement of the bulk photovoltaic effect (BPV) and the relative gradient of resistivity . . . . . 4.4. Measurement of the resistivity distribution . . . . . . . . . . 4.5. The Green function and the potential distribution in thin samples . . . . . . . . . . . . . . . . . . . . . . . . .
. . .
320 320 322
. .
325 327
.
329
The BPV—potential distributions . . . . . . . . . . . . . . 5.1. The method of images . . . . . . . . . . . . . . . . . 5.2. Half-space and space between two parallel half-planes . 5.3. Infinitely long plate . . . . . . . . . . . . . . . . . . . 5.4. Cuboid with electrodes at two opposite faces . . . . . 5.5. Slices . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1. Half-plane and infinite strip . . . . . . . . . . 5.5.2. Circular slice . . . . . . . . . . . . . . . . . . 5.5.3. Absolute measurements of grad r in a circular slice . . . . . . . . . . . . . . . . . . . . . . 5.5.4. Circular slice with extended electrodes . . . . .
. . . . . . . .
331 331 332 335 337 339 339 342
. . . . . . . .
343 347
Bulk photoelectromagnetic effect . . . . . . . . . . . . . . . . . . . 6.1. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . .
350 350 357
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
358
A. Derivation of Eq. (2.17) . . . . . . . . . . . . . . . . . . . . . . .
358
B.1. B.2.
Derivation of (2.22) . . . . . . . . . . . . . . . . . . . . . . . . Derivation of (2.23) . . . . . . . . . . . . . . . . . . . . . . . .
360 361
C.1. Calculation of grad n0 =ni . . . . . . . . . . . . . . . . . . . . . . C.2. Calculation of kB T=q grad ln ni . . . . . . . . . . . . . . . . . .
362 363
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
364
3.
4.
5.
6.
Bulk photovoltaic effect (BPV) . . . . The hole and electron current densities Continuity equations . . . . . . . . . . Stationary transport equation system .
. . . .
. . . .
. . . .
. . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
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1. Introduction The bulk photovoltaic effects, resulting from the existence of an inhomogeneity of continuous (non-abrupt) character, have long been ignored and, what is more, no advantage was gained from their occurrence. Initially, studies on the photovoltaic effect due to a continuous inhomogeneity were inspired by its disadvantageous influence upon the results of other electrical measurements, such as e.g., the measurement of the Hall mobility or resistivity. Tauc [1] in 1955 and Trousil [2] in 1956, who discovered the bulk photovoltaic phenomenon (BPV) and formulated its strict theory, explained the mechanism of this influence. The paper of Lashkarev and Romanov [3] devoted to BPV too was published in the next year. These authors discovered BPV independent of Tauc and Trousil, but their results could be published in USSR only after a considerable delay. These studies initiated investigations of the photovoltaic phenomena in inhomogeneous semiconductors, where the relative change of certain material parameters along the diffusion length is well below unity. These parameters include in the first place the concentration of impurities and the chemical composition. The carrier mobility and the width of the band gap, as well as their variability, are associated with these parameters and with the temperature. The inhomogeneous semiconductors have also other features that distinguish them from junction structures, namely, the effect of rectification and excess carrier injection are small and can be neglected. The BPV phenomenon has found its application in the measurements of the resistivity gradient and has become a widely used diagnostic method to which many papers have been devoted. At present, interest in the photovoltaic effects in inhomogeneous semiconductors is inspired by the development of new technologies designed for producing regions with varying band gap, which, when formed in semiconductor structures, greatly improve their performance parameters. The generalised theory of the photovoltaic effects permits modelling the operation of semiconductor devices in a more precise way, and enables us to take advantage of the effects, induced by built-in inhomogeneities, or produced purposely for diagnostic examinations. The present study synthesises the achievements of earlier and more recent studies devoted to the photovoltaic effects active in inhomogeneous semiconductors. Section 2 discusses in detail the classical BPV phenomenon and introduces a new formalism that permits us to interpret it in clear physical terms. Section 2.2 is devoted to the derivation of the continuity equations, which constitute a system of transport equations. Although these equations are confined to the description of the BPV effect here, they have a general significance. Section 3 reports on certain new results based on an extended concept of inhomogeneity in the presence of a temperature gradient. The equations derived in this chapter classify and describe the photovoltaic effects that can occur in materials with such inhomogeneity. Based on the equations obtained in Section 2, Section 4 contains the equations that define the potential distribution created by point illumination of the semiconductor surface. These equations, obtained using the Green function
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formalism, can be used for determining the potential distribution in samples of various shapes. The generality of their mathematical form justifies the application of these equations to the measurement of resistivity. Section 4 also includes the general mathematical foundations of the procedure used by Blackbourn and Larrabee for determining the absolute value of the resistivity gradient. Section 5 provides the general principles and examples of calculating the BPV potential for various sample geometries, based on the method of images and using again the Green function formalism. The potential distributions for 3D cases (cuboidal sample) and 2D cases (circular slices) are also shown. Section 6 describes the bulk photoelectromagnetic (BPEM) effect that was discovered during theoretical and experimental studies on the photovoltaic phenomena in inhomogeneous semiconductors. The theory of BPEM and its experimental confirmation are discussed here in detail. 2. Photovoltaic effects. Transport equations 2.1. Transport of excess carriers in a homogeneous semiconductor Equations describing the carrier transport in a homogeneous semiconductor were derived in a strict form by van Roosbroeck [4,5]. His results are still valid and are utilised by many scientists. The starting point of the Roosbroeck theory are equations, used by Shockley [6] in 1949 that describe the densities of the hole and electron currents J p and J n J p ¼ sp grad V qDp grad p ¼ sp grad V kB Tmp grad p;
ð2:1Þ
J n ¼ sn grad V þ qDn grad n ¼ sn grad V þ kB Tmn grad n;
ð2:2Þ
where sn and sp are the electron and the hole conductivities, q is the elementary charge, Dn and Dp are the electron and the hole diffusion coefficients, mn and mp are the electron and the hole mobilities, kB is the Boltzmann constant and T is the temperature. The hole and electron concentrations (p and n, respectively) include the concentration of the excess carriers (Dp) generated as a result of the semiconductor being illuminated. The potential V corresponds to the intrinsic Fermi level Efi : It is normally found near the centre of the band gap. The potential Efi is the Fermi level in the case when the equilibrium concentration of holes is equal to the concentration of electrons. This is why the distance of Efi to the valence band edge and the distance to the conduction band are approximately the same. The small deviation of Efi from the band gap centre results from the difference between effective masses of the holes and electrons and is of the order of kB T. Eqs. (2.1) and (2.2) have been playing a very important role in semiconductor electronics. Based on the monograph written by Chandrasekhar [7], Roosbroeck considers in Ref. [4] the equations for Jp and Jn to be the Smoluchowski equations adapted to describe the flow of holes and electrons. Both Eqs. (2.1) and (2.2) defining the hole and electron currents contain the ohmic and diffusion terms. The diffusion term can be expressed in two forms, which involve
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the diffusion coefficient or the mobility. This possibility results from the relationship discovered by Eintstein and independently by Smoluchowski. The diffusion coefficient is chosen according to the D1 formulation, which is simpler especially as regards the Einstein–Smoluchowski relationship and the current densities, as pointed out by Landsberg [8]. In the theory of the p–n junction, still another form of Eqs. (2.1) and (2.2) played an important role. Shockley noticed [6] that the mobilities and concentrations of charge carriers were involved in both the conductivities sp and sn and in the term describing diffusion. Hence, if we introduce two Fermi potentials jp and jn ; which correspond to the hole and electron Fermi levels we can express the two currents J p and J n in the form of a single term. Then we obtain the equations: J p ¼ sp grad jp ;
ð2:3Þ
J n ¼ sn grad jn :
ð2:4Þ
These equations are also valid for degenerate semiconductors, and thus are more general than Eqs. (2.1) and (2.2). Eqs. (2.3) and (2.4) are also known in the thermodynamics of irreversible processes [9], where grad jp and grad jn are referred to as thermodynamic forces. When describing photovoltaic phenomena, it is of crucial importance to formulate the equation that defines the density of the total current J: By adding Eqs. (2.3) and (2.4) we obtain J ¼ sEth ;
ð2:5Þ
where s is the total conductivity, and Eth is a thermodynamic force defined by sp sn ð2:6Þ Eth ¼ grad jn grad jn : s s Unlike the potential forces grad jp and grad jn their linear combination Eth may contain generally a non-potential component, and this is the origin of the electromotive force responsible for the occurrence of the photovoltaic effect. The present paper is devoted to the photovoltaic effects in inhomogeneous semiconductors with no abrupt changes of the material parameters (as would be the case in e.g., p–n and l–h junctions). Therefore, the samples examined obey the principle of quasi-neutrality, which says that the concentration of excess holes is equal to the concentration of excess electrons. This is written as p p0 ¼ n n0 ¼ Dp;
ð2:7Þ
where p0 and n0 are the equilibrium concentrations of the holes and electrons, and Dp is the concentration of the excess carriers. The equations presented thus far are valid for any inhomogeneous semiconductor with p0 and n0 dependent on position. However in this section, we will confine our attention to a homogeneous semiconductor, where p0 and n0 are independent of position. Using the grad operator in Eq. (2.7), we obtain grad p ¼ grad n ¼ grad Dp:
ð2:8Þ
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Since this result cannot be directly substituted in Eqs. (2.3)–(2.5), it is necessary to use (2.1) and (2.2) where p and n occur in explicit forms. Adding these equations gives J ¼ s grad V þ kB Tmn grad n kB Tmp grad p:
ð2:9Þ
Substituting (2.8) into (2.9) we obtain J ¼ s grad V þ kB Tðmn mp Þgrad Dp:
ð2:10Þ
In this equation, the carrier mobility, temperature and equilibrium concentrations n0 and p0 are independent of position, and, thus, we can derive from it the equation for the thermodynamic force Eth (as defined by (2.6)): ! J k B T m n mp s ln : ð2:11Þ Eth ¼ ¼ grad V þ grad s q m n þ mp s 0 This was done using the formula " # " # Ds s grad Dp ¼ grad ¼ grad qðmn þ mp Þ qðmn þ mp Þ " # ln s ¼ s grad qðmn þ mp Þ " # ln s=s0 ¼ s grad ; qðmn þ mp Þ
ð2:12Þ
and introducing the equilibrium conductivity s0 so as to avoid the use logarithms for a dimensional quantity. Since s0 is constant here, this substitution does not change the value of the gradient. As will be shown later, even in the case s0 aconst; the ln s=s0 term remains unchanged. Then, Eq. (2.11) can be written in the form J ð2:13Þ Eth ¼ ¼ grad Chom ; s where Chom ¼ V
kB T mn mp s ln q mn þ mp s0
ð2:14Þ
is the potential introduced by Roosbroeck [5, Eq. (19)]. Eq. (2.11) describes the thermodynamic force Eth which occurs in inhomogeneous semiconductors. We can see that Eth has two components. One is the gradient of the potential V and the other is the gradient of a certain function which contains the difference between the carrier mobilities mn mp : The latter component is known as the Dember field and, thus, it is also a potential component. Therefore, the thermodynamic field Eth is defined by the gradient of the single potential Chom ; determined by Eq. (2.13). Hence, we can infer that the photoconductivity in an inhomogeneous semiconductor is not accompanied by the bulk photovoltaic effect which would have acted as origin of the electromotive force.
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2.2. Transport of excess carriers in a semiconductor with position-dependent doping 2.2.1. Bulk photovoltaic effect (BPV) If a semiconductor is doped so that the concentrations of donors ND and acceptors NA are position-dependent and the concentration of excess charge carriers Dp is equal to zero, the total equilibrium concentrations of holes and electrons are also position-dependent and equal to p0 ðx; y; zÞ and n0 ðx; y; zÞ; respectively. Under these conditions, the total current J is zero. Then by substituting s ¼ s0 ; V ¼ V0 ; p ¼ p0 ; n ¼ n0 in the general Eq. (2.9), we obtain 0 ¼ s0 grad V0 þ kB Tmn grad n0 kB Tmp grad p0 :
ð2:15Þ
This equation can be understood as the flow of the drift current s0 grad V0 ; which is reduced by the hole and electron diffusion currents, so that the total current is equal to zero. The term grad V0 represents an electric field called the ‘built-in field’ which is an important parameter that characterises semiconductors doped in a position-dependent manner. By subtracting (2.15) from (2.9) and dividing the result by s we obtain J s0 kB T ðmn mp Þgrad Dp: Eth ¼ ¼ grad V þ grad V0 þ ð2:16Þ s s s This can be done because Eq. (2.9) is also valid for inhomogeneous semiconductor. After several transformations, given in Appendix A, we obtain ! J kB T mn mp s Eth ¼ ¼ gradðV V0 Þ þ grad ln s q mn þ mp s0
2qm0 Dp grad V0 ; s
ð2:17Þ
where m0 ¼
sp0 sn0 mn þ m s0 s0 p
ð2:18Þ
is the equilibrium ambipolar mobility. The concept of the ambipolar diffusion coefficient was introduced by Roosbroeck [5] (see Eq. (2.45)). This coefficient is related to the ambipolar mobility through the Einstein–Smoluchowski relationship. These parameters appear in the equations that describe the transport of excess carriers, i.e. holes and electrons that occur simultaneously and have the same concentrations (Dp ¼ Dn). We therefore deal with the ambipolar transport. It can easily be seen from Eq. (2.18) that, in an n-type semiconductor where sn bsp ; the ambipolar mobility m0 ¼ mn : Similarly m0 ¼ mp for the p-type. This means that the minority carriers ‘impose’ the mobility on the majority carriers. This apparent paradox has led to the use of the erroneous term ‘minority carriers’ instead of ‘the excess carriers’. It is interesting to note that Roosbroeck [5] used the term ‘added current carriers’. Continuing the reasoning that led to Eq. (2.17), let us note that, after introducing the built-in field grad V0 (which results from the inhomogeneity of the semiconductor), this equation has become a generalised form of Eq. (2.11). Eq. (2.17) can also be
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written in the form Eth ¼
J 2qm0 Dp ¼ grad C þ grad V0 ; s s
ð2:19Þ
where C V V0
k B T m n mp s ln þ j0 q m n þ mp s 0
ð2:20Þ
is a generalised form of the potential Chom defined by (2.14). The potential C is normalised by using the constant j0 ; which is the value of the single Fermi potential occurring in a given semiconductor when it is in the state of equilibrium. This means that J ¼ 0 and Dp ¼ 0 at any point. If this condition is not fulfilled, i.e., Ja0; Dp ¼ 0 and s ¼ s0 at certain points within the semiconductor, such as e.g. point r0 with the co-ordinates x0 ; y0 ; z0 ; then by Eq. (2.20) we have Cðr0 Þ ¼ V ðr0 Þ V0 ðr0 Þ þ j0 :
ð2:21Þ
An analysis of this equation, given in Appendix B, leads to the conclusion that Cðr0 Þ ¼ jðr0 Þ:
ð2:22Þ
Therefore, the normalised potential C at the point r0 ; where Dp ¼ 0; is equal to the Fermi potential jðro Þ and hence these two potentials are here equivalent from the point of view of the measurement of the potential C and of the photovoltage. According to the principles of non-equilibrium thermodynamics, the Fermi potential jM of an arbitrary galvanic measuring probe at these points will acquire a potential equal to the Fermi potential of the given semiconductor. The potential jM of the probe will thus represent the potential Cðr0 Þ: How the two sets of charge carriers within the contact region are brought to thermodynamic equilibrium and the physical sense of the Fermi level are explained in Ref. [10]. The above analysis of the situation when Dp ¼ 0 permits us to conclude that the Dember component of the thermodynamic force Eth is not involved here. This suggests that the Dember effect occurs only when contact phenomena are in question. In the regions where Dpa0 we have two quasi-Fermi potentials jp and jn : It would be interesting to know how the potential C is related to them. It is demonstrated in Appendix B that for small Dp we have sp0 sn0 C¼ j þ j : ð2:23Þ s0 p s0 n Hence we can see that the value of C lies somewhere between the values of jp and jn ; rather nearer to the Fermi potential of the majority carriers. The field Eth has a potential and non-potential components defined by Eq. (2.19). We should ask whether this decomposition of the field Eth into components is unique, or, in other words, whether this field can have another non-potential component. When we however assume that C cannot be a direct function of the coordinates since it is a function of state, i.e. a function of V0 and Dp; it appears that its non-potential component is well defined by Eq. (2.19).
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In order to illustrate the equations and results discussed above let us consider the simplest case of a cuboidal sample in which all the parameters depend on the coordinate x alone, as shown in Fig. 1. When adapting Eq. (2.19) to this onedimensional case we have Jx dC 2qm0 Dp dV0 Ethx ¼ ¼ ð2:24Þ dx s s dx and hence dC Jx 2qm0 Dp dV0 ¼ ; ð2:25Þ dx s s dx which permits us to calculate the voltage at the sample contacts as Z B Z B Z B dC dx 2qm0 Dp dV0 UBA ¼ dx ¼ CB CA ¼ Jx dx: ð2:26Þ dx s s dx A A A In order to interpret Eq. (2.26) let us use the concept of the electromotive force Z B 2qm0 Dp dV0 dx; ð2:27Þ ðEMFÞ ¼ s dx A the current outflowing from the sample I ¼ Jx s and the internal resistance of the sample Z 1 B dx ; Rint ¼ s A s
ð2:28Þ
ð2:29Þ
where s is the sample’s cross-section, we can write Eq. (2.26) in the form UBA ¼ ðEMFÞ Rint I: When the sample is in open circuit, the current I is zero and we have Z B 2qm0 Dp dV0 dx ¼ ðEMFÞ: UBA ¼ UOC ¼ s dx A
ð2:30Þ
ð2:31Þ
Fig. 1. Illuminated inhomogeneous sample with internal field grad V0 : A; B—supply electrodes. Point electrodes illustrate physical meaning of the potentials jp ; jn and C:
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We can see from this equation that, upon illumination of the semiconductor, the built-in field grad V0 becomes the origin of the electromotive force, which results in the occurrence of the photovoltaic phenomenon. The short-circuit current ISC in the sample may be calculated form Eq. (2.30) by assuming UBA ¼ 0: Then we obtain RB 2qm Dp=s ðEMFÞ ISC ¼ ¼ A R B0 : ð2:32Þ Rint 1=S dx=s A
The above reasoning is based on the fact that, in the non-illuminated region, the potential C is equal to the Fermi level j: Any metallic electrode or probe placed in this region will acquire a potential equal to the potential C: Within the illuminated region, the potential of the electrode depends on the structure formed in the contact area. If it is a p–n junction, the potential of the metallic contact will be equal to the quasi-Fermi potential of the minority carriers jmin : A point contact 1 (Fig. 1) of this type is for example formed when a point probe is used for measuring the so-called floating potential, which is just equal to the Fermi potential of the minority carriers as shown by Bardeen [11] and Henisch [12, Eq. (833.12)]. The point-probe measurements played an important role at the beginnings of the development of semiconductor electronics. An electrode of another type is the probe 2 shown in Fig. 1, which contains an l–h structure. It can be shown that it acquires a potential equal to the quasi-Fermi potential of the majority carriers jmaj : If the sample has been prepared so that it possesses a non-illuminated outgrowth 3, the probe of this type will acquire the potential equal to the potential C: This can be proved on the assumption that the length of the outgrowth exceeds the diffusion length LD of the excess carriers. This reasoning indicates that when measuring the parameters characteristic of the fields, potentials and currents in an illuminated semiconductor that contains excess carriers, it is very important to be aware of the method employed for the measurements. When measuring the voltage UBA under open-circuit conditions, i.e., measuring the electromotive force, Eq. (2.31) gives us the built-in field dV0 =dx: In practice however we are rather interested in the gradient of the equilibrium resistance r0 : To analyze this parameter, we should first determine the derivative of the carrier concentration as a function of dV0 =dx: In an inhomogeneous non-degenerate semiconductor, the concentrations n0 and p0 are given by n0 ¼ ni exp
q ðV0 j0 Þ; kB T
p0 ¼ ni exp
q ðj V0 Þ: kB T 0
ð2:33Þ
Hence, if n0 and p0 are functions of the co-ordinate x alone, we have dn0 q dV0 ¼ n0 ; kB T dx dx
dp0 q dV0 ¼ p0 kB T dx dx
ð2:34Þ
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since j0 ¼ const: By multiplying the first of Eqs. (2.34) by qmn; the other equation by qmp and then adding we obtain d ds0 q dV0 ðsp0 þ sn0 Þ ¼ ¼ ðsn0 sp0 Þ ð2:35Þ dx kB T dx dx and hence dV0 kB T 1 ds0 kB T 1 1 dr0 ¼ ¼ q sn0 sp0 dx q sn0 sp0 r20 dx dx kB T s20 dr0 : ¼ q sn0 sp0 dx
ð2:36Þ
Then substituting this result into Eq. (2.27), taking sEs0 ; and denoting Ds ¼ qðmn þ mp ÞDp; we obtain the electromotive force EMF as Z B 2qm0 Dp dV0 dx ðEMFÞ ¼ s dx A Z B kB T 2qm0 Dp s20 dr0 dx ¼ q A s sn0 sp0 dx Z kB T B 2m0 Ds s0 dr0 E dx: ð2:37Þ q A mn þ mp sn0 sp0 dx (1) In an n-type semiconductor m0 Emp according to (2.18) and thus we have m0 =ðmn þ mp ÞEmp =ðmn þ mp Þ ¼ 1=ðmn =mp þ 1Þ ¼ 1=ðb þ 1Þ; where b is the ratio of mobilities. Moreover, since here sn bsp ; we have sn sp Es0 and then Z B kB T 2 dr ð2:38Þ ðEMFÞn ¼ Ds 0 dx: q bþ1 A dx (2) In a p-type semiconductor, the relationship between the mobilities is m0 =ðmn þ mp ÞEmn =ðmn þ mp Þ ¼ ðmn =mp Þ=ðmn =mp þ 1Þ ¼ b=ðb þ 1Þ and sp bsn: and sn sp E s0 : Hence we obtain Z B kB T 2b dr ð2:39Þ ðEMFÞp ¼ Ds 0 dx: q bþ1 A dx These results are fully consistent with Eqs. (3.25) and (3.24) in Ref. [13] and with Eqs. (7) in Ref. [3]. Eqs. (2.38) and (2.39) are useful in determining experimentally dr0 =dx; and also in determining the precise course of the r0 ðxÞ distribution based on the bulk photovoltaic effect. However, Eq. (2.27) is more general since it describes both n-type and p-type semiconductors and also semiconductors with a continuous (without the barrier) transition from n-type to p-type. 2.2.2. The hole and electron current densities In order to formulate the equations that describe the transport of excess carriers in a complete form, it is necessary to transform the equations for J p and J n : Let the starting point be a current which we will call the internal current. The density of this
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current J is defined as sn sp J ¼ J n J p: s s
ð2:40Þ
The densities of the electron and hole currents calculated from this equation are sp J p ¼ J þ J ; ð2:41Þ s Jn ¼
sn J J : s
ð2:42Þ
Now we add these two equations and see that the current J ; which is a component of the current due to the holes, and the current J ; which is a component of the electron current, cancel each other and the resultant is the total current J: This is the explanation why the current J has been called ‘internal’: it does not contribute to the total current J: In physical terms, these two current components cancel each other, since the excess holes in the current J move in the same direction as the excess electrons in the current J : It can be interpreted as an electrically neutral flux of electron–hole pairs. Substituting Eqs. (2.1) and (2.2) into Eq. (2.40) and ordering the terms we obtain s sp n mp grad p þ mn grad n : J ¼ kB T ð2:43Þ s s As can be seen, J is determined by the diffusion components, whereas the terms that contain grad V0 cancel each other. It follows from (2.7) that p ¼ p0 þ Dp and n ¼ n0þ Dp. Using these relationships in Eq. (2.43) gives s kB T sp n ðsn mp grad p0 þ sp mn grad n0 Þ kB T J ¼ mn þ mp grad Dp s s s kB T ðsn mp grad p0 þ sp mn grad n0 Þ kB Tm grad Dp; ð2:44Þ ¼ s where m is the ambipolar mobility in an illuminated semiconductor, and also fulfils the equation for the ambipolar diffusion coefficient D: D ¼ kB Tm;
m¼
sn sp mp þ mn : s s
ð2:45Þ
To proceed further, let us differentiate Eqs. (2.33) and (2.34). When this is done we have grad p0 ¼
q p0 grad V0 ; kB T
grad n0 ¼
q n0 grad V0 : kB T
Substituting these results into Eq. (2.44) gives s sp n J ¼ q mp p0 mn n0 grad V0 kB Tm grad Dp: s s
ð2:46Þ
ð2:47Þ
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Now, if we adopt the quasi-neutrality principle (2.7), the bracketed expression that appears in the grad V0 term can be transformed to take the form sn sp 1 m p0 mn n0 ¼ ½sn mp ðp DpÞ sp mn ðn DpÞ s s p s
1 1 1 sn spn sn sp sn snp mp mnp Dp ¼ s q q s s ¼ mn Dp;
ð2:48Þ
where sn sp m m s p s n is known as the group mobility. Using (2.48) in Eq. (2.47)we obtain m ¼
ð2:49Þ
J ¼ qm Dp grad V0 kB Tm grad Dp:
ð2:50Þ
We can infer from this equation that, in an illuminated inhomogeneous semiconductor, the internal current is composed not only of the ambipolar diffusion current, which is proportional to the gradient of the excess carrier concentration Dp; but also contains the drift current of the excess carriers induced by the built-in field grad V0 : This is thus another type of the neutral flux of the electron–hole pairs, which is a result of the inhomogeneously distributed doping in the semiconductor. It is worthwhile to analyse the behaviour of this neutral flux under the action of a magnetic field. Thanks to the Lorentz force, the magnetic field tends to split the flux along the direction perpendicular to itself and to the flux axis, thereby generating an electromotive force. The result of the action of the magnetic field on the component kB Tm grad Dp (Eq. (2.50)) is known as the photoelectromagnetic effect [14,15]. The action of magnetic field B upon the component qm grad V0 is the bulk photomagnetoelectric effect, which will be discussed later in Section 6. Utilising Eqs. (2.19), (2.41), (2.42) and (2.50) we can write equations for the densities of the hole and electron currents in an inhomogeneous semiconductor that contains excess carriers. These equations are J p ¼ sp grad C qD grad Dp
2qm0 sp Dp grad V0 qm Dp grad V0 ; s
ð2:51Þ
2qm0 sn Dp grad V0 þ qm Dp grad V0 : ð2:52Þ s By putting grad V0=0 in these equations we obtain the equations derived by Roosbroeck [5] for homogeneous semiconductors. J n ¼ sn grad C þ qD grad Dp
2.2.3. Continuity equations To describe the transport of charge carriers with allowance made for the excess carriers, it is necessary to start with two continuity equations: first equation for the space charge and the second equation for the concentration of the excess carriers. When adopting the quasi-neutrality principle, the space charge and its time derivatives are equal to zero, from which it follows that the field of the total current
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density vector is solenoidal. Hence, the first continuity equation is div J ¼ 0:
ð2:53Þ
This equation may also be used for time-dependent processes, but we must remember that, there, it is an approximation valid only when the displacement currents are negligible. For stationary processes, Eq. (2.53) is strictly fulfilled even at high concentration Dp; irrespective of whether or not the quasi-neutrality principle is satisfied. Since J ¼ J p þ J n ; we can easily see that div J p ¼ div J n :
ð2:54Þ
The second continuity equation contains the excess carrier concentration Dp: qDp 1 1 ¼ div J p u þ g ¼ div J n u þ g; qt q q
ð2:55Þ
where u is the recombination rate and g is the generation rate of the electron–hole pairs. In view of the fact that the equilibrium carrier concentrations are independent of time, it is only the time derivative of the excess carrier concentration placed on the left-hand side of Eq. (2.55), which should be taken into account. An important assumption is that the carrier generation rate g due to an external factor, such as light, is the same for both the holes and the electrons. If we assume that the recombination mechanism is of the RSH (Shockley–Read–Hall) type, we obtain the well-known form [16] u¼
pn n2i ; tp ðn þ n1 Þ þ tn ðp þ p1 Þ
ð2:56Þ
where ni is the intrinsic concentration, whereas tn ; tp ; n1 and p1 are the time and the concentration parameters of the recombination centres, respectively. The RSH type is the basic mechanism of recombination in Ge and Si. To put Eq. (2.55) into the form that meets the practical application requirements, we should calculate div J p (or div J n ) using Eq. (2.41) (or (2.42)) and basing on the first continuity Eq. (2.53). When this is done we have s s sp p p div J p ¼ div J þ J ¼ div J þ J grad s s s sp ð2:57Þ þ div J ¼ 0 þ J grad þ div J ; s which when substituted into (2.55) gives qDp 1 sp ¼ div J þ J grad u þ g: qt q s
ð2:58Þ
In order to obtain a more detailed form of this equation, we should rearrange and transform this equation so as to calculate: grad sp =s: grad
sp sn0 sp þ sp0 sn q qm ¼ grad V grad Dp: þ 0 kB T s s2 s
ð2:59Þ
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2.2.4. Stationary transport equation system The results obtained thus far permit us to write a system of three equations that are fulfilled by the three fundamental quantities: (i) The total current density J; (ii) the potential C and (iii) excess carrier concentration Dp: These equations are J ¼ s grad C 2qm0 Dp grad V0 ;
ð2:60Þ
div J ¼ 0;
kB T m grad Dp þ m Dp grad V0 div q
sn0 sp þ sp0 sn m kB T grad V0 grad Dp J þ s2 s Dpðp0 þ n0 þ DpÞ þ g ¼ 0: tp ðn þ n1 Þ þ tn ðp þ p1 Þ
ð2:61Þ
ð2:62Þ
Eq. (2.60) follows immediately from Eq. (2.17). Eq. (2.61) is the continuity Eq. (2.53), and Eq. (2.62) is a detailed form of the continuity Eq. (2.58) and is composed of four members. The first member can be obtained by acting upon J ; as given by Eq. (2.50), with the operator div. The coefficient accompanying J in the second member of this equation is gradðsp =sÞ given by (2.59). The third member determines the recombination rate u given by Eq. (2.56), and the fourth represents the externally induced generation. The continuity Eq. (2.62) is the time-independent generalised form of the widely known Eq. (12) (given by Roosbroeck in Ref. [5]). That this is so can be seen by putting grad V0 ¼ 0 in Eq. (2.62). Compared to the equation which describes the behaviour of a homogeneous semiconductor, Eq. (2.62) has a very characteristic feature. If we assume g ¼ 0 (no illumination) and Ja0 then substituting Dp ¼ 0 gives the left-hand side of Eq. (2.62) proportional to (grad V0 ) J which cannot be zero. Hence the solution of Eq. (2.62) should be Dpa0: The obtained result shows that in an inhomogeneous semiconductor without illumination the current injection of excess carriers takes place, just as in the case in the neighbourhood of a p–n junction. In order to obtain a unique solution, we should specify the boundary conditions. Let us consider the case when the semiconductor sample has two metallic contacts (Fig. 2). On the surface S3 ; the boundary conditions for the hole and electron currents are J p n ¼ qsDp qgs ;
ð2:63Þ
J n n ¼ qsDp þ qgs
ð2:64Þ
where n is the normal unit vector, s is the surface recombination velocity and gs is the hole-electron pair generation rate per unit surface area. The normal components J p n and J n n of vector J p and J n define the charge of the holes and electrons that flow into the unit surface area. By adding these equations two-side, we obtain the first
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Fig. 2. Sample with arbitrary shape. S1 ; S2 —electrodes, S3 —the remaining area of the surface, Dp— concentration of excess carriers, TB—bulk of the sample.
boundary condition Jn ¼ 0;
ð2:65Þ
which means that the normal component of the total current J is equal to zero, thanks to which no resultant electric charge accumulates there. Now, multiplying the first of the equations by sn =s; the second equation by sp =s; and then adding, we obtain the second boundary condition J n ¼ qsDp qg:
ð2:66Þ
These boundary conditions for an inhomogeneous semiconductor may be formulated in a more detailed form by using Eqs. (2.50) and (2.60). Then, they become
qC 2qm0 Dp qV0 ¼ ; qn s qn
ð2:67Þ
qDp qV0 m Dp ¼ sDp gs : ð2:68Þ qn qn The operator q=qn means the derivative in the direction normal to the sample surface. D
3. Transport of excess charge carriers in a semiconductor with extended inhomogeneity in the presence of temperature gradient 3.1. Theoretical foundation The description of the excess carrier transport in an inhomogeneous semiconductor, with due allowance made for the position dependence of its band gap EG ; requires resorting to a sufficiently general theory. The most valuable contributions to such a theory were made by Tauc [13], Marshak [17–21], van Vliet [22,23], van Ruyven [24], Emtage [25] and Kishore [26]. Based on the results obtained by these
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investigators, the present author [27] formulated equations that described the transport of energy and of charge carriers with account of the presence of excess carriers. Then the concept of semiconductor inhomogeneity was extended [28] to include not only the position-dependent doping but also the position-dependent band gap EG : In later publications [29,30], the transport equations were generalised to include the position-dependence of temperature (grad Ta0). The generalised equations permit us to identify all the photovoltaic effects that occur under these conditions. As in Section 2.1, we begin the description of the carrier transport with formulating the general equations that describe the hole and electron current densities. According to Marshak [20], we have Efp J p ¼ sp grad Lp grad T; ð3:1Þ q J n ¼ sn grad
Efn Ln grad T; q
ð3:2Þ
where Efp and Efn are the Fermi levels for the holes and electrons, respectively, and Lp and Ln are the coefficients which define the thermoelectric currents. Eqs. (3.1) and (3.2) are analogous to Eqs. (2.3) and (2.4), except that the potentials jn and jp are replaced by the Fermi levels Efn and Efp divided by q; and that certain members proportional to grad T appear. In the paper of Marshak and Vliet [20] the coefficients Lp and Ln are defined as the ratios of the integrals of energy functions, Fermi functions and carrier relaxation constants. The integrals cover the first Brillouin zone. The authors mentioned above assumed that the relaxation constants depend on the energy alone and that the energy depends on the wave vector k in a spherical-parabolic manner. Based on these assumptions, the following results are obtained:
kB EV Efp ep L p ¼ sp ; ð3:3Þ q kB T
kB Efn EC en ; ð3:4Þ Ln ¼ sn q kB T where EV and EC are the edges of the valence and conduction bands, Efp and Efn are the hole and electron Fermi levels, and ep and en are the hole and electron thermoelectric coefficients, respectively. When the phonon scattering prevails, then for non-degenerate semiconductor with parabolic-spherical band, we have ep ¼ en ¼ 2: (Appendix 1 in Ref. [27]). By combining Eqs. (3.1) with (3.3) and Eqs. (3.2) with (3.4) we obtain
sp EV Efp grad Efp þ grad kB T ep grad kB T ; ð3:5Þ Jp ¼ q kB T
sn Efn EC grad Efn grad kB T þ en grad kB T : Jn ¼ ð3:6Þ q kB T
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Now we aim at transforming Eqs. (3.5) and (3.6) so as to give them a Shockley-like form. The Shockley potential V [6], which was already introduced in Section 2.1, is related to the intrinsic Fermi level Efi by the relationship V ¼
Efi : q
ð3:7Þ
We should also use certain equations from the Maxwell–Boltzmann statistics which involve the energy level Efi ; the effective densities of states NV in the valence band and NC in the conduction band, and the intrinsic concentration ni : These equations are: p Efi Efp ¼ kB T ln ; ni
EV Efp p ¼ ln ; NV kB T
ð3:8Þ
n Efn Efi ¼ kB T ln ; ni
Efn EC n ¼ ln : NC kB T
ð3:9Þ
Now, by using Eqs. (3.7)–(3.9) in Eqs. (3.5) and (3.6) and applying certain transformations (described in detail in Appendix 1 to our earlier paper [29]), we obtain the equations for Jp and Jn as J p ¼ sp grad V mp kB T grad p
kB T ni kB T grad ln ni sp ep ln ; þ sp grad q q NV J n ¼ sn grad V þ mn kB T grad n
kB T ni kB T grad ln ni þ sn en ln : sn grad q q NC
ð3:10Þ
ð3:11Þ
Putting grad ln ni ¼ 0 and grad T ¼ 0 in these equations gives Eqs. (2.1) and (2.2), a result which proves that our Eqs. (3.5) and (3.6) are Shockley-like equations. If we regard Eqs. (3.10) and (3.11) as belonging to the system of transport equations, we can consider the currents J p and J n ; the concentrations p and n and the potential V to be unknown quantities, with the remaining quantities as given. We assume that the distribution Tðx; y; zÞ is known. If the distribution of the chemical composition of the material is given (e.g. SiGe), then the intrinsic concentration ni ¼ ni ðT; x; y; zÞ; the mobilities mn and mp ; the effective densities of states NV and NC are all functions of temperature and co-ordinates. The introduction of the concentrations p and n into Eqs. (3.8) and (3.9) permits us to apply the neutrality principle, which is very useful for the photovoltaic phenomena that take place in inhomogeneous semiconductors. Now by adding Eqs. (3.10) and (3.11) we obtain the total current density J as kB T grad ln ni J ¼ s grad V þ ðmn grad n mp grad pÞkB T þ ðsp sn Þ q
ni ni kB T ; ð3:12Þ þ sn en ln sp ep ln grad q NC NV
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which, when assuming grad ln ni ¼ 0 and grad T ¼ 0; becomes Eq. (2.9). Hence, we can see that (3.12) is a generalised form of (2.9). The presence of the term grad ln ni results from the allowance made for the position-dependent chemical composition of the semiconductor, whereas the non-zero value of the grad T reflects the effect of varying temperature upon the flow of the current and upon the distribution of the electric field. For the sake of precision, it should be added that grad ln ni may be also related with the temperature gradient. In further transformations of Eq. (3.12), we will use certain relationships and notations consistent with the neutrality principle, such as p ¼ p0 þ Dp;
n ¼ n0 þ Dp;
sp ¼ sp0 þ Dsp ;
ð3:13Þ
sn ¼ sn0 þ Dsn ;
s ¼ s0 þ Ds;
ð3:14Þ
where Dsp ¼ qmp Dp;
Dsn ¼ qmn Dp;
Ds ¼ qðmn þ mp ÞDp:
ð3:15Þ
Moreover, it appeared necessary to utilize the relationship between the equilibrium concentrations p0 and n0 and the Fermi potential under conditions of equilibrium (no illumination). This relation is defined by the equation p0 q n0 ðj V0 Þ ¼ ln ; ð3:16Þ ln ¼ ni k B T 0 ni which results from the Maxwell–Boltzmann statistics adequate for the nondegenerate semiconductors. The presence of other members in Eq. (3.12) permits us to confirm that, apart from the bulk photovoltaic effect, other photovoltaic phenomena and, in addition, the Seebeck effect also take place in the semiconductor. To analyse these phenomena we need to transform Eq. (3.12) using Eqs. (3.13)–(3.16), which requires a series of algebraic operations to be done. These were described in detail in Ref. [29]. These transformations give the equation for the thermodynamic force Eth similar to Eq. (2.17): J ð3:17Þ Eth ¼ ¼ grad C þ EBVP þ EMPV þ ETPV þ ETEV ; s where ! mn mp kB T s grad C ¼ grad V V0 j0 ln ; ð3:18Þ s0 mn þ mp q 2qm0 Dp gradðV0 j0 Þ; s 1 s0 s qðmn mp Þðn0 p0 Þ ¼ 2 ln kB T s0 Ds s0 s ! mn mp m Dp grad ln n ;
mn þ mp mp
EBPV ¼ EMPV
ð3:19Þ
ð3:20Þ
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q en þ ep EG sn0 sp0 n0 s0 s ðmn mp Þ þ ln ln s 2 2kB T s0 ni Ds s0
en ep 3 m c þðmn þ mp Þ þ ln 2 mv 2 kB T ;
Dp grad q
1 n0 p0 kB T : ¼ sn0 en ln sp0 ep ln grad s q NC NV
ETPV ¼
ETEV
ð3:21Þ ð3:22Þ
These equations are valid for the inhomogeneous semiconductor, with positiondependent doping, chemical composition and temperature. The excess carriers generated in the material by illumination induce the photovoltaic effects described by Eqs. (3.17)–(3.22). The components of the thermodynamic force Eth are: grad C is defined by Eq. (3.18) which differs from Eq. (2.20) only in that j0 is position-dependent whereas in the former it is not; EBPV is the component representing the photovoltaic effect, analogous to the second term of Eq. (2.19), except that it does not include the varying potential j0 : An analysis of this component shows that, in spite of its apparent simplicity, it has a complex structure (to be discussed later in the text); EMPV is the photovoltaic component resulting from the position dependence of the electron-to-hole mobility ratio; ETPV is the component of the thermophotovoltaic effect, which occurs wherever the simultaneous presence of a photoconductivity and temperature gradient takes place; ETEV is the component of the Seebeck classical thermoelectric effect, which does not belong to the group of photovoltaic phenomena. In order to examine the internal structure of the generalised bulk photovoltaic effect (Eq. (3.19)), we should derive an equation which will define the built-in field gradðV0 j0 Þ; and reflect the role of the position-dependent doping, positiondependent temperature and position-dependent chemical composition in the occurrence of this built-in field. To do this we proceed as follows. We can see from (3.16) that
kB T n0 n0 kB T kB T n0 ln þ grad ln : ð3:23Þ gradðV0 j0 Þ ¼ grad ¼ ln grad q q q ni ni ni The calculations were performed based on certain assumptions and using additional notations. These are: (a) The resultant concentration of the ionised impurities N does not depend on temperature and is only a function of the co-ordinates x; y; z; which can be expressed as þ N ¼ Nðx; y; zÞ ¼ ND NA ;
ð3:24Þ
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þ where ND and NA are the concentrations of the ionized donors and acceptors, respectively. (b) The energy band gap depends explicitly on the co-ordinates, because of the position-dependent composition, and on temperature, which is written as
EG ¼ EG ðT; x; y; zÞ
ð3:25Þ
and (c) similarly the product of the effective densities of states in the conduction and valence bands is a function of the co-ordinates x; y; z and temperature so that NC NV ¼ NC NV ðT; x; y; zÞ:
ð3:26Þ
This product satisfies the equation NC NV ¼ 3=2 lnðmc mv Þ þ 3 ln T þ const;
ð3:27Þ
where mc is the effective mass of an electron and mv is the effective mass of a hole. The term gradðV0 j0 Þ which represents the built-in field (see Eq. (3.19)) is derived in Appendix C (Eq. (C.15)) using Eqs. (3.23)–(3.27). This permits us to write the components of the generalised bulk photovoltaic effect in the form VC T EBPV ¼ EN BPV þ EBPV þ EBPV ;
ð3:28Þ
where EN BPV
2qm0 Dp kB T grad N s q p0 þ n0
results from the position-dependent doping, 2qm0 Dp p0 n0 3 kB T EG grad lnðm m Þ grad EVC c v xyz xyz BPV s p0 þ n0 4 q 2q
ð3:29Þ
ð3:30Þ
results from the position-dependent composition of the material, and
2qm0 n0 p0 n0 3 q 3 1 qEG EG T lnðmc mv Þ þ ln þ þ EBPV ¼ 2 2kB qT s ni p0 þ n0 4 qT 2kB T kB T ð3:31Þ Dp grad q is the thermo-photovoltaic component, which can be combined with the component ETPV (3.21) if we wish to estimate the contribution of the temperature gradient to the photovoltaic effects. 3.2. Numerical example In order to illustrate the theoretical results presented in this chapter, let us consider an inhomogeneous rectangular semiconductor plate equipped with two contact electrodes, as shown in Fig. 1. All the material parameters are assumed to
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depend on the co-ordinate x alone. The excess carriers, whose concentration is constant and equal to Dp; are generated as a result of uniform illumination in such a way that their concentration in the vicinity of the electrodes is equal to zero. To determine the electromotive force of the voltaic effect, we will use Eqs. (3.17)– (3.22) adapted to the one-dimensional case. The calculations are performed for the two cases: (A) when the inhomogeneity is due to the position dependence of the þ chemical composition and doping (grad EG a0 and gradðND NA Þa0), and (B) when it is due to the position-dependent doping density (grad EG ¼ 0 and þ gradðND NA Þa0). The temperatures taken in the calculations are 300 and 400 K. In group A the germanium–silicon samples are considered provided the chemical composition changes from 100% Ge to 100% Si. The position dependence of the doping density is also taken into account. Certain material parameters, such as the þ energy band gap EG ; doping density ND ; effective masses and mobilities, are assumed to be linear versus co-ordinate x. So the values of jgrad EG j ¼ 0:5 eV/cm þ and jgrad ND j ¼ 3 1023 m–4 are adapted to perform calculations. The parameters of group B concern pure Ge with constant energy gap EG ; where þ grad ND ¼ 3 1023 m–4, and average concentration NDC ¼ 2 1021 m–3 (the same as that in group A). The calculated variation of the total photovoltage Upv as a function of the concentration of excess carriers is shown in Fig. 3. The thick solid and dashed curves show the values of Upv for the parameters of group A. The thin solid and dashed curves represent the variation of the B-group parameters. The solid curves were taken at a temperature of 300 K and the dashed curves, at a temperature of 400 K. Curves a and b are obtained when grad EG and grad ND have the same signs, whereas curves c and d, when their signs are opposite. In examining case B; we can þ see that when grad EG =0, the sign of Upv changes together with the sign of grad ND (curves e,f and g,h). These results show that grad EG can give rise to considerable photovoltage.
3.3. Experimental investigation of thermophotovoltaic effect The results reported in our earlier papers [29,30] show that the phenomena predicted theoretically, in particular the thermophotovoltaic effect, need to be verified experimentally. A schematic diagram of the set-up used for the measurements is shown in Fig. 4. The samples, made of silicon, were illuminated with a light spot of a diameter of 1 mm. The temperature gradient within the samples was established using an appropriate heater and Peltier-cooler. The temperature distribution was measurement by three Pt micro-resistors, spaced at a distance of 1 cm from each other. The temperature control system maintains constant temperature TC in the light spot place irrespective of the temperature gradient. Fig. 5 shows the measured magnitudes of the thermophotoelectric voltage as a function of 9grad T9 along the direction of grad ND and in the direction opposite to it, at various initial values of the BPV voltage. The thermophotoelectric unit voltages
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Fig. 3. Total photovoltages calculated from Eqs. (3.17) to (3.22) as a function of Dp for a Ge Si alloy sample with ND ¼ 1 1021 m3, grad ND ¼ 2 1023 m4 and gradwEG ¼ 0:5 eV/cm (lines a, b, e, f). Lines c, d, g, h grad ND ¼ 2 1023 m4, grad EG ¼ 0:5 eV/cm. Lines e, f grad ND ¼ 2 1023 m4, g, h grad ND ¼ 2 1023 m4 and grad EG ¼ 0 (Ge samples). Temperatures: 300 K—solid lines, 400 K— dashed lines.
Fig. 4. Schematic diagram of the set-up for measuring the photovoltaic effects.
obtained during the experiment were of the order of 1 108 Vm K1. Two n-type semiconductor silicon samples with rE4:5 O cm were used. Fig. 6 shows the results of the theoretical calculations. We can easily see from this figure that the magnitude of the thermophotoelectric voltage greatly depends on the þ sample resistivity. These results clearly show how the concentration ND within the þ light spot region, the gradient ND ; and temperature TC in the centre of the light spot influence the photovoltage of the sample with different grad T:
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Fig. 5. Measured values of the thermophotoelectric voltage as a function of jgrad Tj. Directions of grad T identical and opposite to grad ND : Different initial values (grad T ¼ 0) of the voltages differ because of the bulk photovoltaic effect in two different n-type Si samples with rE4:5 O cm.
4. Photovoltaic electrical fields 4.1. Basic equations Section 2.2.1 discussed the simplest one-dimensional case. The results are given by Eqs. (2.24) and further, under the assumption that the excess carrier concentration Dp is constant or that it depends on the co-ordinate x alone. In Section 2.2.3, we formulated the continuity Eq. (2.54) for the total current density, and the continuity Eq. (2.58) fulfilled by Dp: These two equations, together with adequate boundary conditions, permit us to determine the distribution of the potential Cðx; y; zÞ and the concentration Dpðx; y; zÞ: We must mention here that when dealing with a threedimensional case, we face serious mathematical problems. When measuring the resistivity gradient, we use a light spot, which can be treated as nearly point-type illumination. This is a 3D problem, but, in this case, it is sufficient to solve solely the continuity equation for the total current J; which determines the distribution of grad C. We may simplify the problem in this way, provided that the length of the illuminated region is comparable with the diffusion length LD and that the sample dimensions greatly exceed LD : If these conditions are satisfied, we may assume that the distribution Dpðx; y; zÞ in a given small region does not affect the distribution outside this region. If this is so, the total number of the electron–hole pairs is the decisive factor. Analysing the general form of the continuity equation for the total current J and applying several approximations, one obtains a simple result, suitable for practical applications. The mathematical reasoning permits specifying the conditions under
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Fig. 6. Photothermoelectric voltages obtained for various values of grad T and grad ND using (3.17)– (3.22) in turn. (EG depend only on T). (a) NDC ¼ 1 1022 m3, grad NDC ¼ 2 1023 m4, (b) NDC ¼ 1 1022 m3, grad NDC ¼ 2 1023 m4, (c) NDC ¼ 1 1022 m3, grad NDC ¼ 4 1023 m4, (d) NDC ¼ 6 1021 m3, grad NDC ¼ 2 1023 m4, (e) NDC ¼ 1:2 1022 m3, grad NDC ¼ 2 1023 m4, (f) NDC ¼ 1 1022 m3, grad NDC ¼ 5 1023 m4, (g) NDC ¼ 1 1022 m3, grad NDC ¼ 0 (h) NDC ¼ 5 1022 m3, grad NDC ¼ 0: TC ¼ 300 K for a, b, c, d, e, g, h. TC ¼ 350 K for f. grad T range 200 to +200 K/m. Index C indicates the centre of the light spot.
which this simple result can be considered to be a sufficiently good approximation. The theory presented below gives the solutions that determine the distributions of the potential C and the voltage generated between the sample electrodes in many practical cases. The basic method of measuring the photovoltaic effects uses chopped light, thanks to which the thermoelectric component ETEV (occurring in Eq. (3.22)), which does not follow the modulation, is eliminated from the measurement and does not affect the measurement result. Now we can seek the continuity equation for the current J PV ; due only to the photovoltaic effects. From Eqs. (3.17) and (3.28) with the neglect of the component ETEV and introducing a new vector M; we obtain J PV ¼ s grad C þ sDpM;
ð4:1Þ
where C is that portion of the potential which corresponds to the current J PV ; follows the modulation and thus can be measured. The vector M has four components: M ¼ M N þ M E þ M T þ M M;
ð4:2Þ
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which can be expressed as 1 2m0 kB T grad N M N ¼ EN ¼ Dp BPV sðp0 þ n0 Þ 1 VC 2qm0 p0 n0 3 kB T EG ME ¼ E gradxyz ln mc mv gradxyz ¼ ; Dp BPV s p0 þ n 4 q 2q
ð4:3Þ ð4:4Þ
1 T 1 EBPV þ ETPV Dp Dp
2qm0 n0 p0 n0 3 q 3 1 qEG EG lnðmc mv Þ þ ln þ þ ¼ 2 2kB qT s ni p0 þ n0 4 qT 2kB T
q en þ ep EG sn0 sp0 n0 s0 s þ ln þ ðmn mp Þ ln s 2 2kB T s0 ni Ds s0
en ep 3 mc kB T ; ð4:5Þ þ ðmn þ mp Þ þ ln grad 2 mv q 2
MT ¼
1 EMPV Dp mn mp 1 s0 s q m ¼ 2 ln ðmn mp Þðn0 þ p0 Þ kB T grad ln n ; s0 Ds s0 s mn þ mp mp
MM ¼
ð4:6Þ
where the vectors E are taken from Eqs. (3.29)–(3.31), (3.21) and (3.20), respectively. The vector M N is associated with the classical bulk photovoltaic effect, M E with the position dependence of the band gap EG and of the effective masses (when the variable composition case is in question), M T determines in total the thermophotoelectric effect and M M is proportional to the variation, due chiefly to the positiondependent composition, of the mobilities ratio. What we examine is the sum M of these vectors and our aim is to get information on the properties of the material, especially when the position-dependent doping is accompanied by a positiondependent composition. The continuity equation derived directly from Eq. (4.1) is div J PV ¼ sDC s grad C þ div sDpM ¼ 0;
ð4:7Þ
which, after rearranging, gives DC þ grad ln s grad C ¼
divðsDpMÞ ; s
ð4:8Þ
where D is the Laplace operator. 4.2. General solution for an open-circuit sample We will examine a sample of arbitrary shape (Fig. 7), equipped with two contact electrodes (of an arbitrary surface area or point-type), illuminated in such a way that in the vicinity of the electrodes the concentration Dp is zero (so as to avoid contact effects). We consider only the open-circuit arrangement.
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Let us denote S1 ; S2 are the surface area of the electrodes and S3 is thesurface area of the sample beyond the electrodes. In our case, the boundary conditions are: 1. On the surface S1 C ¼ 0:
ð4:9Þ
2. On the surface S3 C ¼ const: 3. The total current is equal to zero that is Z Jn dS2 ¼ 0:
ð4:10Þ
ð4:11Þ
S2
4. The normal component of grad C on surface S3 fulfils the equation ðgrad CÞn ¼ DpM;
ð4:12Þ
where n is the unit vector perpendicular to this surface. It follows from the theory of the elliptical partial differential equations that Eq. (4.8) with the boundary conditions (4.9)–(4.12) can be solved using the Green function that satisfies the equation DA GðA; BÞ þ grad ln sðAÞgradA ðA; BÞ ¼ dðA BÞ;
ð4:13Þ
where A and B are points of the bulk of the sample and dðA BÞ—is the threedimensional Dirac delta function. Function G must in addition satisfy the boundary conditions that follow from Eqs. (4.9)–(4.11) and the equation ½gradA GðABÞ n ¼ 0;
ð4:14Þ
which replaces the boundary condition (4.12).
Fig. 7. Inhomogeneous semiconductor sample of arbitrary shape. S1 ; S2 electrodes. S3 is the remaining area. The light beam generates the excess electron–hole pairs Dp: TB—bulk of the sample.
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That there exist the solutions of Eqs. (4.13) and (4.14), we can see by constructing an electric model of the function GðA; BÞ; shown in Fig. 8. In this model, the current I ¼ sðBÞ is supplied to the sample at point B. The potential distribution established in the sample is a function of points A and B and is equal to the Green function. The solution of Eq. (4.8) with the boundary conditions (4.9)–(4.12) is the function
C ðx; y; zÞ ¼ C ðAÞ ¼
Z GðA; BÞ TB
þ
Z
divðsDpMÞ dtB s
GðA; BÞDpMn dSB ;
ð4:15Þ
S
where TB is the bulk of sample, dtB is the volume element, S is the total surface area of the sample and dSB is a surface element. Eq. (4.15) may be transformed to the form
C ðAÞ ¼
Z TB
gradB GðA; BÞMðBÞDp dtB
Z GðA; BÞMðBÞgrad ln s dtB ; TB
ð4:16Þ which is easier to interpret and to transform further into an approximate equation. Eq. (4.16) indicates that the potential distribution C ðAÞ results from the dipole sources whose distribution is MðBÞDpðBÞ and single sources with the distribution MðBÞgrad ln sðBÞDpðBÞ: When the illumination is of the point type, we can seek an approximate equation, since Dpa0 only in a small region of the light spot. An analysis of the Green function GðA; BÞ using the Legendre polynomials shows that a first approximation of Eq. (4.16) is C ðAÞEgradB GðA; B0 ÞMðB0 ÞP GðA; B0 ÞMðB0 Þgrad ln sðB0 ÞP;
Fig. 8. Electrical model of Green function GðA; BÞ.
ð4:17Þ
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where B0 is the centre of light spot and P is the total number of excess carriers given by Z P¼ DpðBÞ dtB : ð4:18Þ TB
Both the terms MðBÞ and grad ln s appearing in Eq. (4.17) are proportional to the degree of inhomogeneity. If the inhomogeneity is not significant, MðBÞgrad ln s can be neglected as small compared to MðB0 Þ: Even with the proposed simplifications, we still face the problem of determining the function GðA; BÞ; since it must satisfy Eq. (4.13), containing function grad ln sðBÞ: When we assume that this function is known, we obtain the equation for G0 ðA; BÞ in the form of the infinite series [32] GðA; BÞ ¼ G0 ðA; BÞ þ G1 ðA; BÞ þ G2 ðA; BÞ þ ?;
ð4:19Þ
in which the function G0(A,B) satisfies both the boundary conditions and the equation DA G0 ðA; BÞ ¼ dðA BÞ:
ð4:20Þ
The function G1 and the subsequent functions are found in a recurrent manner to be Z G0 ðA; CÞgrad ln s grad Gn ðC; BÞ dtC : ð4:21Þ Gnþ1 ðA þ BÞ ¼ TB
The condition for the convergence of this series is Rjgrad ln sjmax o0:1;
ð4:22Þ
where R is the maximum diameter of the semiconductor sample. The latter inequality roughly defines the range of the series convergence, even though we only take into account the first term G0 ðA; BÞ of the series. This reasoning permits us to write Eq. (4.17) in the approximate form C ðAÞ ¼ Cðx; y; zÞEgradB G0 ðA; Bsr ÞMðB0 ÞP;
ð4:23Þ
which describes the potential C and its gradient as a dipole field with the moment MP: Based on the derivation of the approximate Eq. (4.23) and on the complete Eq. (4.17) we can estimate the errors incurred during the measurements and to assess the suitability of the experimental methods employed. 4.3. Fundamentals of the measurement of the bulk photovoltaic effect (BPV) and the relative gradient of resistivity In this section we confine our attention to the bulk photovoltaic effect alone, since in practical situations, it forms the basis for determining the relative distribution of the resistivity gradient in semiconductor samples. Fig. 7 shows schematically the measurement of the photovoltage between the electrodes of an illuminated semiconductor sample with a resistivity gradient established in it. According to Eq. (4.23), the voltage UBPV is equal to UBPV ðB0 Þ ¼ gradB dGðB0 ÞM N ðB0 ÞP;
ð4:24Þ
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where dGðB0 Þ G0 ðC; B0 Þ G0 ðD; B0 Þ
ð4:25Þ
and the dipole moment M N is defined by (4.3). In order to find a simple relationship between this moment and the resistivity gradient r, we consider n-type semiconductor where n0 bp0 ; NEn0 ; m0 ¼ mp : Substituting these into Eq. (4.3), we obtain MN ¼
2mp kB T grad ln n0 : s
ð4:26Þ
Then multiplying n0 under the logarithm by the coefficient qmn ; (in the case of an ntype semiconductor this product is equal to s) and using the equality sr0 Es0 r0 ¼ 1; we can write 2mp kB T 2mp kB T grad ln s0 ¼ grad ln r0 s s 2mp kB T grad s0 ¼ E2mp kB T grad r0 ; s r0
MN ¼
ð4:27Þ
Substituting (4.27) into Eq. (4.22) gives UBPV ðB0 Þ ¼ 2mp kB TP grad dGðB0 Þgrad r0 ðB0 Þ ¼ 2mp kB TPjgrad dGðB0 Þj grad r ðB0 Þcos a; 0
ð4:28Þ
where a is the angle between the vectors grad dG and grad r0 : Knowing the geometry of the sample and its electrodes determines grad dG; therefore the projection of the vector grad r0 in the direction of the known vector grad dG may be expressed as UBPV ðB0 Þ grad r cos a ¼ : ð4:29Þ 0 B0 2mp kB T jgrad dGðB0 ÞjP By illuminating the sample at various points and measuring UBPV there, we can find the relative distribution of 9grad r0 9cos a; provided that the sum of the pairs P of the excess carriers is the same at all points B0 ; which is true when the carrier lifetime t is the same at all these points. When the angle a ¼ 0; and we are able to confirm this, and the measurement is performed along a straight line taken to be the axis x; then we can find the relative distribution dr=dx: The absolute distribution rðxÞ can be determined by measuring r at two points of the x axis, e.g. by means of the 4-point probe. In sufficiently long samples grad G0 may be assumed to be constant and formulae (4.24) and (4.26) define the relative distribution of grad r0 ; which can be determined by the measurements of UBPV : An early example of application of the bulk photovoltaic effect is the work of Oroshnik and Many [31]. The method used there was photovoltaic scanning to evaluate the homogeneity of germanium single crystals. In course of time the ! interesting results are obtained by Swiderski. In the conference paper publication [32] the micro-inhomogeneities were observed in germanium which enabled to investigate the details of discontinuities taking place in the process of single crystals production.
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4.4. Measurement of the resistivity distribution A very convenient method for determining the resistivity distribution rðxÞ in rectangular samples consists of the following three steps: (1) Measurement of UBPV along the x-axis R x of the sample. (2) Calculation of the integral IðxÞ ¼ x0 UBPV ðx0 Þ dx0 (x0 is arbitrarily chosen) which is the linear function A IðxÞ þ B of IðxÞ; where the two constants A and B are determined by measuring, with a 4-point probe, the resistivity at the two points x1 ; x2 : It is desirable to choose x1 ; x2 where UBPV has an extremum. (3) Having determined these quantities, we can express the distribution rðxÞ in the form rðxÞ ¼ fr1 ½IðxÞ Iðx2 Þ r2 ½IðxÞ Iðx1 Þ gðI1 I2 Þ1 :
ð4:30Þ
The method just described has been tested practically at our Institute. Fig. 9 shows the results of our experiments. The diagram at the top of the figure is UBPV plotted as a function of x; and at the bottom is the distribution rBPV ðxÞ; where the solid line represents the distribution calculated from the integral IðxÞ and formula (4.30), and the squared line the distribution measured with the 4-point probe. We can see that the two resistivity distributions, measured and calculated, are in good agreement, except in a certain region near x ¼ 35 mm. This difference may result from the poor geometrical resolution of the 4-point probe compared to that achieved in the light spot measurement, and from the contribution of the concentration gradient directed
Fig. 9. Distribution of the photovoltage measured in a rectangular inhomogeneous Ge sample (solid line in top figure). The distribution of resistivity obtained from the BPV effect (thick solid line); resistivity distribution obtained from 4-point probe measurements (squares).
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perpendicularly to the sample juncture line of the contacts; the value of UBPV only depends on the grad ND component that is parallel to the juncture line of the contacts. The relative distribution of the resistivity gradient can only be determined when the carrier lifetime in the semiconductor can be assumed to be independent of position. To obviate this restriction, Blackburn et al. [33] proposed that the bulk photovoltage and the photoconductivity should be measured at the same illuminated point, which permits determining the absolute resistivity gradient dr=dx with the position dependence of the carrier lifetime taken into account. These authors carried out the measurements on circular semiconductor slices using a small-diameter light spot. The formula determining the correction due to the circular geometry of the sample was derived using the conformal transformation. An improved method of Blackburn et al. [33] is presented by Larrabee and Blackburn [34]. The method based on the Green function presented in Section 4.1 makes it possible to develop a more general theory which may also be applied to wafers with a geometry different from circular even in the 3D case. To enable the photoconductivity measurement, a constant current Ics is forced through the sample using a current source, and the sample is illuminated by modulated light. During the period without illumination the current density in the sample is J 0 ¼ s0 grad C0 :
ð4:31Þ
During the period where the sample is illuminated the current density is described by (2.60), which, in the case of an n-type semiconductor (m0 Emp ), takes the form J ¼ s grad C 2qmp Dp grad V0 :
ð4:32Þ
Subtracting (4.30) from (4.31) we obtain J J 0 ¼ s grad dC Ds grad C0 2qmp Dp grad V0
! 2mp ¼ s grad dC qðmn þ mp Þ grad C0 þ grad V0 Dp; m n þ mp
ð4:33Þ
where dC ¼ C C0
ð4:34Þ
while the increase in the conductivity was assumed to be Ds ¼ qðmn þ mp ÞDp. The form of (4.33) makes it possible to notice that the components determining the final solution are grad C0 and 2ðm0 =mn þ mp Þgrad V0 : If (4.31) is to be simplified, the following condition must be met jgrad C0 jbjgrad V0 j
ð4:35Þ
because 2m0 =ðmn þ mp Þ is of the order of 1. The above condition may be fulfilled in practice because V0 is a given quantity, while grad C0 is proportional to the current Ics flowing through the sample and thus may be adjusted. If it is assumed that (4.33) is applicable, then (4.31) takes the form J J 0 E s grad dC qðmn þ mp ÞDp grad C0 :
ð4:36Þ
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The difference J J 0 is solenoidal, therefore the divergence of the right-hand side of the above equation is equal to zero. Thus divðs grad dCÞ ¼ div½qðmn þ mp ÞDp grad C0 :
ð4:37Þ
Transforming the left-hand side and dividing the equation by s one obtains
qðmn þ mp Þgrad C0 1 div grad dC þ grad ln s grad dC ¼ div sDp : s s ð4:38Þ This equation has the same form as (4.8), where M should be replaced by q=sðmn þ mp Þgrad C0 : According to (4.21) the solution of such an equation is q ð4:39Þ dCðAÞ ¼ dCðx; y; zÞ ¼ grad G0 ðA; B0 Þ ðmn þ mp Þgrad C0 ðB0 ÞP: s It should be noted that G0 is the proper function corresponding to an open sample because the supply current Ics is the same regardless of whether the sample is illuminated or not. As a consequence, G0 fulfils the boundary condition at the sample electrodes. The increase of the potential dC induced by the illumination causes a negative increase of the voltage between the sample electrodes. Thus, the resistance appearing in Eqs. (38) and (39) is changed by DR: If the solution of (4.37) is known, the value dUR of the measured voltage may be found in a manner analogous to (4.28): q dUR ðB0 Þ ¼ ðmn þ mp ÞP grad dGðB0 Þgrad C0 s q ¼ ðmn þ mp ÞPjgrad dGðB0 Þjjgrad C0 jcos b; ð4:40Þ s where b is the angle between vectors grad dG and grad C0 : Similar to (4.28) the term grad dG appears also in the above formula. Another important fact is that P is the total number of excess carrier pairs, is the same in Eqs. (4.40) and (4.28) because it corresponds to the same point B0 : This means that the factor P will be eliminated if (4.40) is divided by (4.28) and we have UBPV 2kB Ts mp grad r0 cos a : ð4:41Þ ¼ q mn þ mp jgrad C0 j cos b dUR The above leads to m þ mp UBPV grad r cos a ¼ q r n ; jgrad C0 jcos b 0 kB T 0 2mp dUR
ð4:42Þ
where s has been replaced by 1=r0 ; which in practice is a well-founded assumption. If 7grad C0 7cos b is known, Eq. (4.42) makes it possible to determine the projection of the gradient of r0 onto the direction of vector grad dG0 : 4.5. The Green function and the potential distribution in thin samples If we assume that the properties of the material and the concentration Dp at a given point do not vary along the co-ordinate z perpendicular to the sample surface,
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the mathematical problem becomes two-dimensional. The Green function G2D fulfils the equation analogous to (4.19) DG2D ðA; BÞ ¼ d2D ðA BÞ; where D¼
q2 q2 þ ; qx2 qy2
ð4:43Þ
d2D is the 2D Dirac delta funciton:;
Points A and B have the co-ordinates x and y. As in Section 4.3, we only consider the BPV effect in an n-type semiconductor. The dipole moment at point B0 is defined by Eq. (4.27), which, as we remember, has the form (with index N omitted) M ¼ 2mp kB T grad r0 :
ð4:44Þ
At a given dipole moment the distribution of the potential C is given by Eq. (4.21). The 2D-form of this equation is P C2D ðAÞ ¼ Cðx; yÞ ¼ gradB G2D ðA; B0 ÞM : ð4:45Þ d This equation may be used for determining the distribution of the potential and then calculating the voltage established between the sample electrodes. Having determined these, we can find the distribution of the resistivity gradient within the sample. Points A and B0 are defined by the two co-ordinates x and y and, thus, their positions may be expressed as the complex number z ¼ x þ iy: This permits us to introduce a function of the complex variable that describes grad G2D : Then the field generated by a single dipole in space is described by the function 1 # ð4:46Þ GðzÞ ¼ 2pz the real part of which fulfils Eq. (4.43). If the vector MP appearing in Eq. (4.45) is described in terms of the vector # ¼ Dx þ iDy ¼ M x þ iM y P; D ð4:47Þ then the potential due to this dipole will be # D : ð4:48Þ Cðx; yÞ ¼ Re 2pdz # given by (4.47) into Eq. (4.48) and taking z ¼ x þ iy; we obtain Now, substituting D Cðx; yÞ ¼ Re
ðDx þ iDy Þz ðDx þ iDy Þðx iyÞ Dx x þ Dy y ¼ Re ¼ ; 2pdðzz Þ 2pdjzj2 2pdjzj2
ð4:49Þ
which is the known equation that describes the potential due to a dipole with the components Dx ; Dy ; positioned at the origin of the co-ordinate system. In the case when the dipole is positioned at z0 ¼ x0 þiy0 ; Eqs. (4.45) and (4.46) may be easily transformed to take the form Dx þ iDy Dx ðx x0 Þ þ Dy ðy y0 Þ ¼ Cðx; yÞ ¼ Re : ð4:50Þ 2pdðz z0 Þ 2pdjz z0 j2
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In the next chapter, we derive equations describing the distribution of the BPV potential in thin slices of selected shapes. The results will be given in the form of the equipotential line plots.
5. The BPV—potential distributions 5.1. The method of images The basic equations and relationships describing the distribution of the field induced by BPV under conditions of point illumination have been given in the previous chapter. The equation of fundamental significance is Eq. (4.23). Its righthand side developed into the components is C ¼ Dx
qG0 qG0 qG0 þ Dy þ Dz ; qxB qyB qzB
ð5:1Þ
where we have introduced the notations that relate the components of vector D with those of vector M Dx ¼ Mx P;
Dy ¼ My P;
Dz ¼ Mz P:
ð5:2Þ
The potential C is a function of A, i.e., of the co-ordinates x; y; z: The components of the dipole D are defined at point B0 in the centre of light spot. The co-ordinates of this point are x0 ; y0 ; z0 : Formula (5.1) is valid for any shape of the sample. To determine the potential C in the definite case it is necessary to know the Green function G0 ; which is determined by the shape of the sample under consideration. The fundamental and simplest case of the Green function is 14prAB0 ; where rAB0 is the distance between points A and B0 : This function fulfils Eq. (4.20), which can be verified by using an appropriate substitution, and satisfies the boundary conditions in the infinity, thereby vanishing when r-N: So the function 14prAB0 is valid for the boundless sample. Taking the above into account and in accord with Eq. (4.23), we can write the potential induced by a single dipole MP positioned at point B0 in the infinite three-dimensional space C ðAÞ ¼
1 ½Dx ðx x0 Þ þ Dy ðy y0 Þ þ Dz ðz z0 Þ : 4pr3B0
ð5:3Þ
The potential C given by this equation vanishes when rAB0 -N thereby satisfies the boundary condition in the infinity. Unlike the case concerning the infinite space, the potential of the dipole situated in the half-space must fulfil following boundary conditions on the boundary plane. (1) In a semiconductor surface coated with a metal layer, whose conductivity is several times as high as that of the semiconductor, the boundary conditions is C¼ const: The same problem is encountered in electrostatics, and, there, we create the field of a point charge and of its image which is its reflection in the boundary plane. The charge and its image have opposite signs, as shown in Fig. 10a. We treat dipoles as two-charge objects, so we can say that the components perpendicular to
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Fig. 10. Charge q with its image, and dipole D with its image D0 in the case of (a) Metal–semiconductor interface and (b) semiconductor–insulator interface.
Fig. 11. Dipole at the surface of half-space.
the boundary plane of both the dipoles have the same directions, whereas the parallel components have opposite directions (see Fig. 10a). (2) When the semiconductor surface is not covered with metal we deal with a semiconductor/insulator boundary and the boundary condition is that the perpendicular to the boundary plane component of the field should be equal to zero. As a consequence, the directions of the dipole components are opposite to those that these components had in the previous case. Fig. 10b shows that the perpendicular components have opposite directions, whereas the parallel components have the same directions. Further in this section, we discuss the potential distributions determined in samples of various shapes by the method of images. 5.2. Half-space and space between two parallel half-planes Fig. 11 shows a co-ordinate system with a dipole positioned at its origin on the surface of a half-space; the dipole has the components Dx and Dy (Dz ¼ 0). In this situation, the image of the dipole is identical to the dipole itself. Using Eqs. (5.2) and making appropriate substitutions, we obtain C¼
1 ðDx x þ Dy yÞ: 2pr3AB0
ð5:4Þ
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Fig. 12. Dipole at the surface of the region between two parallel half-planes.
Fig. 13. Sequence of images for the case shown in Fig. 33.
Fig. 14. Components of the dipole sequence of Fig. 13. (a) Dipoles parallel to both planes and (b) dipoles perpendicular to the planes.
The region between two parallel half-planes and the position of the dipole are shown in Fig. 12. Because of the presence of the two mutually parallel planes, we must form images given by these planes, then images of these images, and so on. As a result we obtain the infinite sequence of dipoles ‘‘hinted at’’ Fig. 13. This sequence can be formed of the two other simple sets of dipoles shown in Figs. 14a and b. The Green function G0 of the sequence of single dipole sources placed at points B2 ; B1 ; B0 ; B1 and B2 has the form G0 ¼
N 1 X 1 : 4p n¼N rA;Bn
ð5:5Þ
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The function, which describes the components of the dipole sources (Fig. 14a), is the derivative of G0 with respect to n, which will be denoted as qG0 ¼ jð3Þ x ðx; y; zÞ: qx
ð5:6Þ
The dipole sources in the direction y (Fig. 14b) are described by the derivative of G0 with respect to y; which is qG0 ¼ jð3Þ y ðx; y; zÞ: qy
ð5:7Þ
To determine the function G0 which describes the potential of the single sources we should only calculate the potential induced by a sequence of point charges distributed periodically on a straight line. This was done by Madelung [35] who gave a formula in the form of a rapidly converging series, easy to differentiate. The ð3Þ differentiation gives the equations, which define the functions jð3Þ x and jy as
N 2x X 2pnr 2pny x þ ðx; y; zÞ ¼ nK ð5:8Þ cos jð3Þ 1 x rd 2 n¼1 d d 2pdr2 as well as jð3Þ y ðx; y; zÞ
N 2 X 2pnr 2pny ; ¼ 2 nK0 sin d n¼1 d d
ð5:9Þ
where K0 i K1 are the modified Hankel functions of imaginary argument and r is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5:10Þ r ¼ ðx2 þ z2 Þ: The assumption that the dipole has only two components Dx and Dy means that we neglect the inhomogeneity in the direction z: An analysis of the relationship between the two sets of dipoles shown in Figs. 13 and 14 and yields the equation for the distribution of the potential C within the region considered: ð3Þ Cð3Þ ðx; y; zÞ ¼ 2Dx ½jð3Þ x ðx; y; zÞ þ jx ðx; y 2a; zÞ ð3Þ þ 2Dy ½jð3Þ y ðx; y; zÞ þ jy ðx; y 2a; zÞ :
ð5:11Þ
The coefficients 2 appearing in both the terms of this equation indicates that the dipole images were taken to be identical with the dipoles themselves, just as in the half-space case. Fig. 15 shows the distribution of the potential C3 ðx; y; 0Þ induced by a dipole placed on the surface of the sample in the middle of its width; the moment of this dipole is given by the relationship Dx =d 2 ¼ 1 V, Dy ¼ 0: We can see that, at a certain distance from the dipole, the potential depends on x alone, the equipotential lines are straight lines and the potential may be expressed as 2Dx x Cð3Þ D : ð5:12Þ pdr2
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Fig. 15. Potential distribution in the x y plane of the dipole directed along the axis of the sample and placed symmetrically in the region shown in Fig. 12.
Fig. 16. Dipole localised at the surface of an infinitely long sample with finite width and thickness.
The photovoltage measured does not depend on the position of the measuring electrodes, provided that they are far enough from the light spot. 5.3. Infinitely long plate Fig. 16 shows an infinitely long sample and a dipole D that has the two components Dx and Dy (Dz ¼ 0). The position of the dipole on the sample surface (z ¼ 0) is defined by the distance a; b=2 and c=2 are the width and the thickness of the sample, respectively. The solutions can be obtained based on the distributions of the potentials induced by the dipoles parallel to the x-axis and parallel to the y-axis. The dipoles are placed on the plane z ¼ 0 at the nodes of the infinite orthogonal flat lattice shown in Fig. 17. This primary lattice should be accompanied by an image of secondary lattice (not shown in Fig. 17) shifted by 2a in parallel to the y-axis. The coordinates of the nodes of the primary lattice are (0; bl; cn), where l and n are integers within the interval (N, +N). Using these quantities we can determine the distance
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Fig. 17. Dipole lattices for the case of Fig. 16. (a) Dipoles parallel to the sample edge and (b) dipoles perpendicular to the sample edge.
ren between the points on the sample surface and the nodes of the primary lattice. This distance can be calculated from ren ¼ ½x2 þ ðy blÞ2 þ ðcnÞ2 1=2 :
ð5:13Þ
By substituting y bl 2a in the place of y bl in this equation we obtain r0l;n concerning the secondary lattice r0l;n ¼ ½x2 þ ðy bl 2aÞ2 þ ðcnÞ2 1=2 : In the reasoning which follows, we will use the function N X ðx; y; 0Þ ¼ ½x2 þ ðy bÞ2 þ ðcnÞ2 3=2 : jð4Þ x
ð5:14Þ
ð5:15Þ
l;n¼N
According to the method of images, the potential on the sample surface can be found using Eq. (5.3), which defines the dipole field in a general way. This potential is given by Dx x ð4Þ Cð4Þ ðx; y; 0Þ ¼ ½j ðx; y; 0Þ þ jð4Þ x ðx; y 2a; 0Þ 4p x Dy y ð4Þ ½j ðx; y; 0Þ þ jð4Þ þ y ðx; y 2a; 0Þ 4p y Dx x þ Dy y ð4Þ Dx x Dy y ð4Þ j ðx; y; 0Þ þ j ðx; y 2a; 0Þ: ð5:16Þ ¼ 4p 4p As in the previous case (5.12), the potential Cð4Þ for large values of x depends neither on y nor on z and may be expressed as 2Dx : ð5:17Þ Cð4Þ ðN; y; zÞ ¼ bc The potential distribution at xo0 is anti-symmetrical to the distribution at x > 0 and, thus, Cð4Þ ðN;y;zÞ ¼ 2Dx =bc: Hence the voltage between the sample far ends is U¼
4Dx Dx ¼ ¼ g; bc s
ð5:18Þ
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Fig. 18. Potential distribution of the dipole in an infinitely long sample with the square cross-section. Dipole parallel to the edge placed symmetrically.
where s is the cross-sectional area of the sample. This voltage does not depend practically on the dipole position, but it only depends on the dipole moment per unit cross-sectional area g: This observation is of importance in the photovoltaic diagnostics of the material. An analysis of Eqs. (5.13) and (5.14) indicates that when measuring the potential at a point with the co-ordinate x; and estimating the moment Dx ; we make the relative error equal to ! xd px exp pffiffi ð5:19Þ s s from which we can evaluate how long the sample should be in order that Eq. (5.17) might be used. The distribution of the potential Cð4Þ ðx; y; 0Þ is shown in Fig. 18. A dipole with the components Dx =s ¼ 1 V, Dy ¼ 0 is placed on the sample surface at its centre. The sample cross-section is square (b ¼ c). 5.4. Cuboid with electrodes at two opposite faces The distribution of the potential in a cuboidal sample equipped with two contact electrodes at its two end faces, shown in Fig. 19, is determined using the components of Eq. (5.15) that define the potentials of the dipole iDx and jDy in an infinitely long sample. These two potentials may be written as ð4Þ Fx ðx; y; zÞ ¼ 2Dx ½jð4Þ x ðx; y; zÞ þ jx ðx; y 2a; zÞ ;
ð5:20Þ
ð4Þ Fy ðx; y; zÞ ¼ 2Dy ½jð4Þ y ðx; y; zÞ þ jy ðx; y 2a; zÞ :
ð5:21Þ
The potential may be written as ð5Þ Cð5Þ ¼ Cð5Þ x þ Cy
ð5:22Þ
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Fig. 19. Cuboidal sample with electrodes covering the whole width of the sample.
where the first component related with the dipole iDx is N X Cð5Þ ðx; y; zÞ ¼ Fx ðx 2nf d; y; zÞ þ Fx ðx 2nf þ d; y; zÞ x
ð5:23Þ
n¼N
and the second component related with the dipole jDy is N X Cð5Þ ðx; y; zÞ ¼ Fy ðx 2nf d; y; zÞ þ Fy ðx 2nf þ d; y; zÞ: y
ð5:24Þ
n¼N
Eqs. (5.23) and (5.24) together with Eqs. (5.13) and (5.14) permit calculating the ð5Þ potential at any arbitrary point of the sample. In the case of Cð5Þ x and Cy ; the result is a series with a double index. In practical applications, it is interesting to know the value of Cð5Þ at two points in the vicinity of the electrodes, namely Cð5Þ x ð0; y; zÞ and ð5Þ ð5Þ Cð5Þ x ðf ; y; zÞ; and also Cy ð0; y; zÞ and Cy ðf ; y; zÞ: In view of the symmetry of the function, i.e., Fx ðx; y; zÞ ¼ Fx ðx; y; zÞ; the algebraic calculation of these quantities with the use of Eq. (5.23) yields the very simple result Cð5Þ x ð0; y; zÞ ¼ 0:
ð5:25Þ jð4Þ x
In the other case, we should also utilise the value of calculated in the previous section. The result is 4Dx : ð5:26Þ Cð5Þ x ðf ; y; zÞ ¼ bc By performing similar calculations based on the equation ð4Þ jð4Þ y ðx; y; zÞ ¼ jy ðx; y; zÞ
ð5:27Þ
we obtain Cð5Þ y ð0; y; zÞ ¼ 0;
Cð5Þ y ðf ; y; zÞ ¼ 0:
ð5:28Þ
This result means that, under open circuit conditions, the component Dy does not contribute to the generation of the voltage at the electrodes of the sample and, thus, the voltage is given by 4Dx Dx ð5Þ ¼ ¼ g: ð5:29Þ U ð5Þ ¼ Cð5Þ x ð0; y; zÞ Cx ðf ; y; zÞ ¼ bc s
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This is the same result as that obtained in the previous section (5.17), it however goes further. In a long sample not equipped with appropriate electrodes, Dx cannot be determined when the light spot is incident near a small-size electrode. When dealing with a cuboidal sample where the electrodes cover the whole surfaces of its end walls, the voltage is always Dx =s; irrespective of the position of the point considered, which is of great practical significance and cannot be predicted intuitively. We should of course remember the condition, which is still valid, that the distance between the illuminated points and the electrodes must exceed the length of the diffusion length. 5.5. Slices 5.5.1. Half-plane and infinite strip #z ¼ Let us consider a half-plane x > 0 that borders on an insulator. The dipole D Dx þ iDy (Fig. 20) is positioned at the point z0 ¼ x0 þ iy0 : Its image in the insulator # z ¼ Dx þ iDy situated at z ¼ x0þ iy0 : According to Eq. (4.50), the is the dipole D 0 resultant potential is
Dx þ iDy Dx þ iDy 1 Re C1 ðx; yÞ ¼ þ : ð5:30Þ 2pd z z0 z þ z0 In order to find the potential distribution within the infinite strip 0oxoh where h is the strip width, we have to consider the infinite sequence of the pairs D# z ; D# z shown in Fig. 21. The spacing between the recurring pairs of dipoles is 2h; and z0 is assumed to be equal to x0 : The potential is expressed as an infinite sum of the fields given by Eq. (5.30), shifted by a multiple of 2h: This is written as "
# N X Dx þ iDy Dx þ iDy 1 Re lim Cðx; yÞ ¼ þ : ð5:31Þ N¼N 2pd z x0 2kh z þ x0 þ 2kh k¼N
Fig. 20. Dipole Dz in a half-plane x > 0 and its image D0z .
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Fig. 21. The infinite sequence of dipole pairs Dz ; D0z for a strip with finite width h:
In this equation, the infinite sum is composed of two sums, which may be replaced by cotangents of adequate arguments. When this is done, we have lim
N¼N
N X
1 p p ¼ ctg ðz7x0 Þ: z7x 2h 2h 72kh 0 k¼N
ð5:32Þ
Substituting these functions into Eq. (4.49) we obtain h 1 p Re ðDx þ iDy Þctg ðz x0 Þ Cstr ðx; yÞ ¼ 4dh 2h i p þðDx þ iDy Þctg ðz þ x0 Þ : ð5:33Þ 2h The equipotential lines of the field induced by the dipoles Dx ; iDy and Dx þ iDy ; determined from this equation are shown in Figs. 22a–c. Fig. 22d shows the surface plot of the potential of the dipole Dx : Dx =dh and Dy =dh are equal to 1 V. We can see from the potential distribution shown in Fig. 22a that, at a sufficiently long distance from the dipole, the potential is constant and is equal to 0.5 V at the upper end and 0.5 V at the lower end. Fig. 22b shows that, in distant regions, the perpendicular dipole generates zero potentials, so that no potential difference is established at the ends. These observations are confirmed quantitatively by Eq. (5.33) when calculating the voltage established at the ends of a sufficiently long sample. The potentials of the points distant from the dipole are 1 Re½ðDx þ iDy Þ þ ðDx þ iDy Þ ctgð7NÞ Cstr ðx; 7NÞ ¼ 4dh
72Dy Dy 1 1 Re 72iDy ¼7 : ð5:34Þ ¼ ¼ 4dh i 4dh 2s Hence the voltage at the ends of a long sample, with the thickness d; width h and cross-sectional area s ¼ d h is Dy Cstr ðx; NÞ Cstr ðx; NÞ ¼¼ ¼ g: ð5:35Þ s This result is identical to that obtained for the sample with a finite thickness. The respective equations given in Ref. [35] permit evaluating the relative error incurred in the measurement of the resistivity gradient in rectangular plates as not exceeding 3 expð2pDx=hÞ; provided that the distance Dx between the light spot and the edge
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Fig. 22. Potential distribution in an infinitely long thin sample with finite width. (a) Dipole Dz parallel to the sample length, (b) dipole perpendicular to the sample length, (c) dipole deviated at 45 , (d) surface plot of potential distribution for the deviated dipole.
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Fig. 23. Circular slice with measuring arrangement and the location of the dipole.
where the measuring electrode is installed is below 0.64h. This also applies to the point-type electrode. 5.5.2. Circular slice Fig. 23 shows a circular slice of radius R; a plane of complex numbers z ¼ x þ iy # z ¼ Dx þ iDy : The circle jzjpR is surrounded by an insulator. We and the dipole D # w that satisfies the boundary condition seek the potential distribution of the dipole D jzj ¼ R at the circle periphery. To do this we utilise the solution (with adequately changed notation of the co-ordinates), obtained for a half-plane given by Eq. (5.30) and perform a transformation according to Rþz ; ð5:36Þ w¼ Rz where the point w0 corresponds to point z0 ; w0 ¼
R þ z0 ; R z0
w0 ¼
R þ z0 : R z0
# w is related to dipole D # z in the following: The dipole D
#w ¼D # z dw # z 2R : D ¼D dz z¼x0 ðR x0 Þ2 Substituting (5.36) and (5.37) into Eq. (5.30) we obtain 1 R Cðx; yÞ ¼ pd ðR z0 Þ2 Dx þ iDy
Re ðR þ zÞ=ðR zÞ ðR þ z0 Þ=ðR z0 Þ
Dx þ iDy þ : ðR þ zÞ=ðR zÞ þ ðR þ z0 Þ=ðR z0 Þ
ð5:37Þ
ð5:38Þ
ð5:39Þ
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By adding, in an appropriate proportion, the potentials due to the perpendicular dipole Dy and to the parallel dipole Dx ; we can determine the distribution of the potential C for any dipole positioned within the slice. The potential and force lines can be found from the real and imaginary parts of equation, respectively, (5.39). Fig. 24 shows these distributions for the dipoles with the intensity Rd 1V, directed along (a) u-axis, (b) v-axis and (c) deviated at an angle of 45 . The dipoles Du and Dv are located on the u-axis. The position of the Du þ iDv dipole is arbitrary. Let us now consider the experimental results. The voltage UBPV was measured using the set-up shown in Fig. 4. The sample was a typical circular n-type silicon slice. The resistivity of the material was 4.5 O cm. The measurements are performed along the diameter of the slice under open-circuit conditions. The measured values of the UBPV versus the position of the light spot are plotted in Fig. 25 (thin solid line). The thick solid line represents the values corrected using the geometrical factor ðR x0 Þ2 found from Eq. (5.39). The absolute values of rðxÞ can be determined from the corrected BPV distribution, if we in addition measure the resistivity at two appropriately matched positions using a 4-point probe. 5.5.3. Absolute measurements of grad r in a circular slice Studies on the inhomogeneity in circular semiconductor samples using the BPV effect were reported by Blackburn et al. [33] and Larrabee and Blackburn [34]. The latter investigators used the most advanced methods for scanning the light spot over the diameter of a large circular slice. The slice is equipped, at its periphery, with eight electrodes installed at eight points 1–8, as shown in Fig. 23. The voltage may be measured between various pairs of the contacts. If the light is scanned along the diameter 5–1, we measure the photovoltaic voltages U51 and U73 between the contacts 5–1 and 7–3. The method described above is referred to by the authors as the X–Y Geometry variant. Another variant—the ‘45 geometry’—consists of the measurements of U62 þ U48 ; U62 U48 ; and dURð64Þ in the presence of the current flow I82 : Here we shall only describe the X–Y Geometry variant, considering it to be sufficiently illustrative of this very valuable measurement idea. In order to find the voltage U51 we need to calculate the potentials CðR; 0Þ and CðR; 0Þ. Substituting w ¼ R into Eq. (5.39) results in an infinity in the denominators, which leads to Y ðR; 0Þ ¼ 0:
ð5:40Þ
Then, from Eq. (5.39) we obtain
Dx þ iDy Dx þ iDy 1 R CðR; 0Þ ¼ þ 2 Re 2R 2pd R x0 RðR x0 Þ R x0 R 2 R Dy ¼ pd R2 x0 R
ð5:41Þ
and hence we obtain the voltage U51 ¼ CðR; 0Þ CðR; 0Þ ¼
2RDx : pdðR2 x20 Þ
ð5:42Þ
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Fig. 24. Potential distribution and force lines, in a circular sample with point contacts. (a) Dipole parallel to the sample diameter, (b) dipole perpendicular to the sample diameter, and (c) dipole beyond the diameter, deviated at 45 with respect to the diameter. Contour and surface plot.
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Fig. 25. Measured distribution of the BPV voltage (thin solid line) in a circular 300 Si wafer. The distribution of the BPV voltage calculated with account of the circular geometry and point contacts (thick solid line). The distribution of the resistivity along the sample diameter (dotted line) determined from the corrected distribution of BPV.
The voltage U73 is equal to the difference of the potentials Cð0; iRÞ Cð0; iRÞ: Thus, using Eq. (5.39) we obtain U73 ¼ Cð0; iRÞ Cð0; iRÞ
R R iR R þ iR Re ðDx þ iDy Þ ¼ 2pdðR x0 Þ RðiR x0 Þ RðiR x0 Þ
2RDy R iR R þ iR þ ðDx þ iDy Þ 2 2 : ¼ R iRx0 R þ iRx0 pdðR2 þ x20 Þ
ð5:43Þ
Eqs. (5.42) and (5.43) are consistent with Eqs. (16) and (17) given by Larrabee and Blackburn [34]. According to notation (5.2) and Eq. (4.27), we have for n-type material qr ð5:44Þ Dx ¼ Mx P ¼ 2mp kB T 0 P: qx Substituting this equation into Eq. (5.42) and performing appropriate transformations we obtain
qr0 pdðR2 x20 Þ U51 : ð5:45Þ ¼ 4R qx x0 mp kB TP Similarly, Eq. (5.43) gives
qr0 pdðR2 þ x20 Þ U73 : ¼ 4R qy x0 mp kB TP
ð5:46Þ
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It should be remembered that these equations are only valid when we deal with ntype semiconductors. In the case of p-type semiconductors mp should be replaced by mn : It follows from the equations derived above that the voltages U51 and U73 only permit determining the approximate relative distributions of the derivative of r0 ; since in practice we are unable to determine precisely the number of carriers, P; generated by the light spot. This number depends on the excess carrier lifetime at a given point, which is unknown. This spoils the accuracy of the estimate of the existing inhomogeneity. Having however known Eqs. (5.45) for both the components of the grad r0 ; we can find its precise direction (Fig. 23) as tgY ¼
qr0 qr0 R2 þ x20 U73 : ¼ 2 : qy qx R x20 U51
ð5:47Þ
Larrabee et al. [34] and Blackburn [33] proposed to measure additionally the photo-induced resistance change DR; which permitted them to eliminate the dependence of the results on the carrier number P: They measured this change by passing a constant current I51 through the sample. The light spot was focused at point x0 and modulated, thanks to which the change of the voltage dU51 (established between electrodes 1 and 5) due to the photoconductivity of the light spot region could easily be measured. Section 4 described the generalised fundamentals of this method and gave Eqs. (4.40) that determined the voltage dUR ; which now will be denoted dU51 : Another equation derived there was (4.42) which permitted determining the value of the individual components of the grad r as functions of grad C0 ; the UPBV voltage (now denoted as U51 ), and dUR ¼ dU51 : Therefore, the only quantity to be found is the potential that occurs in the presence of the current I51 : In the case of a single source positioned at the edge of a half-plane, the potential distribution is of the 1=z type, and we have a singularity at the point z ¼ 0: Then, transforming the half-plane into a circle with the radius R and centre at z ¼ 0; we obtain I51 r0 R þ z C0 ¼ ; ð5:48Þ ln Rz pd where the function has a singularity at z ¼ 7R: In terms of complex numbers, gradient C0 ðx; yÞ is determined by the derivative with respect to z: dC0 I51 r0 2R ¼ : dz pd R2 z2
ð5:49Þ
Let us consider an arbitrary point x0 positioned on diameter 1–5. Substituting x ¼ x0 into Eq. (5.49), we obtain a real number with no imaginary part, which leads to qC0 ðx0 ; 0Þ I51 r0 2R ¼ ; qx pd R2 x20
ð5:50Þ
qC0 ðx0 ; 0Þ ¼ 0: qy
ð5:51Þ
The absence of the component perpendicular to the 1–5 diameter means that, in Eq. (4.42), cos a ¼ 1 and cos b ¼ 1: In this equation, the value of jgrad C0 j is defined
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by Eq. (5.50). Hence, we can conclude that the component of gradient r0 parallel to 1–5 diameter can be determined from
qr0 q 2 mp þ mn R I51 r ¼ U51 : ð5:52Þ qx x0 kB T 0 mp R2 x20 dU51 It should be noted that, in this equation, 1=DR appears as I51 =dU51 : It follows from the general theory that the equation that determines qr0 =qy cannot be found in this way. We may however use Eq. (5.47). By multiplying by sides Eqs. (5.47) and (5.52) we obtain
qr0 q 2 mp þ mn RðR2 þ x20 Þ I51 r ¼ U73 : ð5:53Þ qy x0 kB T 0 mp ðR2 x20 Þ2 dU51 Eq. (5.52) is consistent with Eq. (28) given by Larrabee [34], Eq. (28), and Eq. (5.53) is also confirmed there [34, Table 1, column III]. It should be noted that Eqs. (5.52) and (5.53) are valid for n-type semiconductor. 5.5.4. Circular slice with extended electrodes The problem of extended electrodes will first be considered for the case of a halfstrip shown in Fig. 26. The initial dipole D# z ¼ Dx þ iDy placed at the point z0 and its image D# z are placed at the point z0 : In order to obtain the equation for the potential induced by the original dipole placed at z0 ; we should transform Eq. (5.33) by subtracting iy from z: This means that in the first term x0 is replaced by z0 and in the second term by z0 : When this is done we obtain n h i h io 1 p p Re D# z ctg ðz z0 Þ D# z ctg ðz þ z0 Þ : Cor ðx; yÞ ¼ ð5:54Þ 4dh 2h 2h
Fig. 26. Dipole Dz situated at point z0 of the half-strip with a metal electrode and its image Dz placed at point z0 :
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The potential of the image D# z one can obtain by substituting into the above equation D# z instead of D# z and D# z instead of D# z : The point z0 should be replaced by z0 and vice versa. Then we have n h i h p io 1 p Re D# z ctg ðz z0 Þ þ D# z ctg ðz þ z0 Þ : Cim ðx; yÞ ¼ ð5:55Þ 4dh 2h 2h The potential in the half-strip is the sum of Cor and Cim n h i 1 p p Re D# z ctg ðz z0 Þ þ ctg ðz þ z0 Þ Cst=2 ðx; yÞ ¼ 4dh h 2h 2h io p p D# z ctg ðz þ z0 Þ þ ctg ðz z0 Þ : ð5:56Þ 2h 2h This equation forms the basis for calculating the potential distribution in a circular slice with one extended electrode. The first step is to formulate a conformal transformation so that the electrode segment becomes the (tg a; tg a) segment on the real axis of the half-plane and the sides of the half-strip pass on the remaining parts of the real axis. The second transformation converts the half-plane into a circle with the radius R positioned in the complex plane w ¼ u þ iv. The point electrode should be placed at w ¼ R: The extended electrode lies on an arc of the circle between the points Re2ia and R2ia . Thus, the size of the electrode is defined by 4a: Performing these two transformations, using the Christoffel–Schwarz integral, and transforming the circle to the y-axis we obtain h h i Rw : ð5:57Þ z ¼ þ arcsin 2 p tg a R þ w To perform transformation (5.57) we should find an equation that defines the relation between the dipoles D# z and D# w (placed at the point w0 ). This equation is
dz 2hRi # # qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: Dz ¼ Dw ¼ D# w ð5:58Þ dw w0 pðR þ w0 Þ tg2 aðR þ w0 Þ2 þ ðR w0 Þ2 The system of Eqs. (5.55)–(5.57) enables calculating the potential distribution of BPV (under open-circuit conditions) in a circular slice with one extended electrode. The values of the slice radius R; the angle a; the dipole D# w and its position w0 should be assumed. The value of z0 occurring in Eq. (5.54) can be calculated from Eq. (5.57). Fig. 27 shows the potential distribution established by the dipoles of intensities R d 1 V directed along: (a) u-axis, (b) n-axis and (c) deviated at 45 . The resistivity distribution determined based on the measurement of BPV bears an error dependent on the geometry of the sample and its electrodes. Typically, the measurements are made on circular slices with point or extended electrodes. A circular slice with two point electrodes is shown in Fig. 23. The variation of the BPV voltage with the light spot position is described by Eq. (5.39). Using the correction divisor BðR x0 Þ2 we can find the distribution of grad r: The BPV distribution in a circular slice with one point electrode and one extended electrode is defined by Eqs. (5.56)–(5.58). The more advanced calculations in the case of a circular slice with two extended electrodes were made. Assuming a1 ¼ j1 =4 and a2 ¼ j2 =4; the dipole placed on the u-axis of the electrodes and directed parallel to this diameter
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Fig. 27. Potential distribution in a circular sample with one from two extended electrode. Dipoles directed (a) along the axis of the electrodes, (b) perpendicular to the axis, and (c) beyond the sample diameter, deviated at 45 with respect to the diameter. Contour and surface plot of potential distribution. Angular width of the electrode is 60 .
one obtains UðR;RÞ 4Du C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ¼ h ; ð5:59Þ h i h i i2 hu i2 R 2 2 u u u 0 0 0 0 1 þ þ1 1 þ þ 1 tg2 a2 tg2 a1 R R R R where u0 is the distance between the light spot and centre of slice and Z p p=2 db qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: C¼ 2 0 ð1 ½tga1 tga2 2 sin2 bÞ
ð5:60Þ
The results of the calculation are shown in Fig. 28 in the form of a family of curves for various angular widths of the electrodes. Each curve represents the dependence of
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Fig. 28. Normalized photovoltage as a function of the position of a unit dipole directed along the diameter of a circular sample. Angular widths of both extended electrodes (a) 1 , (b) 5 , (c) 10 , (d) 20 , (e) 30 and (f) 60 .
the unit dipole voltage between the electrodes on its position. Two extended electrodes are placed symmetrically on both sides of circular slice. The unit dipole is placed in the u-axis of the electrodes being a diameter of the circle. The direction of the dipole is parallel to the diameter. Every curve is calculated for a specified angular size j ¼ 4a; the same for both electrodes. The voltages presented by the curves in Fig. 28 are obtained assuming a constant intensity of the dipole, so the voltage value for a definite position may serve as a correction coefficient. It is interesting to note that at j ¼ 4a ¼ 30 ; the voltage is almost constant. This means that in this case the correction coefficient is constant and equal to unity.
6. Bulk photoelectromagnetic effect 6.1. Theory The action of the magnetic field on the charge carriers in metals and semiconductors leads to a variety of interesting phenomena from magnetoresistance and thermomagnetic effects to the Hall effect. In view of the fact that excess carriers may be present in semiconductors, an additional group of photoelectromagnetic effects has to be considered. The well-known phenomenon, namely the photoelectromagnetic effect was discovered by Kikoin and Noskov [14] and was investigated by Bulliard [36] in germanium and silicon. The rigorous theory of this effect was given by van Roosbroeck [15]. In the publications mentioned above the authors assume that the semiconductor is homogeneous. In the case of an inhomogeneous semiconductor exposed to a magnetic field, another effect takes place, which was discovered and discussed by the present autor in Refs. [37–39]. This effect was called bulk photoelectromagnetic effect (BPEM).
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This chapter is devoted to the derivation of the transport equations for the excess current carriers in an inhomogeneous semiconductor exposed to a magnetic field, and to the theory of the bulk photoelectromagnetic effect and its experimental confirmation. It should be noted that the considerations presented throughout this chapter do not take into account any magnetoresistance effects, since only linear (with respect to the magnetic field) theory is considered here. According to the laws governing the action of the magnetic field on charge carriers, the additional hole J Bp and electron J Bn current appear J Bp ¼ ðJ 0p bÞYp ;
ð6:1Þ
J Bn ¼ ðJ 0n bÞYn ;
ð6:2Þ
where J 0p ; J 0n are hole and electron current densities in the absence of magnetic field, b is unit vector parallel to the magnetic induction B and the Hall angles Yp ; Yn are defined as Yp ¼ mpH B;
ð6:3Þ
Yn ¼ mnH B;
ð6:4Þ
where B is the absolute value of magnetic induction, and mpH ; mnH are the Hall mobilities of holes and electrons, respectively. The hole and electron current densities, J p and J n ; in the presence of magnetic field can be obtained by adding current (6.1) and (6.2) to the J p0 and J n0 : J p ¼ J 0p þ ðJ 0p bÞYp ;
ð6:5Þ
J n ¼ J 0n þ ðJ 0n bÞYn :
ð6:6Þ
In order to transform further these equations it will be useful to introduce the internal current J defined by (2.40). Then we obtain the form analogous to (2.41) and (2.42): sp J 0p ¼ J 0 þ J ; ð6:7Þ s sn J 0n ¼ J 0 J ; ð6:8Þ s where J 0 ¼ J 0p þ J 0n :
ð6:9Þ
Substituting (6.7) and (6.8) into Eqs. (6.5) and (6.6) one obtains the formulae that describe the hole and electron currents: sp sp J p ¼ J 0 þ J þ Yp J 0 b þ Yp J b; ð6:10Þ s s sn sn J n ¼ J 0 þ J þ Yn J 0 b þ Yn J b: ð6:11Þ s s
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Adding these formulae and combining the corresponding components one obtains the formula for the total current J s sn p Yp þ Yn ðJ 0 bÞ þ ðYp Yn ÞJ b: ð6:12Þ J ¼ Jp þ Jn ¼ J0 þ s s The introduction of the Hall angle sp sn Yp þ Yn s s and the angle of the photomagnetoelectric effect YH ¼
ð6:13Þ
Y ¼ Yp Yn ¼ ðmpH þ mnH ÞB
ð6:14Þ
enables formula (6.12) for the total current to be expressed in the simple form J ¼ J 0 þ YH J 0 b þ YJ b: 0
ð6:15Þ
The currents J and J present in Eq. (6.15) may be replaced by the equation derived in Section 2.2.2, which defines the same quantities. J 0 is the same as the total current density in the absence of the magnetic field; and thus one may use (2.60). Similarly J is defined by (2.50). Substituting these expressions into Eq. (6.15) and rearranging the terms appropriately one obtains an important equation defining the total current density in an inhomogeneous semiconductor that contains excess carriers and is exposed to magnetic field: J ¼ s grad Cð1 þ YH bÞ qDY grad Dp b 2qm0 Dp grad V0 ð1 þ YH bÞ qm DpY grad V0 b:
ð6:16Þ
Multiplying both sides of this equation by 1 YH b and neglecting the terms that include Y2H and YH Y give Jð1 YH bÞ ¼ s grad C qDY grad Dp b 2qm0 Dp grad V0 qm DpY grad V0 b:
ð6:17Þ
After rearranging the terms and dividing the equation by s one obtains the equation for gradient of the potential C J YH J b YqD 2qm0 Dp þ grad Dp b þ grad V0 grad C ¼ s s s s qm DpY grad V0 b: þ ð6:18Þ s An analysis of Eq. (6.18) enables each component of grad C to be linked to a certain law or phenomenon: (1) (2) (3) (4) (5)
ðJ=sÞ grad V0 ohmic component, YH J b=s Hall effect, ðqDY=sÞ grad Dp b Kikoin–Noskov photoelectromagnetic effect, ð2qm0 Dp=sÞ grad V0 bulk photovoltaic effect and ðqm DpY=sÞ grad V0 b bulk photoelectromagnetic effect.
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Fig. 29. Illuminated rectangular sample in the presence of a magnetic field. The gradient of carrier concentration and built-in field are perpendicular to the longest edge of the sample.
In order to describe the bulk photoelectromagnetic effect undisturbed by other phenomena, the case of a thin rectangular plate placed in a perpendicular magnetic field B (shown in Fig. 29) will now be considered. The internal field grad V0 is directed perpendicularly to the length of the plate. Hence the conductivity and the remaining parameters of the material are functions of the co-ordinate y alone. The steady state and a homogeneous magnetic field are assumed. As the illumination is homogeneous and the thickness of the plate is small, the assumption of Dp¼ const: is justified within the entire illuminated region. The non-illuminated region is wide enough to prevent the excess carriers from reaching the electrodes. The foregoing assumptions concerning the geometry of the sample, and the direction of magnetic and internal fields yield b ¼ k;
grad V0 ¼ k
dV0 ; dy
J ¼ iJx :
ð6:19Þ
Substituting these formulae into Eq. (6.16) one obtains the equation system:
qC Jx qm DpY dV0 Jx ¼ þ ¼ þ EBPEM ; qx s s dy s
ð6:20Þ
qC YH Jx 2qm Dp dV0 YH Jx ¼ þ ¼ þ EBPV ; qy s s dy s
ð6:21Þ
qC ¼ 0; qz
ð6:22Þ
where EBPEM and EBPV are the EMF—fields of the bulk photoelectromagnetic and bulk photovoltaic effects, respectively. The derivative q=qxðqC=qyÞ taken from (6.21) vanishes because the variable parameters only depend there on y. Consequently, q=qyðqC=qxÞ calculated from (6.20) is also equal to zero and hence qC Jx ¼ þ EBPEM ¼ const: qx s
ð6:23Þ
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Solving this for Jx yields
qC þ EBPEM : Jx ¼ s qx
ð6:24Þ
The above equations show that, generally, Jx must depend on y; but detailed computation proves that the assumption Jx Econst:
ð6:25Þ
results in an insignificant error, provided that grad V0 is sufficiently small. Using formulae (6.20) and (6.21) we will explain (i) the short-circuit, (ii) compensation and (iii) open-circuit measuring methods. (i) If the sample electrodes are short circuited, the Isc current may be calculated by substituting qC=qz ¼ 0 into Eq. (6.20) to obtain Isc dV0 ¼ qm DpY ¼ sEBPEM ; s dy
ð6:26Þ
where s is the cross-section of the sample. The above equation indicates that the short-circuit current flowing in the external circuit is proportional to the excess carrier concentration Dp and to the magnetic field (through the angle Y). The group mobility m is an important parameter in this equation. This parameter is positive in an n-type semiconductor, negative in a p-type one and is equal to zero in an intrinsic semiconductor. Thus the direction of the short-circuit current flow depends on the semiconductor type. It is interesting to calculate the distribution of the electric field in the direction y: It follows from (6.26) that Jx is proportional to Y; and thus substituting its value into Eq. (6.21) we obtain a term that contains B squared and proportional to Y YH ; which may be neglected. Thus in the case of a short-circuited sample we obtain
qC 2qm0 Dp dV0 ¼ ¼ EBPV : qy s dy
ð6:27Þ
It follows from (6.27) that only the bulk photovoltaic effect acts in the direction y: The component in the direction z does not exist nor does it exist in the cases considered below. The component dC=dy can be measured if additional electrodes (shown in Fig. 30) are used, and the electrodes A and B are short circuited.
Fig. 30. Components of the bulk photomagnetoelectric effect and the measurement contacts.
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(ii) To realise the compensation method a modulated illumination is used and a constant current I ¼ Jx s is passed through the electrodes A, B. Two modulated photovoltages are then observed. The first one is due to the BPEM and the second is related to the modulated photoconductivity. The current I (and thus Jx ) must then be chosen so as to suppress the resultant of both signals to zero. During the illumination the current is given by qC dV0 qm DpY ð6:28Þ J x ¼ ½s0 þ qðmn þ mp ÞDp qx dy and without illumination, by
qC J xc ¼ s0 ; qx C
ð6:29Þ
where s0 is the equilibrium conductivity and qðmn þ mp ÞDp is the photoconductivity. Subtracting (6.29) from (6.28) and taking into acount that Jx ¼ Jxc and qC=qx ¼ ðqC=qxÞC ; one obtains
qC dV0 ð6:30Þ 0 ¼ qðmn þ mp ÞDp þqm DpY qx C dy and hence
Jxc qC m dV0 : ¼ ¼ Y qx C s0 mn þ mp dy
ð6:31Þ
If the parameters of the semiconductor are known and Jxc is measured, the presented theory of BPEM can be proved using this equation. Using definitions (6.5), (6.6) and (6.14) one can transform Eq. (6.31) to obtain mnH þ mpH dV0 Jxc dV0 ¼ BkH m ; ð6:32Þ ¼ B m s0 mp þ mn dy dy where kH is the ratio of the Hall mobility and the drift mobility, m is the group mobility defined by (2.49). This result indicates the possibility of measuring the minority carrier mobility mm in the n- or p-type material, where m ¼ 7mm ; respectively. (iii) Considering the open-circuit case shown in Figs. 30 and 31 one should assume Jx ¼ 0
ð6:33Þ
according to the result (6.25). Substituting this value into Eqs. (6.20) and (6.21) one obtains the component of grad C in the direction x as
qC qm DpY dV0 ¼ ¼ EBPEM qx s dy
and the component in the direction y as qC 2qm0 DpY dV0 ¼ ¼ EBPV : qy s dy
ð6:34Þ
ð6:35Þ
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Fig. 31. Distribution of the electric field in an open-circuit, n-type sample with finite dimensions.
Experimental studies of the BPEM may be based on the measurement of both fields EBPEM and EBPV and the determination of their ratio which is tga ¼
EBPEM Y m ¼ : 2 m0 EBPV
ð6:36Þ
The parameter a is the angle between the direction of grad C and qC=qy or it is the inclination of equipotential lines with respect to the plate edge (Fig. 31). To calculate tga given by (6.36) one should only know the magnetic field B; the two Hall mobilities (according to (6.5) and (6.6)) and the ratio m =m0 : This last value is equal to +1, for n-type and 1 for p-type semiconductor. Thus formula (6.36) is more convenient than (6.31). The agreement between the values of tg a determined experimentally and theoretically proves that the theory is correct. Eq. (6.36) may be used for determining the sum of Hall mobilities. According to (6.14), (2.49) and (2.18) one can write 1 m 1 tga ¼ BðmpH þ mnH Þ ¼ 7 BðmpH þ mnH Þ; 2 2 m0
ð6:37Þ
where plus for n-type and minus for p-type hold. The above equation enables us to determine the sum of Hall mobilities and then the minority carrier mobility if the majority carrier mobility is known from classical Hall measurements. Thus, the BPEM measurements give two possibilities to measure the minority carrier mobility. The first possibility is demonstrated by (6.32) and the second is represented by Eq. (6.36). To realize the measurement of tg a according to Eq. (6.37) it is necessary to determine EBPEM and EBPV : It is seen from (6.34), (6.35) and Fig. 30 that UBPEM ; ð6:38Þ EBPEM ¼ l EBPV ¼
UBPV ; a
ð6:39Þ
where l and a are the length and width of the illuminated region, respectively, UBPEM is the voltage between the electrodes A, B, and the voltage UBPV is measured between the point electrodes C, D.
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6.2. Experiment Investigations were carried out using germanium p- and n-type plates [37] made from single crystals with a sufficiently high, though uniform, resistivity gradient. The plates were cut in such a direction so as to obtain an internal field grad V0 directed perpendicularly to he length of the plate. As a result, the transverse built-in field in the samples was at least an order of magnitude higher than the lateral component of this field. The thickness of the investigated samples ranged from 0.2 to 0.5 mm, the width was between 6 and 7 mm and the length between 12 and 15 mm. The sample was illuminated using an optical set-up with a tungsten bulb as the source of light. The light beam was directed to the sample using a mirror. A rectangular light spot with approximate dimensions of 5 5 mm was created at the surface of the sample. Both the sample (held appropriately) and the mirror were placed between the poles of an electromagnet capable of providing magnetic induction up to approximately 0.7 T. The placements of the sample and mirror, as well as the path of the light are shown in Fig. 32. The measurements enabled the voltages UBPEM and UBPV and thus the fields EBPEM and EBPV to be determined. The field EBPEM was measured versus the magnetic field B and the intensity of illumination. Fig. 33 shows the shape of EBPEM ¼ f ðBÞ for n-type germanium sample 1 (0.9 O cm) and n-type sample 2 (5.8 O cm) at constant illumination. The dependence of EBPEM on the magnetic field B was linear within the experimental error in agreement with the theory of small angles yp ; yn : The dependence of EBPEM on the illumination was also found to be linear. The magnitudes of EBPEM at B ¼ 0:7 T (points K, L in Fig. 33) were 0.40 mV (sample 1) and 0.30 mV (sample 2) at full voltage on the lamp. Moreover EBPV across points C, D was measured. In the sample 1 EBPV ¼ 2:0 mV, whereas in sample 2 EBPV ¼ 1:12 mV. Thus the computation of tg a was possible (Eqs. (6.34)–(6.36)) yielding in sample (1Þtg a ¼ 0:15; in sample (2) tg a ¼ 0:27: A rough theoretical computation yielded the value tg a ¼ 0:26; which was accepted as satisfactory.
Fig. 32. Experimental arrangement for measuring the bulk photomagnetoelectric effect; (a) side view of the sample, (b) front view; 1—Ge sample, 2—mirror.
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Fig. 33. UBPEM versus the magnetic induction B for two (1, 2) n-type Ge samples with different resistivities.
Acknowledgements The authors like to acknowledge Prof. Dr. Jaroslaw. Swiderski for the valuable discussions.
Appendix A. Derivation of Eq. (2.17) The equations involved in the calculations are: Ds ¼ s s0 ¼ qðmn þ mp ÞDp; grad Ds ¼ s grad ln
s Ds þ grad s0 ; s0 s0
kB T grad s0 ¼ ðsn0 sp0 Þgrad V0 : q
ðA:1Þ ðA:2Þ ðA:3Þ
Eq. (A.1) follows from the definition of the conductivity. To prove that Eq. (A.2) is correct we should apply the grad operator to the right side of it. Eq. (A.3) may be derived based on the formulae for the equilibrium concentrations p0 and n0 : q ðj V0 Þ ; p0 ¼ ni exp ðA:4Þ kB T 0 q n0 ¼ ni exp ðV0 j0 Þ ; ðA:5Þ kB T
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where j0 is the equilibrium Fermi level, which is position independent. By calculating the gradient of the two concentrations we obtain q grad p0 ¼ p0 grad V0 ; ðA:6Þ kB T grad n0 ¼
q n0 grad V0 : kB T
ðA:7Þ
Then, based on the assumption that mn and mp are position independent, we multiply both sides of these equations by qmp and qmn to obtain kB T grad sp0 ¼ sp0 grad V0 ; q
ðA:8Þ
kB T grad sn0 ¼ sn0 grad V0 ; q
ðA:9Þ
where sn0 and sp0 are the equilibrium conductivities. Summing Eqs. (A.8) and (A.9) gives (A.3). Now, let us use Eqs. (A.1) and (A.2) to transform Eq. (2.16). J Ds kB T mn mp ¼ grad V þ grad V0 grad V0 þ grad Ds s s qs mn þ mp Ds grad V0 ¼ gradðV V0 Þ s
k B T m n mp s Ds þ s grad ln þ grad s0 qs mn þ mp s0 s0 kB T mn mp s ¼ gradðV V0 Þ þ grad ln q mn þ mp s0 ! 1 mn mp 1 kB T grad s0 grad V0 Ds: þ s mn þ mp s0 q
Eth ¼
ðA:10Þ
By combining the first and second members of this equation, inserting Ds as defined by (A.1), and using the assumption that T; mp and mn are position independent, we obtain ! kB T mn mp s Eth ¼ gradðV V0 Þ þ grad ln q mn þ mp s0 " # 1 m n mp 1 þ ðsn0 sp0 Þgrad V0 grad V0 Ds s mn þ mp s0 ! kB T mn mp s ln ¼ gradðV V0 Þ þ grad q mn þ mp s0
q sn0 sp0 þ ðmn mp Þ ðA:11Þ ðmn þ mp Þ Dp grad V0 : s s0 s0
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Transforming the coefficient enclosed in the square brackets gives
sn0 sp0 ðmn mp Þ ðmn þ mp Þ s0 s0
sn0 sp0 sn0 sp0 1 mp þ1 ¼ mn s0 s0 s0 s0 mp mn ¼ ðsn0 sp0 s0 Þ ðsn0 sp0 þ s0 Þ s0 s0 sp0 sn0 m 2 m ¼ 2m0 ; ¼ 2 s0 n s0 p
ðA:12Þ
which is consistent with definition (2.18). Using these results we may write ! J kB T mn mp s Eth ¼ ¼ gradðV V0 Þ þ grad ln s q mn þ mp s0 2qm0 Dp grad V0 : s This is exactly Eq. (2.17).
ðA:13Þ
Appendix B. Derivation of Eqs. (2.22) and (2.23) Eqs. (2.22) and (2.23) are derived based on the Boltzmann statistics. The reasoning is as follows. B.1. Derivation of (2.22) If Dpa0 and Ja0; the concentrations p and n are given by q p ¼ ni exp ðj V Þ; kB T p n ¼ ni exp
q ðV jn Þ: kB T
ðB:1Þ ðB:2Þ
If Dp ¼ 0; but Ja0; then jp ¼ jn ¼ j and p ¼ p0 ; n ¼ n0 : In this case, Eqs. (B.1) and (B.2) become q ðj V Þ; ðB:3Þ p0 ¼ ni exp kB T n0 ¼ ni exp
q ðV jÞ kB T
ðB:4Þ
kB T ni kB T ni ln ¼ ln : q q p0 n0
ðB:5Þ
and hence V j¼
In the particular case when Dp ¼ 0 and J ¼ 0; the term V j is replaced by the term V0 j0 ; which is equal to the former term since it depends on p0 and n0 in the same
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way. Thus we have V0 j0 ¼ V j:
ðB:6Þ
Eq. (2.21) can now be transformed so that Cðr0 Þ ¼ V ðr0 Þ V0 ðr0 Þ þ j0 ¼ ½V ðr0 Þ jðr0 Þ ½V0 ðr0 Þ j0 þ jðr0 Þ:
ðB:7Þ
The Eq. (B.6) for point r0 is V ðr0 Þ jðr0 Þ ¼ V0 ðr0 Þ j0 ;
ðB:8Þ
which means that the terms in square brackets cancel one another and we arrive at Eq. (2.22): Cðr0 Þ ¼ jðr0 Þ:
ðB:9Þ
B.2. Derivation of (2.23) Because of (B.6), (B.3) and (B.4) for the point r0 can be written as q p0 ¼ ni exp ðj V0 Þ; kB T 0 n0 ¼ ni exp
q ðV0 j0 Þ: kB T
Now, dividing (B.1) by (B.10) and (B.2) by (B.11) gives p q ðj V j0 þ V0 Þ; ¼ exp p0 kB T p n q ðV jn V0 þ j0 Þ: ¼ exp n0 kB T Hence, we can derive the equation kB T p kB T n ln ¼ jn þ ln : V V0 þ j0 ¼ jp q p0 q n0 For small values of Dp; we have kB T Dp kB T Dp V V0 þ j0 Djp ¼ jn þ : q p0 q n0
ðB:10Þ ðB:11Þ
ðB:12Þ ðB:13Þ
ðB:14Þ
ðB:15Þ
Using Eq. (A.1) and knowing that sp0 =s0 þ sn0 =s0 ¼ 1; we obtain from (B.15)
sp0 kB T Dp sn0 kB T Dp V V0 þ j0 ¼ jp jn þ q p0 q n0 s0 s0
sp0 sn0 kB T Dp sn0 sp0 ¼ jp þ jn þ q s0 n0 s0 s0 p0 sp0 sn0 Dp ¼ j þ j þ kB T ðmn mp Þ s0 s0 p s0 n ! sp0 sn0 kB T mn mp Ds ¼ j þ j þ ðB:16Þ q mn þ mp s0 s0 p s0 n
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and hence sp0 sn0 kB T mn mp Ds jp þ jn ¼ V V0 : q mn þ mp s0 s0 s0
ðB:17Þ
For small values of Ds; the right-hand side of this equation is, by (2.20), equal to the potential C: Thus, in this case sp0 sn0 CD jp þ j : ðB:18Þ s0 s0 n This is exactly Eq. (2.23).
Appendix C. Calculation of gradðV0 j0 Þ appearing in Eq. (3.23) and in equations following (3.28) C.1. Calculation of grad n0 =ni The expression to be found contains the term grad ln
n0 grad n0 grad ni ¼ : ni n0 ni
ðC:1Þ
The equilibrium concentrations n0 and p0 are related to each other through the neutrality principle and the mass-action law: n0 p0 ¼ N;
ðC:2Þ
p0 n0 ¼ n2i
ðC:3Þ
which results in ðp0 þ n0 Þ2 ¼ N 2 þ 4n2i :
ðC:4Þ
Applying grad to (C.4) and using (C.3) yields ðp0 þ n0 Þgradðn0 þ p0 Þ ¼ 2N grad N þ 4n0 p0 grad ln ni :
ðC:5Þ
where N was replaced by (p0 n0 ) in accordance with (C.2). The application of grad to (C.2) and subsequent multiplication by (p0 þ n0 ) yields ðp0 þ n0 Þgradðn0 p0 Þ ¼ ðn0 þ p0 Þgrad N:
ðC:6Þ
Adding (C.5) and (C.6) and dividing the result by n0 one obtains ðp0 þ n0 Þgrad ln n0 ¼ grad N þ 2p0 grad ln ni : Subtracting the term ðp0 þ n0 Þgrad ln ni from both sides of (C.7) yields n0 ðp0 þ n0 Þgrad ln ¼ grad N þ ðp0 n0 Þgrad ln ni ; ni
ðC:7Þ
ðC:8Þ
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which leads to grad ln
n0 grad N p0 n0 ¼ þ grad ln ni : ni p0 þ n0 p0 þ n0
ðC:9Þ
C.2. Calculation of kB T=q grad ln ni The formula defining intrinsic concentration
pffiffiffiffiffiffiffiffiffiffiffiffiffi EG ni ¼ NC NV exp 2kB T
ðC:10Þ
leads to kB T kB T T EG grad ln ni ¼ grad lnðNC NV Þ grad q 2q 2q T
kB T T EG EG grad lnðNC NV Þ grad 2 grad T ¼ 2q 2q T T kB T 1 EG kB T ¼ grad lnðNC NV Þ grad EG þ : grad 2q 2q q 2kB T
ðC:11Þ
In order to take into account the dependence of NC NV and EG on the temperature and position we proceed in accordance with (3.25) 3 grad lnðNC NV Þ ¼ grad lnðmC mV Þ þ 3 ln T þ const 2 3 q 3 3 grad T ¼ lnðmC mV Þgrad T þ gradxyz lnðmC mV Þ þ 2 qT 2 T
ðC:12Þ
and 1 1 qEG kB T 1 grad EG ¼ þ gradxyz EG ; grad 2q 2kB qT q 2q
ðC:13Þ
where the operator gradxyz only refers to the co-ordinates. Substituting results (C.12) and (C.13) into Eq. (C.11) we obtain kB T grad lnni q kB T 3 q 3 3 grad T lnðmC mV Þgrad T þ gradxyz lnðmC mV Þ þ ¼ 2q 2 qT 2 T 1 qEG kB T 1 EG kB T gradxyz EG þ : grad grad 2kB qT q 2q q 2kB T
ðC:14Þ
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Substituting (C.14) into Eq. (C.9), rearranging the terms and substituting the result into Eq. (3.23) we obtain n0 kB T kB T n0 þ grad ln grad q q ni ni
kB T grad N n0 p0 n0 3T q 3 1 qEG EG þ ¼ þ ln þ lnðmC mV Þ þ ni p0 þ n0 4 qT 2kB T q p0 þ n0 2 2kB qT kB T p0 n0 3kB T EG þ gradxyz lnðmC mV Þ gradxyz
grad ðC:15Þ q p0 þ n0 4q 2kB T
gradðV0 j0 Þ ¼ ln
which is used in formulae (3.29) and (3.31).
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