- Email: [email protected]
Diffusion and Drift of Charges
Chapter 10
Diffusion and Drift of Charges Chapter Outline 10.1 10.2 10.3 10.4
Diffusion of Ions and Electrons The Drift of Ions Electron Drift Motion of Electrons and Holes in Semiconductors 10.4.1 Features of the Motion of Electrons and Holes in Semiconductors
185 187 188 191 191
10.4.2 Effective Mass, Polaron, and Autolocalization of Holes 10.5 The Debye Screening Radius 10.6 Criterion of the Space Charge Formation 10.7 Current Limited by a Space Charge 10.8 Ambipolar Diffusion References
191 192 193 195 195 196
10.1 DIFFUSION OF IONS AND ELECTRONS In the absence of an electric field, charges in matter can undergo diffusion displacements. The basic diffusion equation, the Fick’s equation, shows the relationship between the particle flux density nv and the gradient of their concentration: nv ¼ D grad n;
(10.1)
where D is the diffusion coefficient. The diffusion flux of particles obeys the continuity equation dn=dt þ gradðnvÞ ¼ 0:
(10.2)
Using Eqs. (10.1) and (10.2), one can obtain the nonstationary diffusion equation vn v2 n ¼ D 2. vt vr
(10.3)
If there are n0 particles at the origin and at the instant t ¼ 0 they begin to diffuse through the gas, then for the particle density at time t from the solution of Eq. (4.3) we get: in the one-dimensional case n0 x2 nðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi exp ; 4Dt 4pDt
(10.4)
n0 r2 exp ; nðr; tÞ ¼ 2Dt 4Dt
(10.5)
in the two-dimensional case
in the three-dimensional case
r2 exp . nðr; tÞ ¼ 3=2 4Dt ð4pDtÞ n0
Radiation. https://doi.org/10.1016/B978-0-444-63979-0.00010-0 Copyright © 2019 Elsevier B.V. All rights reserved.
(10.6)
185
186 PART | I Fundamentals
From these distribution functions, one can find the mean square displacement of a particle from the origin: in the one-dimensional case pffiffiffiffi pffiffiffiffiffiffiffiffi x2 ¼ 2Dt ;
(10.7)
pffiffiffiffi pffiffiffiffiffiffiffiffi r 2 ¼ 4Dt ;
(10.8)
pffiffiffiffi pffiffiffiffiffiffiffiffi r 2 ¼ 6Dt .
(10.9)
in the two-dimensional case
in the three-dimensional case
For the three-dimensional case, we point out the quantity of the average displacement pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ 12Dt=p.
(10.10)
Using the obtained relations, one can find the average time during which the particles diffuse by a distance b: in spherical geometry t ¼
b2 . Dp2
(10.11)
in cylindrical geometry t ¼ In the kinetic theory of gases, it is shown that
b2 Dð2:405Þ2
.
(10.12)
D ¼ vL 3:
(10.13)
Expression (10.13) is approximate and agrees with the experimental data only in order of magnitude. The correct values of the diffusion coefficient can be calculated by using the charge distribution function in terms of energies or velocities. In Section 10.3, this problem will be observed in detail. Self-diffusion coefficients in gases are as follows: D(N2) ¼ 2.12$105 m2/s; D(Ar) ¼ 1.57$105 m2/s; and D (CH4) ¼ 2.0$105 m2/s. The diffusion coefficients of positive Dþ and negative De ions in some gases are given in Table 10.1. In liquids, the diffusion coefficient is several orders of magnitude lower than in gases at atmospheric pressure: in nonviscous liquids at 20 C, it has the order of 109 m2/s (in gases, it is [1e10]$105 m2/s). In particular, for air in liquid water D ¼ 2.0$109 m2/s. The diffusion coefficients in organic solids have a considerable spread. The diffusion coefficients of gases in polymers, in particular, depend on many properties of diffusing molecules and polymers, but mainly on the free volume of the polymer (i.e., the difference between the actual volume and the total volume of densely packed molecules) and the heterogeneity of its structure. D(O2 in polyethylene) ¼ 4.6$1011 m2/s; D(N2 in polymethylsiloxane) ¼ 94$1011 m2/s.
TABLE 10.1 The Diffusion Coefficients of Positive DD and Negative De Ions in Some Gases at t [ 298 K and p [ 105 Pa Gas
Air
N2
CO2
H2
O2
Dþ, 106 m2/s
2.8
3.9
2.3
12.3
2.6
De, 106 m2/s
4.3
4.1
2.6
19
3.9
De/Dþ
1.54
1.41
1.13
1.54
1.58
Diffusion and Drift of Charges Chapter | 10
187
If the diffusion occurs in an electric field, then the coefficients of longitudinal diffusion Djj and isotropic (transverse) Dt differ. We indicate a qualitative explanation of this difference. When the cloud of charges moves in an electric field, then the faster charges move ahead of the cloud and behind it the slower ones. Further analysis is determined by the form of the dependence of the scattering cross section of the charges and, consequently, of the drift velocity on the energy of the charges. If the cross section decreases with increasing energy, as is in the case for ions, then the mean free path, and with it the drift velocity, increases and, consequently, the faster charges move even faster, whereas the slower ones lag behind and the cloud of charges stretches along the field and, hence, Djj/Dt > 1. If the cross section increases with increasing energy, as is in the case for electrons in many gases, then on the contrary, the mean free path, and with it the drift velocity decreases with increasing energy; hence, the electrons moving ahead are slow, the ones moving behind accelerate their motion, the cloud contracts and, therefore, Djj/Dt < 1. For electrons in argon in the range E/p w 20e130 V/cm$Pa Djj/Dt w 1/5e1/7.
10.2 THE DRIFT OF IONS In gases, liquids, and often even in solids, one must deal with two types of charges with noticeably different masses: with ions (positive or negative) and with electrons. Because of the difference in mass, the nature of the motion in the electric field for these two groups of charges is noticeably different. The ions, participating in the thermal motion, are simultaneously slightly shifted in the electric field. The ions easily give up the energy that they collect in an electric field in collisions over the mean free path, and, thus, ions do not accumulate ion energy. Under such conditions, the ion drift velocity is much lower than the rate of thermal motion. The drift velocity can be defined as the average displacement velocity in an electric field over the mean free path. Because the ion moves uniformly in the field over the mean free path, vd ¼
v1 þ v2 ; 2
(10.14)
where v1 is the velocity in the field immediately after the collision and v2 is the velocity at the end of the free path before the next collision. In accordance with the initial condition adopted above, by the collision, ion completely loses its velocity in the direction of the field acquired during acceleration by the field at the previous mean free path, i.e., v1 ¼ 0. Introducing a as the acceleration of the ion, we write v2 ¼ at=2;
(10.15) where t is the mean free time. Because a ¼ eF/M, F is the electric field strength, M is the ion mass, and t ¼ L v, then for the drift velocity we obtain the expression vd ¼
eFL . 2Mv
(10.16)
The formula (10.16) is obtained by very simple means. The distribution of free ranges and velocities is considered simply by using their average values. If the distribution of free ranges is more rigorously taken into account, then the drift velocity is found to be twice as large: vd ¼
eFL ; Mv
(10.17)
and simultaneously considering the distribution of velocities, we obtain vd ¼
0; 815eFL pffiffiffiffi . M v2
(10.18)
Formulas (10.16)e(10.18), basically, correctly determine the form of the functional dependence, but one cannot expect quantitative conformity with experiment. The correct values of the drift velocity can be obtained by more rigorous theory, in particular, the ion energy distribution function. The expression (10.17) can be reshaped into the following form: vd ¼ mF;
(10.19)
188 PART | I Fundamentals
TABLE 10.2 Mobility of Some Ions in Their Own Gases Under Normal Conditions m, cm2/s V
HeD
NeD
NeD 2
ArD
ArD 2
KrD
KrD 2
XeD
XeD 2
ND 2
OD 2
OL 2
10.8
4.2
6.5
1.5
1.8
0.9
1.2
0.6
0.85
2.5
2.3
2.0
where m is the proportionality coefficient, called mobility m ¼
eL . Mv
(10.20)
In Table 10.2, the mobility of some ions in their own gases under normal conditions is shown. Generally speaking, the lower the velocity of ion drifts, the larger is their mass. However, this is not so for ions in their own gases. In this case, the charge exchange can perform þ Xþ fast þ Xslow /Xfast þ Xslow .
(10.21)
When foreign ions move in the gas and, in particular, the molecular ions Xþ 2 in a noble gas, the probability of charge exchange appears much less. Because of this, the mobility of molecular ions is higher than that of atomic ions, as can be seen from Table 10.2. Assuming that the mobility of the ions is of the order of 1 cm2/s$V, the ion drift velocity in a typical field of w103 V/cm is 103 cm/s. As is known, the thermal velocity of molecules in gases under normal conditions is w105 cm/s. Thus, the ion drift velocity actually appears much less than the velocity of thermal motion, as was suggested in deriving the expressions (10.16e10.18).
10.3 ELECTRON DRIFT The patterns of ion motion in the electric field, discussed in Section 10.2, are also valid for electrons in very weak fields (<1 V/cm). In large fields, electrons accumulate appreciable energy that is much higher than the energy of thermal motion, due to the fact that in the collision they transfer only a small part of their energy to molecules. The mobility of electrons in a gas does not depend on the field strength, only by the condition that the velocity distribution of electrons is Maxwellian. Due to the accumulation of energy, the energy distribution of electrons ceases to be Maxwellian and the simple relation (10.19) ceases to be satisfied. When discussing diffusion and drift of ions, the calculation of such macroscopic parameters as drift and diffusion, starting from the elementary microscopic parameters of collision processes, requires knowledge of the particle velocity distribution function. This is due to the fact that the probability (frequency, cross section) of the collision is a definite function of the velocity of the colliding particles. The average value obtained in measurements is the result of averaging over the entire spectrum of velocities. The analysis of ion drift at the simplest level yields reasonable results. In the case of electrons, it is necessary to involve deeper representations using the distribution function [1]. When the electron gas is in thermal equilibrium with the medium, the electrons are characterized by the Maxwellian energy distribution sffiffiffiffiffiffiffiffiffiffiffiffiffiffi E E f ðEÞdE ¼ 2 exp . (10.22) 3 kT pðkTÞ In the electric field, the distribution function somewhat changes. If the field is weak and the anisotropy of the electron velocity distribution is small, the change can be considered in the form of a series expansion. Using the expansion in a series of the Legendre polynomials, the distribution function can be written in the form f ðvÞ ¼ f0 þ f1 ðvÞcosq;
(10.23)
where q is the angle between the positive direction of the x axis and the direction of the electric field. f0(v) is the isotropic part of the function, it remains Maxwellian, and the second term of Eq. (10.23) defines the variation of the function under the action of the field, and, besides, this change is weak.
Diffusion and Drift of Charges Chapter | 10
189
The distribution function can be found as the solution of the Boltzmann kinetic equation. Without performing this equation, we remind only that it contains the cross sections and the type of collisions. The solution of the kinetic equation is the Druyvesteyn distribution " # 4; 56 m 3=4 E 1=2 3m E2 f ðEÞdE ¼ (10.24) dE. 3=2 exp Gð3=4Þ M M eFL 2 eFL where G is the gamma function, with all other parameters determined before. Wherein eFL vf0 . f1 ðvÞ ¼ me v vv
(10.25)
Knowledge of the distribution function makes it possible to calculate the basic parameters of the motion of electrons: average energy R N 3=2 E f0 ðEÞdE ; E ¼ R0N 1=2 E f0 ðEÞdE 0
(10.26)
drift velocity 1 vd ¼ 3
rffiffiffiffiffiffi R N Ef1 ðE ÞdE 2 R N0 ; me 0 E 1=2 f0 ðEÞdE
(10.27)
diffusion coefficient R E rffiffiffiffiffiffi N f0 ðEÞdE 0 1 2 sðEÞ RN D ¼ . 3N me 0 E 1=2 f0 ðEÞdE
(10.28)
The specific result depends on what dependence of the collision frequency of the electron with the atoms n ¼ Nsv ¼ Cvn is taken in the calculations. Substituting Eqs. (10.24) and (10.25) into Eqs. (10.26) and (10.27) and assuming that the collision frequency does not depend on the velocity (n ¼ 0, n ¼ C), we get 2 eFL E ¼ . (10.29) 2mv2 vd ¼
eFL . me v
(10.30)
In the other limiting case, when the collision cross section of an electron is independent of the velocity (n ¼ 1, C ¼ 1/Lav), we find E ¼ 0; 427ðM=me ÞeFL; 1=2 eFL . vd ¼ 0; 897 1=4 ðMme Þ
(10.31) (10.32)
The two limiting cases, considered above, are realized in practice. In weak fields, the polarization attraction is the main potential determining the elastic interaction of an electron with an atom. Because the scattering cross section is inversely proportional to the velocity in this case, we have the case of a constant collision frequency and vd w F. Strong fields in the interaction of electrons with atoms are dominated by hard-spheres collisions, i.e., the cross section is independent of velocity. In this case, vd w F1/2. In the derivation of the Druyvesteyn function, the collision cross section is assumed to be constant, and, then, in the final result, the cross section changes with a change in velocity. However, the derivation on the basis of the most complete distribution function in an analytical form is not possible and requires a numerical calculation.
190 PART | I Fundamentals
In all the above formulas, the drift velocity depends on the product eFL. It is clear that this is the energy accumulated by the electron along the mean free path. Because Lw1 p, where p is the pressure, the measure of this energy is the ratio F/p. That is why many parameters characterizing the behavior of electrons in gases are a function of F/p. In those cases, when studies are carried out with gases at elevated pressures, due to the compressibility of the gas, a simple connection between the mean free path and pressure disappears. Then, instead of the parameter F/p, one uses the parameter F/N, where N is the number of molecules per unit volume. For this parameter, a special unit (not included in the SI), called Townsend (Td) is introduced, 1 Td ¼ 1017 V$cm2 ¼ 3.03 V$cm∙mm Hg. Above, only elastic collisions of electrons were considered. However, inelastic collisions play an important role in molecular gases and in mixtures with molecular gases. The modification of the Boltzmann equation to account inelastic collisions was carried out by Holstein. The Holstein integro-differential equation requires a numerical solution. In Fig. 10.1, the dependence of the electron drift velocity on the parameter F/p in some gases is shown. It can be seen that the drift velocity can reach saturation or even decrease with increasing F/p. Qualitatively, this can be understood from the simplest expression for vd, Eq. (10.30), if we take into account that L is the function of the average electron energy and depends on the electric field strength. The actual form of the dependence of the drift velocity on the field is determined by the function L ¼ f (F). In particular, a known method for substantially increasing the drift velocity in noble gases is based on the addition of a molecular impurity. In this case, the dependence L on vav is used. In Fig. 5.26, the dependence of the electron scattering cross section of noble gas atoms on energy is shown. The nonmonotonic character of this dependence is called the Ramsauer effect. In strong fields, the electron energy is set in the region of the curve maximum near 10 eV. The addition of a molecular gas sharply reduces the energy of the electrons due to inelastic collisions that excite the vibrational levels of the molecules. As the energy decreases, the collision cross section also decreases and, consequently, the mean free path increases, and with it the drift velocity also increases. From the simplest expressions for mobility and the diffusion coefficient m ¼ eL mv; D ¼ vL 3; (10.33) the relation can be composed D 2 mv2 ¼ . m 3e 2
(10.34)
It is seen that this ratio is proportional to the energy of the electrons, but not equal to any of the set of possible values (probable, average, or mean square). Therefore, the parameter D/m is sometimes called the characteristic energy of the electrons. Thus, simultaneous knowledge of mobility and the diffusion coefficient makes it possible to determine the average electron energy. In the case when the energy distribution of electrons is Maxwellian, the characteristic energy equals mv2 2 ¼ 3kT 2, and then D=m ¼ kT=e.
(10.35)
In scientific literature, the relation (10.35) is called the NernsteEinstein formula.
FIGURE 10.1 The dependence of the electron drift velocity on the parameter F/p in some gases. 1dAr þ 10% CH4; 2dAr þ 1% N2; and 3dpure Ar.
Diffusion and Drift of Charges Chapter | 10
191
If the energy distribution of the electrons differs from the Maxwellian, then mv2 2 ¼ h3kT 2 and D=m ¼ hkT=e;
(10.36)
where h is the Townsend energy multiplier.
10.4 MOTION OF ELECTRONS AND HOLES IN SEMICONDUCTORS 10.4.1 Features of the Motion of Electrons and Holes in Semiconductors In solids, insulators and semiconductors, as a result of ionization, electrons and holes are formed. The mean free path, i.e., the motion between collisions of an electron in a gas under normal conditions is 105 cm in order of magnitude, and the size of an atom with which an electron collides is 1000 times smaller, i.e., 108 cm. Thus, we can assume that in gases an electron after collision with an atom moves in empty space until the next collision. In solids, atoms practically touch each other, so in a crystal the periodic lattice field acts on the electron, as a result, it acquires some properties that radically distinguish it from the classical particle. The electron passes through a sequence of potential barriers and potential wells, generally randomly oriented relative to the direction of electron motion. It is extremely difficult to describe such a movement. But it turns out that all these difficulties can be ignored if we assume that the charge moves as a free one, but with some effective mass different from the mass of the free electron. An analogue of an ion in a solid is a hole, a specific formation that can be represented as a method for describing the collective motion of electrons in the valence band. This charge motion must be attributed by a negative effective mass. In the process of thermalization, electrons in semiconductors drop to the bottom of the conduction band. Holes, like particles with negative effective mass, float to the ceiling of the valence band. Because the behavior of both electrons and holes in solids differs markedly from the behavior of charges in gases, they are often called charge carriers. In semiconductors in a much wider range of field strengths than in gases, the mobility of electrons and holes does not depend on the field strength. However, even here, as the field increases, the dependence deviates from linearity, gradually becomes more shallow, and in strong fields goes to saturation. The dependence of the drift velocity on the electric field strength is shown in Fig. 10.2. It is important that in semiconductors the charge carriers of both signs have comparable mobilities and, therefore, comparable values of the drift velocity.
10.4.2 Effective Mass, Polaron, and Autolocalization of Holes It is possible to consider that the motion of electrons and holes in the corresponding zones under the action of an electric field occurs as if they move freely, but their effective mass is different from that of the free electrons. Quantum mechanics makes it possible to obtain an expression for the effective mass of charge carriers m* m ¼ Z2 d2 E=dk 2 ; (10.37) where E(k) is the dispersion law.
FIGURE 10.2 The dependence of the drift velocity on the electric field strength for electrons and holes. Solid linedelectrons in Si at 300K; dotted linedelectrons in Si at 77K; and dashed linedholes in Ge at 80K.
192 PART | I Fundamentals
TABLE 10.3 Effective Mass of Electrons and Holes in Some Semiconductors in the Unit of the Mass of a Free Electron Substance
Effective Mass of an Electron
Effective Mass of a Hole
Si
1.06
0.56
Ge
0.22
0.39
Ga As
0.07
0.45
InSb
0.01
0.5
For a free particle, the dispersion law is quadratic, and thus the effective mass is constant and equal to the rest mass. In a crystal, the situation is more complicated and the dispersion law differs from a quadratic one. It is important that the effective mass depends on the direction of motion along the field relative to the crystallographic axes of the crystal and is, in general, a tensor. The values of the effective mass of electrons and holes for some semiconductors are given in Table 10.3. In any substance, electrons polarize surrounding atoms or molecules and, therefore, appear in the field of polarization. In matter, electrons move along with this field, which significantly affects the dynamics of their movement. This effect is most clearly manifested in crystals with a significant proportion of the ionic bond. Due to the Coulomb interaction, electrons attract positive ions, repel negative ions, and find themselves in a deep potential well. The measurements showed that in the ionic crystals, the holes are self-localized, and the electrons forced to move with the field of polarized ions become very heavy and slow. Such electrons are called polarons. Their dynamic characteristics are determined by the effective mass, which is noticeably greater than the mass of a free electron. In substances with a smaller fraction of the ionicity of the bond and even in purely covalent ones, polarization of the charge environment still occurs. Even in liquid helium, a gas bubble is formed around electrons due to the strong electroneatom exchange repulsion, and a sphere with an increased density or possibly solid helium (i.e., Atkins’ snowball) around the positive Heþ (or Heþ 2 ) ion is formed [2]. The described process can be characterized either by the depth of the polaron well or by the effective mass of the polaron. For a classical polaron of small radius, the so-called VK center in alkali-halide crystals, it is 0.2e0.6 eV. In radiation chemistry, the process of getting an electron into a potential well as a result of the polarization of surrounding molecules is called solvation, and the electron is solvated. If the substance is water, a solvated electron is called hydrated one.
10.5 THE DEBYE SCREENING RADIUS In the previous sections, the motion of charges in an external electric field was considered. However, at high intensity of the irradiation, in gas discharge plasma and in electrolytes, there is a high concentration of charges of both signs, which form a space charge. In the electric field of the space charge, the motion of individual charges has important features, on which we now concentrate. The electric field of an individual charge, in a neutral medium consisting of oppositely charged particles, is screened by surrounding charges and therefore its action extends over a certain limited distance. The characteristic exponent of this distance is called the Debye screening radius. Let us obtain an expression for the Debye radius. The change in the potential 4 of the electric field in a quasi-neutral medium, containing positive ions at a concentration of ni and electrons at a concentration ne, is described by the Poisson equation D4 ¼
eðni ne Þ . εε0
(10.38)
We assume that the change in the concentration of charges relative to the equilibrium distribution can be determined only by thermal motion. Then the charge concentrations obey the Boltzmann distribution ni ¼ n0 expðe4=kTÞ; ne ¼ n0 expðe4=kTÞ;
(10.39)
where n0 is the equilibrium concentration of charges in a quasi-neutral medium. Substituting Eq. (10.39) into the Poisson Eq. (10.38), we obtain
Diffusion and Drift of Charges Chapter | 10
D4 ¼
2n0 e e4 sh . εε0 kT
Under the condition e4 kT, sh(e4/kT) z e4/kT. Then 2n0 e2 D4 ¼ 4 . εε0 kT
193
(10.40)
(10.41)
The coefficient on the right-hand side of Eq. (10.41) has the dimension of the square of the reciprocal length. We introduce the notation, the Debye screening radius rD rffiffiffiffiffiffiffiffiffiffiffiffi εε0 kT rD ¼ . (10.42) 2n0 e2 Then Eq. (10.41) takes a view
D4 ¼ 4 rD2 .
Because by definition the field strength is F ¼ d4/dr, the solution of Eq. (10.43) gives r ; F ¼ F0 exp rD where F0 is the field strength at the interface of charges. Thus, rD is the characteristic distance at which charge separation is possible. If the temperatures of ions and electrons differ significantly, then
2 1=2 n0 e 1 1 rD ¼ þ εε0 kTi kTe and if, besides, Te [ Ti, then rD ¼
rffiffiffiffiffiffiffiffiffiffiffiffi εε0 kT . n0 e 2
(10.43)
(10.44)
(10.45)
(10.46)
On irradiation, positive and negative charges are formed in the substance. If this substance is in an electric field, then charges are taken by the field in opposite directions. At high irradiation intensity, these charges screen an external electric field and a situation similar to the considered one is obtained. In this case, the main reason for the separation of charges is not the thermal motion, but the external electric field. The characteristic distance, at which the external field falls off due to screening by separated charges, we obtain from Eq. (10.46), substituting the value e4 instead of kT rffiffiffiffiffiffiffiffiffiffiffiffi εε0 e4 rD ¼ . (10.47) n0 e 2
10.6 CRITERION OF THE SPACE CHARGE FORMATION At low-intensity irradiation of a substance with ionizing radiation, the electric field created by the charges generated by the radiation is negligible compared to the external field and the motion of each elementary charge occurs independent in the external field. With high-intensity irradiation, the electric field of the separated charges is significant, compared to the external one. Such conditions are called space charge conditions. It is customary to introduce the concept of critical charge, or charge concentration, whose value determines the transition from the normal (ohmic) regime to the regime of a space charge. We consider the volume of the irradiated substance with the relative dielectric permittivity ε, enclosed between the flat electrodes with area S and separation d. An estimation of the magnitude of the critical charge can be approached in several ways. 1. The role of the space charge is clearly manifested, when the charge Qcrit, formed in the material, becomes comparable with the charge Qc located on the capacitor C, formed by the chamber electrodes, and held by the potential difference 40 Qcrit ¼ Qc ¼ C40 .
(10.48)
194 PART | I Fundamentals
Because for a flat capacitor filled with matter, the capacitance is C ¼ εε0 S=d;
(10.49)
where ε0 is the electrical constant, then we obtain Qlr ¼ εε0 S40 =d.
(10.50)
2. The space charge starts to play an important role when the Debye screening radius rD, Eq. (10.47), becomes comparable with any characteristic size, e.g., with the distance between the electrodes d rD ¼ d.
(10.51)
qcrit ¼ Qcrit =Sd
(10.52)
After simple transformations, considering that
we obtain the same relation (10.50). 3. The space charge starts to play an important role when the transit time of electrons between the electrodes (cathode and anode) T becomes equal to the dielectric relaxation time tM (the Maxwell time constant) of the equilibrium insulator tM ¼ T.
(10.53)
The Maxwell time constant is RC of matter, where R is the volume resistance of the substance and C is its capacity. It is known that the resistance of a sample is R ¼ rd/S, where r ¼ d/qm40 is the resistivity. Then R ¼ d2/qm40S. The sample capacity is C ¼ εε0S/d. Time of flight T ¼ d2/m40. Substituting these expressions into condition (10.53), we obtain the same relation (10.50). 4. Another estimation of the critical charge can be obtained by solving the Poisson equation for a charge of the same sign, uniformly distributed between electrodes: d2 4 dx2 ¼ q=εε0 (10.54) with the boundary conditions on the electrodes 4ð0Þ ¼ 0; 4ðdÞ ¼ 40 .
(10.55)
An analytic solution of this equation with a uniformly distributed charge density q independent of x makes it possible to obtain values of the field strength at the electrodes Eð0Þ ¼
40 q d 4 q d ; EðdÞ ¼ 0 þ . d εε0 2 d εε0 2
(10.56)
The electric field, at least in one point of the chamber (on the surface of the anode, if the charge is positive), becomes zero at 40 q d . ¼ εε0 2 d
(10.57)
Passing from the charge concentration to the charge value, we obtain Qcrit ¼ 2εε0 S40 =d;
(10.58)
which coincides with formula (10.50) up to a factor of 2. While defining this criterion, the coefficient 2 is not very important. However, the last conclusion more clearly shows the physical meaning of the concept of “critical charge.” If there is a critical charge in the chamber, the electric field at any point of the chamber has not yet dropped to zero, but it is already noticeably different from the initial 40/d. We also note that the field in one point of the chamber (at the anode surface, if the charge is positive) is zero, when the charge in the chamber is equal to twice the critical one only for a uniformly distributed charge. In other cases, the field reaches zero value in the formation of a space charge with a larger total charge. Thus, for example, if the positive charge varies linearly from the anode to the cathode, then the field on the surface of the anode becomes zero, when Q ¼ 6Qcrit. We note that the expression obtained for the critical charge (10.50) is a function of the potential difference between the electrodes 40. The minimum value of the critical charge can be obtained using the relation 40 ¼ kT/e Qmin crit ¼ εε0 SkT=de.
(10.59)
Diffusion and Drift of Charges Chapter | 10
We also note that the critical concentration decreases with the increase of the electrode separation qlr ¼ Qlr V ¼ εε0 40 d2 ¼ εε0 F=d.
195
(10.60)
The obtained expression for the critical concentration (10.60), where the concentration value is inversely proportional to the distance between the electrodes, can cause some perplexity. Thus, it is easy to see that for d w 7.5 km, the critical concentration is equal to one electron in m3 (at 40 ¼ 1 V). Let us clarify this point. The condition (10.60) is obtained for a charge uniformly distributed between the electrodes. The quantum nature of the electric charge makes it possible to use the concept of a uniformly distributed charge only for a sufficiently large number of electrons. In addition, the ratio for the critical charge is obtained for a homogeneous electric field. This means that for d ¼ 7.5 km, the area of electrodes S must be of the order of 10 km. It is reasonable to assume that the arguments used in deriving condition (10.50) are valid for S w d2. Then Qcrit ¼ εε0d40, and the critical charge increases with increasing linear dimensions.
10.7 CURRENT LIMITED BY A SPACE CHARGE When space charge is formed, the space charge limited current (SCLC) is recorded. A wide range of information has been accumulated on the issues of the SCLC, and, to date, the basic regularities of this phenomenon have been firmly established. The most characteristic manifestation of SCLC is the transition of a linear dependence of the current on the voltage into the nonlineardquadratic for solids or to the law of “three-halves power” for vacuum devices (ChildeLangmuir law). Because i ¼ Q/T, then using (10.50) and T ¼ d/v, we get that in the case of SCLC the current depends on the voltage according to the formula i ¼ εε0 Sv40 d2 ; (10.61) where v is the electron velocity. Then the quadratic dependence is obtained, if the carrier velocity in solids is linear in the field v ¼ mE ¼ m40/d: (10.62) i ¼ εε0 Sm d3 420 ; and the law of three-halves power is obtained, if electrons move in vacuum uniformly accelerated under the action of a constant potential v ¼ (2e 40/m)1/2 h pffiffiffiffiffiffiffiffiffiffiffi. i 3=2 i ¼ εε0 S 2e=m d2 40 . (10.63) In liquid noble gases, the dependence of the drift velocity on the field strength goes to saturation, and in liquid xenon the saturation region occupies an especially wide range of field strength. This means that in liquid noble gases and in the case of SCLC, a linear dependence can be observed. Another obvious manifestation of SCLC is the absence of an explicit dependence of the current on the intensity of the charge generation source. Thus, in the space charge regime (not in saturation mode), the current of the vacuum diode does not depend on the cathode heating current, the current to the probe in the probe methods of the gas discharge test does not depend on the charge density at the probe location, the current at intense irradiation (under stationary conditions) should not depend on the absorbed dose. We note that these arguments are valid under stationary conditions, when the space charge has already been formed. If the processes in the space charge formation stage are analyzed, when the currents flowing in the initial stages of the irradiation pulses make a significant contribution to the detected signal, even before the formation of the space charge, the detected signal can depend on the absorbed dose.
10.8 AMBIPOLAR DIFFUSION In the case, when the concentrations of positive and negative charges are large, so that the conditions of the space charge are created, their motion cannot be described by the known relationships. If a charge concentration gradient appears in the irradiated volume, then the charges will first diffuse in accordance with the diffusion laws. However, as a result, the electric neutrality is violated, the space charges and the electric field F caused by them appear. This field has a direction that slow down the charges moving faster (for example, electrons) and accelerate the charges moving more slowly (for example, ions). The diffusion flux density has a view nv ¼ D grad n;
(10.64)
196 PART | I Fundamentals
from which the expression for the rate of diffusion displacement can be written as v ¼ Dðgrad nÞ=n.
(10.65)
Summing up the drift in the resulting field and the diffusion motion, we obtain for the resultant velocities the expressions vþ ¼ ðDþ =nÞgrad n þ mþ F;
(10.66a)
v ¼ ðD =nÞgrad n þ m F.
(10.66b)
After the establishment of a stationary state, the motion of both types of charges is equalized vþ ¼ v ¼ va .
(10.67)
grad n Dþ m þ D mþ va ¼ . n mþ þ m
(10.68)
Then from (10.66), we find
Comparison of Eqs. (10.65) and (10.68) shows that the motion of charges in this case can be regarded as diffusion with a coefficient Da ¼
Dþ m þ D mþ . mþ þ m
(10.69)
Such a joint motion of the charges of both signs is known as ambipolar diffusion.
REFERENCES [1] L.G.H. Huxley, R.W. Crompton, The Diffusion and Drift of Electrons in Gases, Wiley, New York, 1974, 669 p. [2] A.G. Khrapak, W.F. Schmidt, E. Illenberger, Localized electrons, holes and ions. Ch.7, in: W.F. Schmidt, E. Illenberger (Eds.), Electronic Excitations in Liquified Rare Gases, ASP, 2005, pp. 239e273.
Diffusion and Drift of Charges Chapter Outline 10.1 10.2 10.3 10.4
Diffusion of Ions and Electrons The Drift of Ions Electron Drift Motion of Electrons and Holes in Semiconductors 10.4.1 Features of the Motion of Electrons and Holes in Semiconductors
185 187 188 191 191
10.4.2 Effective Mass, Polaron, and Autolocalization of Holes 10.5 The Debye Screening Radius 10.6 Criterion of the Space Charge Formation 10.7 Current Limited by a Space Charge 10.8 Ambipolar Diffusion References
191 192 193 195 195 196
10.1 DIFFUSION OF IONS AND ELECTRONS In the absence of an electric field, charges in matter can undergo diffusion displacements. The basic diffusion equation, the Fick’s equation, shows the relationship between the particle flux density nv and the gradient of their concentration: nv ¼ D grad n;
(10.1)
where D is the diffusion coefficient. The diffusion flux of particles obeys the continuity equation dn=dt þ gradðnvÞ ¼ 0:
(10.2)
Using Eqs. (10.1) and (10.2), one can obtain the nonstationary diffusion equation vn v2 n ¼ D 2. vt vr
(10.3)
If there are n0 particles at the origin and at the instant t ¼ 0 they begin to diffuse through the gas, then for the particle density at time t from the solution of Eq. (4.3) we get: in the one-dimensional case n0 x2 nðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi exp ; 4Dt 4pDt
(10.4)
n0 r2 exp ; nðr; tÞ ¼ 2Dt 4Dt
(10.5)
in the two-dimensional case
in the three-dimensional case
r2 exp . nðr; tÞ ¼ 3=2 4Dt ð4pDtÞ n0
Radiation. https://doi.org/10.1016/B978-0-444-63979-0.00010-0 Copyright © 2019 Elsevier B.V. All rights reserved.
(10.6)
185
186 PART | I Fundamentals
From these distribution functions, one can find the mean square displacement of a particle from the origin: in the one-dimensional case pffiffiffiffi pffiffiffiffiffiffiffiffi x2 ¼ 2Dt ;
(10.7)
pffiffiffiffi pffiffiffiffiffiffiffiffi r 2 ¼ 4Dt ;
(10.8)
pffiffiffiffi pffiffiffiffiffiffiffiffi r 2 ¼ 6Dt .
(10.9)
in the two-dimensional case
in the three-dimensional case
For the three-dimensional case, we point out the quantity of the average displacement pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ 12Dt=p.
(10.10)
Using the obtained relations, one can find the average time during which the particles diffuse by a distance b: in spherical geometry t ¼
b2 . Dp2
(10.11)
in cylindrical geometry t ¼ In the kinetic theory of gases, it is shown that
b2 Dð2:405Þ2
.
(10.12)
D ¼ vL 3:
(10.13)
Expression (10.13) is approximate and agrees with the experimental data only in order of magnitude. The correct values of the diffusion coefficient can be calculated by using the charge distribution function in terms of energies or velocities. In Section 10.3, this problem will be observed in detail. Self-diffusion coefficients in gases are as follows: D(N2) ¼ 2.12$105 m2/s; D(Ar) ¼ 1.57$105 m2/s; and D (CH4) ¼ 2.0$105 m2/s. The diffusion coefficients of positive Dþ and negative De ions in some gases are given in Table 10.1. In liquids, the diffusion coefficient is several orders of magnitude lower than in gases at atmospheric pressure: in nonviscous liquids at 20 C, it has the order of 109 m2/s (in gases, it is [1e10]$105 m2/s). In particular, for air in liquid water D ¼ 2.0$109 m2/s. The diffusion coefficients in organic solids have a considerable spread. The diffusion coefficients of gases in polymers, in particular, depend on many properties of diffusing molecules and polymers, but mainly on the free volume of the polymer (i.e., the difference between the actual volume and the total volume of densely packed molecules) and the heterogeneity of its structure. D(O2 in polyethylene) ¼ 4.6$1011 m2/s; D(N2 in polymethylsiloxane) ¼ 94$1011 m2/s.
TABLE 10.1 The Diffusion Coefficients of Positive DD and Negative De Ions in Some Gases at t [ 298 K and p [ 105 Pa Gas
Air
N2
CO2
H2
O2
Dþ, 106 m2/s
2.8
3.9
2.3
12.3
2.6
De, 106 m2/s
4.3
4.1
2.6
19
3.9
De/Dþ
1.54
1.41
1.13
1.54
1.58
Diffusion and Drift of Charges Chapter | 10
187
If the diffusion occurs in an electric field, then the coefficients of longitudinal diffusion Djj and isotropic (transverse) Dt differ. We indicate a qualitative explanation of this difference. When the cloud of charges moves in an electric field, then the faster charges move ahead of the cloud and behind it the slower ones. Further analysis is determined by the form of the dependence of the scattering cross section of the charges and, consequently, of the drift velocity on the energy of the charges. If the cross section decreases with increasing energy, as is in the case for ions, then the mean free path, and with it the drift velocity, increases and, consequently, the faster charges move even faster, whereas the slower ones lag behind and the cloud of charges stretches along the field and, hence, Djj/Dt > 1. If the cross section increases with increasing energy, as is in the case for electrons in many gases, then on the contrary, the mean free path, and with it the drift velocity decreases with increasing energy; hence, the electrons moving ahead are slow, the ones moving behind accelerate their motion, the cloud contracts and, therefore, Djj/Dt < 1. For electrons in argon in the range E/p w 20e130 V/cm$Pa Djj/Dt w 1/5e1/7.
10.2 THE DRIFT OF IONS In gases, liquids, and often even in solids, one must deal with two types of charges with noticeably different masses: with ions (positive or negative) and with electrons. Because of the difference in mass, the nature of the motion in the electric field for these two groups of charges is noticeably different. The ions, participating in the thermal motion, are simultaneously slightly shifted in the electric field. The ions easily give up the energy that they collect in an electric field in collisions over the mean free path, and, thus, ions do not accumulate ion energy. Under such conditions, the ion drift velocity is much lower than the rate of thermal motion. The drift velocity can be defined as the average displacement velocity in an electric field over the mean free path. Because the ion moves uniformly in the field over the mean free path, vd ¼
v1 þ v2 ; 2
(10.14)
where v1 is the velocity in the field immediately after the collision and v2 is the velocity at the end of the free path before the next collision. In accordance with the initial condition adopted above, by the collision, ion completely loses its velocity in the direction of the field acquired during acceleration by the field at the previous mean free path, i.e., v1 ¼ 0. Introducing a as the acceleration of the ion, we write v2 ¼ at=2;
(10.15) where t is the mean free time. Because a ¼ eF/M, F is the electric field strength, M is the ion mass, and t ¼ L v, then for the drift velocity we obtain the expression vd ¼
eFL . 2Mv
(10.16)
The formula (10.16) is obtained by very simple means. The distribution of free ranges and velocities is considered simply by using their average values. If the distribution of free ranges is more rigorously taken into account, then the drift velocity is found to be twice as large: vd ¼
eFL ; Mv
(10.17)
and simultaneously considering the distribution of velocities, we obtain vd ¼
0; 815eFL pffiffiffiffi . M v2
(10.18)
Formulas (10.16)e(10.18), basically, correctly determine the form of the functional dependence, but one cannot expect quantitative conformity with experiment. The correct values of the drift velocity can be obtained by more rigorous theory, in particular, the ion energy distribution function. The expression (10.17) can be reshaped into the following form: vd ¼ mF;
(10.19)
188 PART | I Fundamentals
TABLE 10.2 Mobility of Some Ions in Their Own Gases Under Normal Conditions m, cm2/s V
HeD
NeD
NeD 2
ArD
ArD 2
KrD
KrD 2
XeD
XeD 2
ND 2
OD 2
OL 2
10.8
4.2
6.5
1.5
1.8
0.9
1.2
0.6
0.85
2.5
2.3
2.0
where m is the proportionality coefficient, called mobility m ¼
eL . Mv
(10.20)
In Table 10.2, the mobility of some ions in their own gases under normal conditions is shown. Generally speaking, the lower the velocity of ion drifts, the larger is their mass. However, this is not so for ions in their own gases. In this case, the charge exchange can perform þ Xþ fast þ Xslow /Xfast þ Xslow .
(10.21)
When foreign ions move in the gas and, in particular, the molecular ions Xþ 2 in a noble gas, the probability of charge exchange appears much less. Because of this, the mobility of molecular ions is higher than that of atomic ions, as can be seen from Table 10.2. Assuming that the mobility of the ions is of the order of 1 cm2/s$V, the ion drift velocity in a typical field of w103 V/cm is 103 cm/s. As is known, the thermal velocity of molecules in gases under normal conditions is w105 cm/s. Thus, the ion drift velocity actually appears much less than the velocity of thermal motion, as was suggested in deriving the expressions (10.16e10.18).
10.3 ELECTRON DRIFT The patterns of ion motion in the electric field, discussed in Section 10.2, are also valid for electrons in very weak fields (<1 V/cm). In large fields, electrons accumulate appreciable energy that is much higher than the energy of thermal motion, due to the fact that in the collision they transfer only a small part of their energy to molecules. The mobility of electrons in a gas does not depend on the field strength, only by the condition that the velocity distribution of electrons is Maxwellian. Due to the accumulation of energy, the energy distribution of electrons ceases to be Maxwellian and the simple relation (10.19) ceases to be satisfied. When discussing diffusion and drift of ions, the calculation of such macroscopic parameters as drift and diffusion, starting from the elementary microscopic parameters of collision processes, requires knowledge of the particle velocity distribution function. This is due to the fact that the probability (frequency, cross section) of the collision is a definite function of the velocity of the colliding particles. The average value obtained in measurements is the result of averaging over the entire spectrum of velocities. The analysis of ion drift at the simplest level yields reasonable results. In the case of electrons, it is necessary to involve deeper representations using the distribution function [1]. When the electron gas is in thermal equilibrium with the medium, the electrons are characterized by the Maxwellian energy distribution sffiffiffiffiffiffiffiffiffiffiffiffiffiffi E E f ðEÞdE ¼ 2 exp . (10.22) 3 kT pðkTÞ In the electric field, the distribution function somewhat changes. If the field is weak and the anisotropy of the electron velocity distribution is small, the change can be considered in the form of a series expansion. Using the expansion in a series of the Legendre polynomials, the distribution function can be written in the form f ðvÞ ¼ f0 þ f1 ðvÞcosq;
(10.23)
where q is the angle between the positive direction of the x axis and the direction of the electric field. f0(v) is the isotropic part of the function, it remains Maxwellian, and the second term of Eq. (10.23) defines the variation of the function under the action of the field, and, besides, this change is weak.
Diffusion and Drift of Charges Chapter | 10
189
The distribution function can be found as the solution of the Boltzmann kinetic equation. Without performing this equation, we remind only that it contains the cross sections and the type of collisions. The solution of the kinetic equation is the Druyvesteyn distribution " # 4; 56 m 3=4 E 1=2 3m E2 f ðEÞdE ¼ (10.24) dE. 3=2 exp Gð3=4Þ M M eFL 2 eFL where G is the gamma function, with all other parameters determined before. Wherein eFL vf0 . f1 ðvÞ ¼ me v vv
(10.25)
Knowledge of the distribution function makes it possible to calculate the basic parameters of the motion of electrons: average energy R N 3=2 E f0 ðEÞdE ; E ¼ R0N 1=2 E f0 ðEÞdE 0
(10.26)
drift velocity 1 vd ¼ 3
rffiffiffiffiffiffi R N Ef1 ðE ÞdE 2 R N0 ; me 0 E 1=2 f0 ðEÞdE
(10.27)
diffusion coefficient R E rffiffiffiffiffiffi N f0 ðEÞdE 0 1 2 sðEÞ RN D ¼ . 3N me 0 E 1=2 f0 ðEÞdE
(10.28)
The specific result depends on what dependence of the collision frequency of the electron with the atoms n ¼ Nsv ¼ Cvn is taken in the calculations. Substituting Eqs. (10.24) and (10.25) into Eqs. (10.26) and (10.27) and assuming that the collision frequency does not depend on the velocity (n ¼ 0, n ¼ C), we get 2 eFL E ¼ . (10.29) 2mv2 vd ¼
eFL . me v
(10.30)
In the other limiting case, when the collision cross section of an electron is independent of the velocity (n ¼ 1, C ¼ 1/Lav), we find E ¼ 0; 427ðM=me ÞeFL; 1=2 eFL . vd ¼ 0; 897 1=4 ðMme Þ
(10.31) (10.32)
The two limiting cases, considered above, are realized in practice. In weak fields, the polarization attraction is the main potential determining the elastic interaction of an electron with an atom. Because the scattering cross section is inversely proportional to the velocity in this case, we have the case of a constant collision frequency and vd w F. Strong fields in the interaction of electrons with atoms are dominated by hard-spheres collisions, i.e., the cross section is independent of velocity. In this case, vd w F1/2. In the derivation of the Druyvesteyn function, the collision cross section is assumed to be constant, and, then, in the final result, the cross section changes with a change in velocity. However, the derivation on the basis of the most complete distribution function in an analytical form is not possible and requires a numerical calculation.
190 PART | I Fundamentals
In all the above formulas, the drift velocity depends on the product eFL. It is clear that this is the energy accumulated by the electron along the mean free path. Because Lw1 p, where p is the pressure, the measure of this energy is the ratio F/p. That is why many parameters characterizing the behavior of electrons in gases are a function of F/p. In those cases, when studies are carried out with gases at elevated pressures, due to the compressibility of the gas, a simple connection between the mean free path and pressure disappears. Then, instead of the parameter F/p, one uses the parameter F/N, where N is the number of molecules per unit volume. For this parameter, a special unit (not included in the SI), called Townsend (Td) is introduced, 1 Td ¼ 1017 V$cm2 ¼ 3.03 V$cm∙mm Hg. Above, only elastic collisions of electrons were considered. However, inelastic collisions play an important role in molecular gases and in mixtures with molecular gases. The modification of the Boltzmann equation to account inelastic collisions was carried out by Holstein. The Holstein integro-differential equation requires a numerical solution. In Fig. 10.1, the dependence of the electron drift velocity on the parameter F/p in some gases is shown. It can be seen that the drift velocity can reach saturation or even decrease with increasing F/p. Qualitatively, this can be understood from the simplest expression for vd, Eq. (10.30), if we take into account that L is the function of the average electron energy and depends on the electric field strength. The actual form of the dependence of the drift velocity on the field is determined by the function L ¼ f (F). In particular, a known method for substantially increasing the drift velocity in noble gases is based on the addition of a molecular impurity. In this case, the dependence L on vav is used. In Fig. 5.26, the dependence of the electron scattering cross section of noble gas atoms on energy is shown. The nonmonotonic character of this dependence is called the Ramsauer effect. In strong fields, the electron energy is set in the region of the curve maximum near 10 eV. The addition of a molecular gas sharply reduces the energy of the electrons due to inelastic collisions that excite the vibrational levels of the molecules. As the energy decreases, the collision cross section also decreases and, consequently, the mean free path increases, and with it the drift velocity also increases. From the simplest expressions for mobility and the diffusion coefficient m ¼ eL mv; D ¼ vL 3; (10.33) the relation can be composed D 2 mv2 ¼ . m 3e 2
(10.34)
It is seen that this ratio is proportional to the energy of the electrons, but not equal to any of the set of possible values (probable, average, or mean square). Therefore, the parameter D/m is sometimes called the characteristic energy of the electrons. Thus, simultaneous knowledge of mobility and the diffusion coefficient makes it possible to determine the average electron energy. In the case when the energy distribution of electrons is Maxwellian, the characteristic energy equals mv2 2 ¼ 3kT 2, and then D=m ¼ kT=e.
(10.35)
In scientific literature, the relation (10.35) is called the NernsteEinstein formula.
FIGURE 10.1 The dependence of the electron drift velocity on the parameter F/p in some gases. 1dAr þ 10% CH4; 2dAr þ 1% N2; and 3dpure Ar.
Diffusion and Drift of Charges Chapter | 10
191
If the energy distribution of the electrons differs from the Maxwellian, then mv2 2 ¼ h3kT 2 and D=m ¼ hkT=e;
(10.36)
where h is the Townsend energy multiplier.
10.4 MOTION OF ELECTRONS AND HOLES IN SEMICONDUCTORS 10.4.1 Features of the Motion of Electrons and Holes in Semiconductors In solids, insulators and semiconductors, as a result of ionization, electrons and holes are formed. The mean free path, i.e., the motion between collisions of an electron in a gas under normal conditions is 105 cm in order of magnitude, and the size of an atom with which an electron collides is 1000 times smaller, i.e., 108 cm. Thus, we can assume that in gases an electron after collision with an atom moves in empty space until the next collision. In solids, atoms practically touch each other, so in a crystal the periodic lattice field acts on the electron, as a result, it acquires some properties that radically distinguish it from the classical particle. The electron passes through a sequence of potential barriers and potential wells, generally randomly oriented relative to the direction of electron motion. It is extremely difficult to describe such a movement. But it turns out that all these difficulties can be ignored if we assume that the charge moves as a free one, but with some effective mass different from the mass of the free electron. An analogue of an ion in a solid is a hole, a specific formation that can be represented as a method for describing the collective motion of electrons in the valence band. This charge motion must be attributed by a negative effective mass. In the process of thermalization, electrons in semiconductors drop to the bottom of the conduction band. Holes, like particles with negative effective mass, float to the ceiling of the valence band. Because the behavior of both electrons and holes in solids differs markedly from the behavior of charges in gases, they are often called charge carriers. In semiconductors in a much wider range of field strengths than in gases, the mobility of electrons and holes does not depend on the field strength. However, even here, as the field increases, the dependence deviates from linearity, gradually becomes more shallow, and in strong fields goes to saturation. The dependence of the drift velocity on the electric field strength is shown in Fig. 10.2. It is important that in semiconductors the charge carriers of both signs have comparable mobilities and, therefore, comparable values of the drift velocity.
10.4.2 Effective Mass, Polaron, and Autolocalization of Holes It is possible to consider that the motion of electrons and holes in the corresponding zones under the action of an electric field occurs as if they move freely, but their effective mass is different from that of the free electrons. Quantum mechanics makes it possible to obtain an expression for the effective mass of charge carriers m* m ¼ Z2 d2 E=dk 2 ; (10.37) where E(k) is the dispersion law.
FIGURE 10.2 The dependence of the drift velocity on the electric field strength for electrons and holes. Solid linedelectrons in Si at 300K; dotted linedelectrons in Si at 77K; and dashed linedholes in Ge at 80K.
192 PART | I Fundamentals
TABLE 10.3 Effective Mass of Electrons and Holes in Some Semiconductors in the Unit of the Mass of a Free Electron Substance
Effective Mass of an Electron
Effective Mass of a Hole
Si
1.06
0.56
Ge
0.22
0.39
Ga As
0.07
0.45
InSb
0.01
0.5
For a free particle, the dispersion law is quadratic, and thus the effective mass is constant and equal to the rest mass. In a crystal, the situation is more complicated and the dispersion law differs from a quadratic one. It is important that the effective mass depends on the direction of motion along the field relative to the crystallographic axes of the crystal and is, in general, a tensor. The values of the effective mass of electrons and holes for some semiconductors are given in Table 10.3. In any substance, electrons polarize surrounding atoms or molecules and, therefore, appear in the field of polarization. In matter, electrons move along with this field, which significantly affects the dynamics of their movement. This effect is most clearly manifested in crystals with a significant proportion of the ionic bond. Due to the Coulomb interaction, electrons attract positive ions, repel negative ions, and find themselves in a deep potential well. The measurements showed that in the ionic crystals, the holes are self-localized, and the electrons forced to move with the field of polarized ions become very heavy and slow. Such electrons are called polarons. Their dynamic characteristics are determined by the effective mass, which is noticeably greater than the mass of a free electron. In substances with a smaller fraction of the ionicity of the bond and even in purely covalent ones, polarization of the charge environment still occurs. Even in liquid helium, a gas bubble is formed around electrons due to the strong electroneatom exchange repulsion, and a sphere with an increased density or possibly solid helium (i.e., Atkins’ snowball) around the positive Heþ (or Heþ 2 ) ion is formed [2]. The described process can be characterized either by the depth of the polaron well or by the effective mass of the polaron. For a classical polaron of small radius, the so-called VK center in alkali-halide crystals, it is 0.2e0.6 eV. In radiation chemistry, the process of getting an electron into a potential well as a result of the polarization of surrounding molecules is called solvation, and the electron is solvated. If the substance is water, a solvated electron is called hydrated one.
10.5 THE DEBYE SCREENING RADIUS In the previous sections, the motion of charges in an external electric field was considered. However, at high intensity of the irradiation, in gas discharge plasma and in electrolytes, there is a high concentration of charges of both signs, which form a space charge. In the electric field of the space charge, the motion of individual charges has important features, on which we now concentrate. The electric field of an individual charge, in a neutral medium consisting of oppositely charged particles, is screened by surrounding charges and therefore its action extends over a certain limited distance. The characteristic exponent of this distance is called the Debye screening radius. Let us obtain an expression for the Debye radius. The change in the potential 4 of the electric field in a quasi-neutral medium, containing positive ions at a concentration of ni and electrons at a concentration ne, is described by the Poisson equation D4 ¼
eðni ne Þ . εε0
(10.38)
We assume that the change in the concentration of charges relative to the equilibrium distribution can be determined only by thermal motion. Then the charge concentrations obey the Boltzmann distribution ni ¼ n0 expðe4=kTÞ; ne ¼ n0 expðe4=kTÞ;
(10.39)
where n0 is the equilibrium concentration of charges in a quasi-neutral medium. Substituting Eq. (10.39) into the Poisson Eq. (10.38), we obtain
Diffusion and Drift of Charges Chapter | 10
D4 ¼
2n0 e e4 sh . εε0 kT
Under the condition e4 kT, sh(e4/kT) z e4/kT. Then 2n0 e2 D4 ¼ 4 . εε0 kT
193
(10.40)
(10.41)
The coefficient on the right-hand side of Eq. (10.41) has the dimension of the square of the reciprocal length. We introduce the notation, the Debye screening radius rD rffiffiffiffiffiffiffiffiffiffiffiffi εε0 kT rD ¼ . (10.42) 2n0 e2 Then Eq. (10.41) takes a view
D4 ¼ 4 rD2 .
Because by definition the field strength is F ¼ d4/dr, the solution of Eq. (10.43) gives r ; F ¼ F0 exp rD where F0 is the field strength at the interface of charges. Thus, rD is the characteristic distance at which charge separation is possible. If the temperatures of ions and electrons differ significantly, then
2 1=2 n0 e 1 1 rD ¼ þ εε0 kTi kTe and if, besides, Te [ Ti, then rD ¼
rffiffiffiffiffiffiffiffiffiffiffiffi εε0 kT . n0 e 2
(10.43)
(10.44)
(10.45)
(10.46)
On irradiation, positive and negative charges are formed in the substance. If this substance is in an electric field, then charges are taken by the field in opposite directions. At high irradiation intensity, these charges screen an external electric field and a situation similar to the considered one is obtained. In this case, the main reason for the separation of charges is not the thermal motion, but the external electric field. The characteristic distance, at which the external field falls off due to screening by separated charges, we obtain from Eq. (10.46), substituting the value e4 instead of kT rffiffiffiffiffiffiffiffiffiffiffiffi εε0 e4 rD ¼ . (10.47) n0 e 2
10.6 CRITERION OF THE SPACE CHARGE FORMATION At low-intensity irradiation of a substance with ionizing radiation, the electric field created by the charges generated by the radiation is negligible compared to the external field and the motion of each elementary charge occurs independent in the external field. With high-intensity irradiation, the electric field of the separated charges is significant, compared to the external one. Such conditions are called space charge conditions. It is customary to introduce the concept of critical charge, or charge concentration, whose value determines the transition from the normal (ohmic) regime to the regime of a space charge. We consider the volume of the irradiated substance with the relative dielectric permittivity ε, enclosed between the flat electrodes with area S and separation d. An estimation of the magnitude of the critical charge can be approached in several ways. 1. The role of the space charge is clearly manifested, when the charge Qcrit, formed in the material, becomes comparable with the charge Qc located on the capacitor C, formed by the chamber electrodes, and held by the potential difference 40 Qcrit ¼ Qc ¼ C40 .
(10.48)
194 PART | I Fundamentals
Because for a flat capacitor filled with matter, the capacitance is C ¼ εε0 S=d;
(10.49)
where ε0 is the electrical constant, then we obtain Qlr ¼ εε0 S40 =d.
(10.50)
2. The space charge starts to play an important role when the Debye screening radius rD, Eq. (10.47), becomes comparable with any characteristic size, e.g., with the distance between the electrodes d rD ¼ d.
(10.51)
qcrit ¼ Qcrit =Sd
(10.52)
After simple transformations, considering that
we obtain the same relation (10.50). 3. The space charge starts to play an important role when the transit time of electrons between the electrodes (cathode and anode) T becomes equal to the dielectric relaxation time tM (the Maxwell time constant) of the equilibrium insulator tM ¼ T.
(10.53)
The Maxwell time constant is RC of matter, where R is the volume resistance of the substance and C is its capacity. It is known that the resistance of a sample is R ¼ rd/S, where r ¼ d/qm40 is the resistivity. Then R ¼ d2/qm40S. The sample capacity is C ¼ εε0S/d. Time of flight T ¼ d2/m40. Substituting these expressions into condition (10.53), we obtain the same relation (10.50). 4. Another estimation of the critical charge can be obtained by solving the Poisson equation for a charge of the same sign, uniformly distributed between electrodes: d2 4 dx2 ¼ q=εε0 (10.54) with the boundary conditions on the electrodes 4ð0Þ ¼ 0; 4ðdÞ ¼ 40 .
(10.55)
An analytic solution of this equation with a uniformly distributed charge density q independent of x makes it possible to obtain values of the field strength at the electrodes Eð0Þ ¼
40 q d 4 q d ; EðdÞ ¼ 0 þ . d εε0 2 d εε0 2
(10.56)
The electric field, at least in one point of the chamber (on the surface of the anode, if the charge is positive), becomes zero at 40 q d . ¼ εε0 2 d
(10.57)
Passing from the charge concentration to the charge value, we obtain Qcrit ¼ 2εε0 S40 =d;
(10.58)
which coincides with formula (10.50) up to a factor of 2. While defining this criterion, the coefficient 2 is not very important. However, the last conclusion more clearly shows the physical meaning of the concept of “critical charge.” If there is a critical charge in the chamber, the electric field at any point of the chamber has not yet dropped to zero, but it is already noticeably different from the initial 40/d. We also note that the field in one point of the chamber (at the anode surface, if the charge is positive) is zero, when the charge in the chamber is equal to twice the critical one only for a uniformly distributed charge. In other cases, the field reaches zero value in the formation of a space charge with a larger total charge. Thus, for example, if the positive charge varies linearly from the anode to the cathode, then the field on the surface of the anode becomes zero, when Q ¼ 6Qcrit. We note that the expression obtained for the critical charge (10.50) is a function of the potential difference between the electrodes 40. The minimum value of the critical charge can be obtained using the relation 40 ¼ kT/e Qmin crit ¼ εε0 SkT=de.
(10.59)
Diffusion and Drift of Charges Chapter | 10
We also note that the critical concentration decreases with the increase of the electrode separation qlr ¼ Qlr V ¼ εε0 40 d2 ¼ εε0 F=d.
195
(10.60)
The obtained expression for the critical concentration (10.60), where the concentration value is inversely proportional to the distance between the electrodes, can cause some perplexity. Thus, it is easy to see that for d w 7.5 km, the critical concentration is equal to one electron in m3 (at 40 ¼ 1 V). Let us clarify this point. The condition (10.60) is obtained for a charge uniformly distributed between the electrodes. The quantum nature of the electric charge makes it possible to use the concept of a uniformly distributed charge only for a sufficiently large number of electrons. In addition, the ratio for the critical charge is obtained for a homogeneous electric field. This means that for d ¼ 7.5 km, the area of electrodes S must be of the order of 10 km. It is reasonable to assume that the arguments used in deriving condition (10.50) are valid for S w d2. Then Qcrit ¼ εε0d40, and the critical charge increases with increasing linear dimensions.
10.7 CURRENT LIMITED BY A SPACE CHARGE When space charge is formed, the space charge limited current (SCLC) is recorded. A wide range of information has been accumulated on the issues of the SCLC, and, to date, the basic regularities of this phenomenon have been firmly established. The most characteristic manifestation of SCLC is the transition of a linear dependence of the current on the voltage into the nonlineardquadratic for solids or to the law of “three-halves power” for vacuum devices (ChildeLangmuir law). Because i ¼ Q/T, then using (10.50) and T ¼ d/v, we get that in the case of SCLC the current depends on the voltage according to the formula i ¼ εε0 Sv40 d2 ; (10.61) where v is the electron velocity. Then the quadratic dependence is obtained, if the carrier velocity in solids is linear in the field v ¼ mE ¼ m40/d: (10.62) i ¼ εε0 Sm d3 420 ; and the law of three-halves power is obtained, if electrons move in vacuum uniformly accelerated under the action of a constant potential v ¼ (2e 40/m)1/2 h pffiffiffiffiffiffiffiffiffiffiffi. i 3=2 i ¼ εε0 S 2e=m d2 40 . (10.63) In liquid noble gases, the dependence of the drift velocity on the field strength goes to saturation, and in liquid xenon the saturation region occupies an especially wide range of field strength. This means that in liquid noble gases and in the case of SCLC, a linear dependence can be observed. Another obvious manifestation of SCLC is the absence of an explicit dependence of the current on the intensity of the charge generation source. Thus, in the space charge regime (not in saturation mode), the current of the vacuum diode does not depend on the cathode heating current, the current to the probe in the probe methods of the gas discharge test does not depend on the charge density at the probe location, the current at intense irradiation (under stationary conditions) should not depend on the absorbed dose. We note that these arguments are valid under stationary conditions, when the space charge has already been formed. If the processes in the space charge formation stage are analyzed, when the currents flowing in the initial stages of the irradiation pulses make a significant contribution to the detected signal, even before the formation of the space charge, the detected signal can depend on the absorbed dose.
10.8 AMBIPOLAR DIFFUSION In the case, when the concentrations of positive and negative charges are large, so that the conditions of the space charge are created, their motion cannot be described by the known relationships. If a charge concentration gradient appears in the irradiated volume, then the charges will first diffuse in accordance with the diffusion laws. However, as a result, the electric neutrality is violated, the space charges and the electric field F caused by them appear. This field has a direction that slow down the charges moving faster (for example, electrons) and accelerate the charges moving more slowly (for example, ions). The diffusion flux density has a view nv ¼ D grad n;
(10.64)
196 PART | I Fundamentals
from which the expression for the rate of diffusion displacement can be written as v ¼ Dðgrad nÞ=n.
(10.65)
Summing up the drift in the resulting field and the diffusion motion, we obtain for the resultant velocities the expressions vþ ¼ ðDþ =nÞgrad n þ mþ F;
(10.66a)
v ¼ ðD =nÞgrad n þ m F.
(10.66b)
After the establishment of a stationary state, the motion of both types of charges is equalized vþ ¼ v ¼ va .
(10.67)
grad n Dþ m þ D mþ va ¼ . n mþ þ m
(10.68)
Then from (10.66), we find
Comparison of Eqs. (10.65) and (10.68) shows that the motion of charges in this case can be regarded as diffusion with a coefficient Da ¼
Dþ m þ D mþ . mþ þ m
(10.69)
Such a joint motion of the charges of both signs is known as ambipolar diffusion.
REFERENCES [1] L.G.H. Huxley, R.W. Crompton, The Diffusion and Drift of Electrons in Gases, Wiley, New York, 1974, 669 p. [2] A.G. Khrapak, W.F. Schmidt, E. Illenberger, Localized electrons, holes and ions. Ch.7, in: W.F. Schmidt, E. Illenberger (Eds.), Electronic Excitations in Liquified Rare Gases, ASP, 2005, pp. 239e273.