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Interband light absorption at a rough interface
Physica E 5 (2000) 142–152
www.elsevier.nl/locate/physe
Interband light absorption at a rough interface L. Braginsky ∗ Institute of Semiconductor Physics, 630090 Novosibirsk, Russia Received 30 August 1999; accepted 26 October 1999
Abstract Light absorption at the boundary of indirect-band-gap and direct-forbidden gap semiconductors is analyzed. It is found that the possibility of the electron momentum nonconservation at the interface leads to essential enhancement of absorption in porous and microcrystalline semiconductors. The e ect is more pronounced at a rough boundary due to enlargement of the share of the interface atoms. ? 2000 Elsevier Science B.V. All rights reserved. PACS: 78.66.−w; 78.55.Mb Keywords: Porous and microcrystalline semiconductors; Quantum dots; Optical properties
1. Introduction It is well known that the direct interband electron transitions are the main mechanism of light absorption in pure semiconductors. These transitions are direct owing to the momentum conservation law for the excited electron. The momentum, which this electron obtains from the light wave (∼ ˜=0 , where 0 is the wavelength of the light), is small in comparison with the electron momentum in the crystal (∼ ˜=, where is the electron wavelength). It is clear, however, that the momentum is not conserved if the absorption takes place at the crystal boundary or at the interface between two crystals. The possibility of indirect electron transitions at the interface results in enhancement of the absorption. This means that considerable enhance∗ Tel.: +7-3832-333264; fax: +7-3832-332771. E-mail address: [email protected] (L. Braginsky)
ment of light absorption has to be expected in porous and microcrystalline semiconductors where the share of the interface atoms is suciently large. This interface mechanism of light absorption becomes most important if indirect electron transitions are more preferable than the direct ones. This happens, rst, in indirect-band-gap semiconductors where the electron transition to the side valley in the bulk should be accompanied by the electron–phonon or electron–impurity interaction. Second, this happens in direct-forbidden-band-gap semiconductors where the direct electron transitions between the top of the valence band and the bottom of the conduction band are prohibited. Both possibilities are considered in this paper. We investigate the frequency dependence of the absorption at the fundamental absorption edge and estimate the relative value of the interface absorption. In addition to the light absorption at a plane interface of
1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 2 5 6 - 8
L. Braginsky / Physica E 5 (2000) 142–152
indirect-band-gap semiconductors considered in Ref. [1], we also consider the interband light absorption at a rough interface. Such an interface is characteristic of the intercrystallite boundary in porous and microcrystalline semiconductors. 2. The model We consider the light absorption at the surface of the semiconductor quantum dot or the crystallite embedded in an insulator media. Suppose that each size of the crystallite considerably exceeds the lattice constant. On this assumption the envelope function approximation is valid. Moreover, this assumption allows us to consider the light absorption as inelastic scattering of the electron at the interface if the electron wavelength is suciently small. We introduce the Cartesian coordinate system, where the z-axis is normal to the interface, and assume that semiconductor occupies the region z ¡ 0. The probability for the photon to be absorbed in the crystallite is 2 9 (2˜)2 e2 P f i (c − v − ˜!); (1) = 2 me c!nS p;q 9z where ˜; e; me ; and c are the fundamental constants, ! is the photon frequency, n is the refraction index, S is the area of the crystallite side z = 0 where the absorption is considered, p and q are the electron momenta, p and q are their z components, c (q) and v (p) are the energies of the electron in the conduction and valence bands. The electric eld of the light is directed along the z-axis; ii and fi are the wave functions of the electron before the excitation (in the valence band) and after it (in the conduction band) correspondingly. Wave functions ii and fi are determined by the band structure of the crystallite and the boundary conditions for the envelope wave functions at the interface between the crystallite and the insulator. To consider the Coulomb interaction, which occurs between the excited electron and the hole in the valence band, we have to input the weight factor ( ) = exp =(sinh ) [where = (qaB =˜)−1 , and aB is the e ective Bohr radius] into sum (1) [2]. The Coulomb interaction becomes important if the electron and the hole are too close to the band extrema, so that their wavelengths are large. In this case the
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electron density at the place where the hole is situated is not small. It is possible to change summation in Eq. (1) by integration over the electron energy and parallel-to-interface components of the momentum q|| . We obtain =
e2 mc mh a2 (2)2 ˜4 m2e !cn Z ( )|Pvc |2 dc d 2 q|| q × q ; 2mh (˜! − c ) − q||2 2mc (c − Eg ) − q||2 (2)
where Pvc = −i
N Ns
9 f i ; 9z
mc and mh are the e ective masses of the electron in the conduction and valence bands respectively, N and Ns are the numbers of atoms in the crystallite and at the interface, a is the lattice constant, and Eg is the gap. The limits of integration are determined by the region where the expressions under the square roots in the integrand are positive. 3. Boundary conditions for the envelope wave function 3.1. Boundary condition at the plane interface 3.1.1. Interface of semiconductors with nondegenerate band structure Let z = 0 be the interface between semiconductor (z ¡ 0) and an insulator (z ¿ 0). For a simple nondegenerate electron spectrum the boundary conditions for the envelope wave function at the plane interface can be written in the form [1] ‘ ( 0‘ ) = b11 r ( 11 ); b21 ‘ ( 21 ) = r ( 0r );
(3)
where ‘ and r are the envelopes corresponding to the di erent, left (‘) or right (r), sides of the interface; |bik | ∼ 1 and | ik | ∼ a are the parameters of the boundary conditions, which relate the envelopes at the di erent sites ik at the interface. It is important that these sites are close to the interface. The generally accepted form of the boundary conditions [3] can be obtained from Eq. (3) if we expand ( ) at the interface
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L. Braginsky / Physica E 5 (2000) 142–152
z = 0 assuming | 0 |.1 (where 0 = 9 =9z). Then we nd ‘ r ˆ = T where Tˆ = ||tik ||; r0 ‘0 t11 =
b11 b21 11 − 0r ; b11 ( 11 − 0r )
t21 = −
b11 b21 − 1 ; b11 ( 11 − 0r )
t12 =
b11 b21 11 21 − 0r 0‘ ; b11 ( 11 − 0r )
t22 = −
b11 b21 21 − 0‘ : b11 ( 11 − 0r )
(4)
The values of bik , ik , and tik are independent of the electron energy and can be used as the phenomenological parameters. The determinant of the tik matrix should be equal to mr =m‘ (where mr and m‘ are e ective masses of the electron on each side of the interface) in order to ensure the conservation of the electron probability ux at the interface. It follows from Eq. (4) that, in general, t11 ∼ 1, t12 ∼ a, t21 ∼ a−1 , and t22 ∼ 1. To understand the physical meaning of this estimation, let us consider the electron scattering at the interface. The envelope wave function of the electron is ( ipz [e + Re−ipz ]eip|| ; z ¡ 0; = T e− z+ip|| ; z ¿ 0; where p|| is the parallel-to-interface component of the electron momentum, and is the coordinate vector in this (XY ) plane; (p. .a−1 ) is the decay exponent of the electron wave function in the insulator, R and T are the re ection and transmission coecients. We can use the boundary conditions (4) to obtain the re ection coecient 2ipt22 : (5) 1+R=− t21 + t11 − ipt22 The pole of this expression determines the energy E of the electron level, which separates from the band at the interface. We can rewrite Eq. (5) as follows: 2ip 1+R=− ; (6) − ip √ where = 2mE = (t21 + t11 )=t22 ' t21 =t22 is the decay exponent of the electron wave function of the interface state. Eq. (6) allows to relate the value of t21 with the energy of the interface level. We see that the parameter t21 indicates the value of separation of the interface level from the band extremum. Indeed, for
Fig. 1. The two ways of the light absorption at the indirect-band-gap semiconductor interface: the immediate electron transition to the side valley Ps , and the vertical transition Pc followed by the conversion to the side valley (dotted arrow). Ev and Ec are the interface levels.
t21 ∼ a−1 E ∼ ˜2 =(2ma2 ), i.e., the separation is of the order of the bandwidth. However, the interface level becomes close to the band extremum when t21 → 0. 3.1.2. Interface of semiconductors with degenerate band structure The boundary conditions for the envelope wave function becomes more complicated if the intervalley or interband degeneracy occurs in the band structure of the semiconductor. For simplicity, let us consider the two-valley conduction band presented in Fig. 1. The boundary conditions for the electron in this band can be written as follows [1]: 1 ( 01 ) = c11 r ( 11 ); 2 ( 02 ) = c22 r ( 22 ); c31 1 ( 31 ) + c32 2 ( 32 ) = r ( 03 );
(7)
where 1 and 2 are the envelopes corresponding to degenerate conduction band in the left semiconductor, and parameters cik ∼ 1 and | | ∼ a are independent of the electron energy. The probability ux conservation holds at the interface for arbitrary electron energies. To ensure this, the parameters of the boundary conditions
L. Braginsky / Physica E 5 (2000) 142–152
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(7) must satisfy the following relations: c11 ( 11 − 03 ) c31 ( 31 − 01 ) ; =− m1 m c22 ( 11 − 03 ) c32 ( 31 − 02 ) ; =− m2 m 11 = 22 ;
(8)
where m1 and m2 are the e ective masses of the electron in the central and side valleys of the conduction band of the semiconductor, and m is the e ective mass in the insulator. An important simpli cation arises when the large band o set occurs at the contact of a two-valley semiconductor and an insulator. If so, then r ˙ exp(− r z), where the r value can be considered as independent of the electron energy. Eliminating r from Eqs. (7), we nd 1 ( ˜11 ) + c˜12 2 ( ˜12 ) = 0; c˜21 1 ( ˜21 ) + 2 ( ˜22 ) = 0;
(9)
where c˜ij ∼ cij and ˜ij ∼ ij are known functions of cij , ij , and r . The boundary conditions (9) look quite similar to Eq. (3). Therefore, Eq. (4) holds in this case too. This allows us to use Eqs. (3) and (4) to consider the light absorption also at the interface of indirect-band-gap semiconductor. The more general boundary conditions that are applicable at an interface with a large band o set has been proposed in Ref. [4]. Our Eq. (9) holds in the e ective mass approximation. This approximation has been used in Ref. [1] to obtain Eqs. (3) and (7) and estimate parameters cij and ij . 3.2. Boundary conditions at a rough interface Nonlocal form of the boundary conditions (3) can be used to obtain the boundary conditions for the envelope wave function at a rough interface. We consider the special form of a rough interface that is presented in Fig. 2. The interface looks like an array of the plane areas of the same crystallographic orientation. The random function z = (r) of the coordinates in XY plane determines the positions of these areas relative to z = 0. We assume (r) = 0, so that z = 0 is the average interface plane. We suppose the average height of roughnesses to be small in comparison with the electron wave-
Fig. 2. The model of the rough interface: side view.
length. Then it is possible to describe the rough interface by means of the correlation function W (r 0 ; r00 ) = (r0 )(r00 ). For the homogeneous rough interface W (r 0 ; r00 ) = W (r 0 − r00 ), i.e., the correlation function is the function of one variable: = r 0 − r00 . There are two parameters that are most important when the statistical properties of a rough interface are considered: 2 = W (0) and the correlation length l – the mean attenuation length of the correlation function. In our model the correlation length l can be associated with the mean size of the plane area (Fig. 2). We shall analyze the relation between the light absorption and the parameters and l. The special form of the rough interface (Fig. 2) allows us to apply the boundary conditions (3), which are applicable at a plane interface, at each plane z = . Note that these boundary conditions are not applicable at the vicinity of the corner points (like point 1 in Fig. 2). We assume the mean size of the plane areas to be large in comparison with the lattice constant, so that the relative number of the corner points is small. To obtain the boundary conditions at the rough interface, it is necessary to expand the envelopes in Eq. (3) at z = (r), instead of z = 0. This means that values in Eq. (4) should be replaced by + , so that the matrix of the boundary conditions Tˆ becomes of the form t − t21 t12 + (t11 − t22 ) − t21 2 ; (10) Tˆ = 11 t21 t22 + t21 where tik are the components of the boundary conditions matrix (4) at the plane interface. It is important that now the boundary condition matrix depends on r. This results in the di use components of the wave functions. We write the envelopes
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L. Braginsky / Physica E 5 (2000) 142–152
as the sum of their average ‘; r and di use ’‘; r components [5]: ‘; r
= ‘; r + ’‘; r
where
‘; r
= ‘; r ;
’‘; r = 0: (11)
To obtain the boundary conditions for ‘; r and ’‘; r , we substitute the envelopes (11) into Eq. (4) using the boundary conditions matrix (10). We also have to average these equations and subtract the average equations from the initial ones. Then we obtain r = t11 ‘ + t12 ‘0 − t21 2 ‘0 −t21 ’‘ + (t11 − t22 )’0‘ ; r0 = t21 ‘ + t22 ‘0 + t21 ’0‘ ; ’r = t11 ’‘ + t12 ’0‘ − t21 ‘ +(t11 − t22 )‘0 − t21 2 ’0‘ ; ’0r = t21 ’‘ + t22 ’0‘ + t21 =‘0 :
(12)
The values 2 − 2 , ’‘ − ’‘ , and ’0‘ − ’0‘ are omitted; they are small if =l.1. We use Eqs. (12) to consider the electron scattering at the rough interface. The envelope wave function can be written in the form ‘
‘ r
‘
‘
‘
= ei(p|| +pz z) + Rei(p|| −pz z) ; = Te
r i(p|| +pzr z)
’r (r) = kz =
p
P k
P k
;
where W () = (r + )(r) is the correlation function of the rough interface, and W˜ is its Fourier transform Z W˜ (k) = W ()e−ik d 2 : For the di use component ’‘ (r) we obtain Z 2pz‘ A(kz )(k − p|| )ei(k−kz z) d 2 k; ’‘ (r) = (2)2 where
kz t21 (t˜22 − ipzr t˜12 ) + (t˜21 − ipzr t˜11 )[kz (t11 − t22 ) + it21 ] : [t21 − ikz (t11 + t22 ) − kz2 t12 ][t˜21 − ipzr t˜11 − ipz‘ t˜22 − pz‘ pzr t˜12 ]
’˜ ‘ (k)ei(k−kz z) ; ’˜ r (k)ei(k+kz z) :
2m − k2 ;
t˜11 = t11 + it21 Z kz [t21 + ikz (t11 − t22 )] ˜ × W (k − p|| ) d 2 k; t21 − ikz (t11 + t22 ) − kz2 t12 t˜12 = t12 − 2 t21 Z 2 + kz2 (t11 − t22 )2 ] [t21 + W˜ (k − p|| ) d 2 k; t21 − ikz (t11 + t22 ) − kz2 t12 2 t˜21 = t21 − t21 Z kz2 W˜ (k − p|| ) d 2 k; × t21 − ikz (t11 + t22 ) − kz2 t12 t˜22 = t22 + it21 Z kz [t21 − ikz (t11 − t22 )] ˜ × W (k − p|| ) d 2 k; t21 − ikz (t11 + t22 ) − kz2 t12 (14)
(15)
A(kz ) = − ’‘ (r) =
then from rst two equations (12). These equations accept the form of Eq. (4) if we introduce the e ective parameters t˜ik of the boundary conditions as follows:
(13)
where is the electron energy, pz;‘;||r and kz and k are z and XY components of the electron momenta. It follows from Eq. (13) that p||‘ = p||r = p|| . We use last two Eqs. (12) to express the di use components ’‘ and ’r as functions of , R, and T . The re ection (R) and transmission (T ) coecients can be obtained
We should emphasize the two-fold in uence of the interface roughness. First, the roughness of the interface results in the di use component of the scattered wave. This component is always small for the long wavelength electrons and vanish when → ∞. Second, the e ective parameters of the boundary conditions depend on the interface roughness. This e ect depends on the relation between the electron wavelength and the correlate length of the rough interface. Indeed, we can assume W˜ (k − p|| ) ∼ 2 (k − k0 ) if .l. Then the corrections to the e ective parameters of the boundary conditions (14), caused by the roughnesses, become proportional to 2 =(l) and vanish when → ∞.
L. Braginsky / Physica E 5 (2000) 142–152
Corrections to tik become independent of when /l. This is clear from Eq. (14) because in this case |p|| |.|k|. It is easy to understand the reason. Formally, the roughnesses, the size of which is less then the electron wavelength, could be taken into account when the parameters bik and ik of the boundary conditions are obtained. In this consideration the roughnesses a ect the value of these parameters but do not make them dependent on [1]. It follows from Eq. (14) that -dependent corrections to t˜ik are of the order of (l=)2 ; this is in agreement with estimations [1]. It should be noted that values of t˜ik are not real at a rough interface. The reason for this is the diffuse component of the scattered wave that spoils the time-inversion symmetry.
4. Interband light absorption at a plane interface 4.1. Interface light absorption in direct-forbidden gap semiconductors Let us consider the light absorption at the interface between semiconductor (z ¡ 0) and an insulator (z ¿ 0). The band structure of the semiconductor is presented in Fig. 3. We can write the electron wave
147
functions as follows: 1 ii = √ N ( [vp (r)eipz + Rv vp∗ (r)e−ipz ]eip|| ; z ¡ 0; z ¿ 0; Tv e− v z+ip|| ; 1 fi = √ N ( ∗ [uq (r)e−iqz + Rc uq (r)eiqz ]eiq|| ; z ¡ 0; Tc e− c z+iq|| ;
z ¿ 0;
(16)
where uq (r) and vp (r) are the Bloch amplitudes in the conduction and valence band respectively; c:v are decay exponents of the wave functions apart o the interface; Rv ; Rc ; Tv , and Tc are the re ection and transmission coecients. It is possible to relate these values with positions of the interface levels (see Eq. (6)) 2ip 2iq ; 1 + Rc = ; (17) v − ip c + iq p √ where v = 2mh Ev and c = 2mc (Eg − Ec ). In direct-forbidden gap semiconductors u0 (r) = 0. We assume uq (r) = 2i sin(qa=2) wq (r), where wq (r) is the periodic function of r that does not vanish at q = 0. Then for the interband matrix element Pvc we obtain
1 + Rv = −
Ns (1 + Rv )(1 + R∗c )U p|| ;q|| ; N
Pvc = where Z U=
0
w0
9v0 3 d r; 9z
(18)
so that |Pvc |2 =
Fig. 3. Schematic band structure of the direct-forbidden gap semiconductor. Here Ev and Ec are the interface levels. Arrows indicate the two ways in which light absorption takes place: direct electron transition 1 → 2, and indirect 3 → 4 that occurs at the interface.
4U 2 Ns2 (c − Eg )(! − c ) p ;q : N 2 (c − Ec )(! − c + Ev ) || ||
We see that energy dependence of |Pvc |2 is sensitive to the position of the interface levels Ev and Ec , whether or not they are close to the corresponding band extrema. If we assume the Coulomb exponent ( ) as independent of q, then the result of integration (2) can be written as follows: ˙ (˜! − Eg ) ;
(19)
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L. Braginsky / Physica E 5 (2000) 142–152
where 3 if ˜! − Eg .min(Ev ; Eg − Ec ); 2 if min(Ev ; Eg − Ec ).˜! − Eg = .max(Ev ; Ec ); 1 if ˜! − Eg /max(Ev ; Eg − Ec ): 4.2. Interface light absorption in indirect-band-gap semiconductors The band structure of the semiconductor is presented in Fig. 1. For simplicity, we assume the nondegenerate valence band. There are two valleys in the conduction band: the central valley with the minimum at the center of Brillouin zone, and the side valley at the edge of it. It is important that the side valley of the conduction band is situated in the direction of the normal to the interface. We write the electron wave functions as follows: 1 (20) ii = √ [v(r)eipz + Rv v∗ (r)e−ipz ]eip|| ; N 1 fi = √ [Rc uc (r)ez + Rs us∗ (r)ei(q−=a)z N + us (r)e−i(q−=a)z ]eiq|| ; where v; uc , and us are the Bloch amplitudes in the valence band, the central and side valleys of the conduction band; p, , and q are z components of the wave numbers in these valleys. The coecients Rv ; Rc , and Rs are determined by the boundary conditions for the envelopes. For the interband matrix element Pvc we nd [1] 4pq 1 CPc + v + Ps ; Pvc = (c + iq)(v − ip) a 2 + p2 2 Z Pc = −i˜
0
Z Ps = −i˜
0
u0∗
9v 3 d r; 9z
uq∗
9v 3 d r; 9z
The energy dependence of |Pvc |2 (for p.) is the same as Eq. (18); therefore, |Pc |2 ˙ (˜! − Eg ) |Ps |2 + ; (22) (a)2 where is determined by Eq. (19), and is the dimensionless parameter, which has arisen from the rst term of expression (21). Two terms in the square brackets of Eq. (22) can be interpreted as follows: the rst term corresponds to the immediate transition of the electron to the side valley at the interface, while the second term corresponds to the excitation of the electron to the virtual interface state of the central valley followed by conversion to the side valley; it prevails if the valleys; minima are close (a.1). The Coulomb interaction between the electron in the conduction band and hole in the valence bands can a ect the frequency dependence of the absorption. Indeed, ( ) is independent of q only if qaB /˜. Otherwise, if qaB .˜, it is proportional to q−1 ; in this case the exponent changes, so that → − 1=2 when ˜! − Eg . e4 =(22 ˜2 ) (where −1 2 −1 = m−1 c + mh ; = n ). Thus, the frequency dependence of the absorption at the fundamental absorption edge essentially depends on the conditions at the interface, namely whether or not the interface electron levels are close to the corresponding bands extrema. 5. Light absorption at a rough interface According to Eq. (11), we write i
= i + ’i
where c and v are the electron energies in the conduction and valence bands, respectively, C is the coecient depending on the interface structure, and 0 is the unit cell.
f
= f + ’f ;
(23)
where i; f and ’i; f are the average and di use components of the corresponding wave functions: ( [vp (r)eipz + Rv vp∗ (r)e−ipz ]eip|| ; z ¡ 0; 1 i = √ N Tv e− v z+ip|| ; z ¿ 0; ’i (r) =
(21)
and
2p √ (2)2 N Z × vk (r)Av (kz )(k − p|| )ei(k−kz z) d 2 k;
1 f = √ N
(
[uq∗ (r)e−iqz
iqz
iq||
+ Rc uq (r)e ]e
Tc e− c z+iq|| ;
(24)
; z¡0 z ¿ 0;
L. Braginsky / Physica E 5 (2000) 142–152
’f (r) =
2q √ (2)2 N Z × uk∗ (r)Ac (kz )(k − q|| )ei(k+kz z) d 2 k;
where Av and Ac are the coecients A(kz ) (15) for the valence and conduction bands respectively. By substituting the wave functions (24) into Eq. (16), we express the matrix element Pvc as follows: (1) (2) (3) + Pvc + Pvc ; Pvc = Pvc
where (1) Pvc
∗ ˆ f H int i
=
(2) = Pvc
and (3) Pvc
Z
Z [ Z
=
dV;
∗ ˆ f H int ’i
(25)
+ ’∗f Hˆ int i ] dV;
’∗f Hˆ int ’i dV:
We have to obtain the square module of Eq. (25) and then to average it over the realizations of the random function (). For |Pvc |2 we have (1) 2 |Pvc |2 = |Pvc | +
(2) 2 |Pvc |
+
(1) (3)∗ Pvc Pvc
+
∗
(3) (1) Pvc Pvc :
From (25) and (25) we obtain Ns (1) Pvc = (1 + Rv )(1 + R∗c )U p|| ;q|| ; N (2) 2 |Pvc | =
(26a)
4Ns2 |U |2 W (p|| − q|| ) N 2S |p(1 + R∗c )Av (kz ) − q(1 + Rv )A∗c (kz )|2 (26b)
and Ns (3) Pvc = 4ipqU p|| ;q|| Z N × Av (kz )A∗c (kz )W (k|| − q|| ) d 2 k:
(26c)
Expressions (26) allow to estimate in uence of the roughness on the value and the frequency dependence (1) value is of the of the absorption. Note that the Pvc same form as that for the plane interface. The main di erence concerns parameters of the boundary conditions, Eq. (14), wherein the in uence of the interface roughness has been taken into account. This results
149
in the shift of the interface levels that manifests itself in the frequency dependence of the absorption. This e ect is most important if the interface electron levels are close to both, valence and conduction, bands or if the interface is smooth; the values of 1 + Rv and 1 + Rc are not small in these cases. Indeed, expressions (26b) and (26c) determine the in uence of the di use components of the scattered waves on the absorption; the values of these components are as small as =. On the contrary, this small parameter is absent in Eq. (26a) that determines the in uence of the average component (i.e., position of the interface electron level) on the absorption. However, Eq. (26a) is proportional to (1 + Rv )(1 + R∗c ). This value is small if the interface level is not close to any band. Then the values of Eqs. (26b) and (26c) may be of the same order or even exceed that of Eq. (26a). Comparing Eqs. (26b) and (26c) with Eq. (26a), we nd that the roughness in uence on the absorption is determined by the value =
2 pq2 ∼ |1 + Rc ||1 + Rv | 4 ∼
q
(v2 + p2 )(c2 + q2 )
2 q mc mh (˜! + Ev − Eg )(˜! − Ec ): 2
(27)
The correlation length l of the rough interface is also important for the light absorption. We can assume W˜ (k) = 2 (k) if l/. It can be shown that (2) 2 (1) (3)∗ (1)∗ (3) |Pvc | + Pvc Pvc + Pvc Pvc = 0 in this case. It is easy to understand the reason. Roughnesses, the mean length of which (Fig. 2) essentially exceeds the electron wavelength, could not a ect the electron properties of the interface. In the opposite limiting case l. the roughness in uence is determined by the term P(3) . Enhancement of the absorption in this case arises due to an increase of the number of the interface atoms Ns → Ns =a, in the vicinity of which the interband absorption with the momentum nonconservation occurs. The interesting situation arises when l ∼ . In this case the di use scattering that is described by the term (2) 2 (1) 2 |Pvc | ∼ c2 2 l2 [2mc|| (˜! − Eg )]|Pvc | leads to the change in the frequency dependence of the absorption. This means the rapid increase of absorption at the high frequencies where 2mc|| (˜! − Eg )l2 ∼ 1, so that exponent changes its value from → + 1.
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6. Discussion We considered the interband light absorption at the interface and found that possibility of the electron momentum nonconservation leads to enhancement of the absorption in the small microcrystallite. To estimate the value of the interface absorption, we have to compare the absorption of the microcrystalline solid obtained from the crystallites under consideration, = =L ˙ w=L , with the light absorption in the bulk semiconductor. Here L is length of the crystallite under consideration, w is the main size of the region at the interface where the interband electron transitions with the momentum nonconservation are possible. To estimate this value, we can use Eqs. (17) and (18): w ∼ if the interface electron levels are close to √ both (conduction and valence) bands extrema, w ∼ a if the interface level is close to any (conduction or valence) band extremum, and w ∼ a if both interface levels separate too far o the band extrema. The small value w=L characterizes the ratio of the number of atoms at this interface region to the number of atoms in the whole of solid. In order that the interface absorption becomes important in Ge and the indirect-band-gap semiconductors of the AIII BV group this small value should exceed the small parameter of electron–phonon interaction g (this value is about 10−3 –10−2 ). In these semiconductors the interface absorption essentially increases due to the intervalley conversion, which is determined by the second term in Eq. (22). That is possible if w=L(a)−2 ¿g. The main mechanism of light absorption in bulk Si is the impurity absorption. In order that the interface absorption becomes signi cant in this material, the number of the interface atoms should exceed the number of impurities. It seems possible that the indirect interband electron transitions at the interface considered here are responsible for an increase of luminescence of the porous Si (see Refs. [6,7] and references therein). Enhancement of the light absorption at the fundamental absorption edge has to be expected in direct-forbidden gap semiconductors, wherein the direct electron transitions between the band extrema are prohibited. The essential di erence in the e ective masses of the valence and conduction bands is favourable for the e ect. Such situation is character-
istic for transition metal oxides semiconductors [8,9]. The conduction band in these materials is composed mainly of d orbitals of the metal. These orbitals are strongly localized. For this reason the conduction band is narrow and even becomes at in a certain direction ( -M for TiO2 ). It is important that direct interband electron transition is dipole-forbidden in this material. There is two-fold advantage from the indirect electron transitions. First, they are allowed, i.e., the dipole matrix element of the indirect transitions is not small. Second advantage arises due to the large density of states for the electron in the conduction band. Let us compare the interband transitions in the bulk of semiconductor 1 → 2 and at the interface 3 → 4 (Fig. 3). The absorption is proportional to the density of electron states in each band. The density of states in the conduction band of TiO2 essentially exceeds that in the valence band. Nevertheless, it is the density of electron states in the valence band which determines the absorption in the bulk of crystal. This happens because of the law of conservation of momentum, which makes the electron states with the large momentums inaccessible for the excited electrons. On the contrary, indirect electron transitions make such states accessible. We can use Eq. (19) to estimate the value of absorption coecient of the microcrystalline direct-forbidden gap semiconductor: 2 2 w mc w Wv ∼ ∼ ; 0 L mh L Wc where 0 is the absorption coecient of the bulk monocrystal, Wv is width of the valence band (Wc ∼ 10 meV and Wv ' 1 eV for TiO2 [8,9]). Thus, the interface mechanism of the absorption becomes comparable with the bulk one for the microcrystalline TiO2 the mean size of the crystallite in which is L6w(Wv =Wc )2 ∼ 100 nm. Signi cant increase of optical absorption has been observed in the microcrystalline BaTiO3 [10]. We found that the absorption value and its frequency dependence at the fundamental absorption edge are sensitive to the structure of the interface. Thus, essential increase of the absorption has to be expected at a smooth interface. For such an interface, the main size of the interface region (i.e., the region at the interface where one crystal structure changes into another) considerably exceeds the lattice constant. It
L. Braginsky / Physica E 5 (2000) 142–152
is found that t21 = 0 at a smooth interface [11], so that c = v = 0, and the interband matrix element Pvc increases, because it no longer proportional to the small ratio (a=)2 [see Eqs. (17) and (18)]. Indeed, signi cant (in 100 times) enhancement of the light absorption has been observed at the interface a-Si=mc-Si [12]. The possible existence of the interface electron levels is essential for the optical properties of a sharp interface. Energy position of these levels depends not only on the bordering materials, but also on the interface itself. Structure of the interface as well as impurities and defects on it a ect positions of these levels. Their positions can be measured in optical experiments as the singularities in the absorption spectrum at the energies below the gap value. It seems that the electron interface level should be close to the valence band at least in wide-gap semiconductors. The interface level becomes empty when it is shifted too far from the top of the valence band. This results in a large surface charge and a strong band bending that is not favourable from the energetical point of view. Nevertheless, the interface level can be shifted as a result of structure reconstruction of the interface. This reconstruction does not essentially a ect the interatomic spacings or angles, but it makes the interface level closer to the top of the valence band. The roughness of the interface is one of the possible ways of such reconstruction. It follows from our consideration (Eq. (14)) that signi cant shift of the interface level occurs if the correlation length of rough interface is small (v l.˜). This causes the interface level to be closer to the band extremum. This is the particular case where the structure reconstruction of the interface decreases the interface energy. If so, then the rough interface becomes more favourable than the plane one. It seems strange that the interface roughnesses result only in the shift of the interface levels, but cannot be the cause for such levels. In other words, t˜21 in Eq. (14) vanishes for an appropriate correlation function W˜ (k), but it always equal to zero if t21 = 0. This is the result of our assumption that the interface is smooth (l/a). Perhaps, the interface level can be separated from the band extremum due to the interface roughnesses if l ∼ a. In this case the interface level arises as a result of the rearrangement of the chemical bonds at the rough interface.
Fig. 4. Interband sorption edge in the in uence of ˜2 (2mc|| l2 )−1 ; Ew
151
light absorption at the fundamental abthe crystallite. The dotted line indicates the interface roughness. Here El = Eg + = Eg + W c .
The value of (Eq. (27)) determines the roughnesses in uence on the absorption. This value is small, ∼ 2 =2 , if the interface electron levels are close to the extremum of both the bands. In this case the main size w of the region at the interface where the momentum nonconservation is possible is of the order of the electron wavelength , so that the small roughnesses cannot a ect the absorption. This is not the case if both interface levels shift far from the bands extrema; then w ∼ a and value is of about or even exceeds unity. The roughnesses in uence on the absorption is signi cant at the low frequencies when ˜!6Eg + ˜2 (2mc l2 )−1 ; enhancement of absorption in this case is due to an e ective increase of the number of the interface atoms (Ns → Ns =a) in the vicinity of which the interface light absorption occurs. The possible deviation of the absorption vs. energy curve at the fundamental absorption edge caused by the interface roughnesses is schematically presented in Fig. 4. In conclusion, we show that the possibility of the momentum nonconservation at the interface leads to enhancement of the interband light absorption in small crystallites, the size of which is about a few 10 nm. The interband absorption is sensitive to the interface electron levels, namely, whether or not they are close to the bands extrema. The in uence of the interface
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roughnesses is essential if these levels are not close to any or both conduction and valence bands. The e ect of the roughnesses is two-fold. First, they result in the di use electron scattering at the interface. This leads to enhancement of the absorption due to the e ective increase of the share of the interface atoms. Second, roughnesses result in e ective shift of the electron interface levels. Acknowledgements The author wishes to thank Prof. E. Ivchenko, who introduced him to the problem of in uence of the interface roughnesses on the absorption, Prof. V. Volkov and Prof. M. Entin for the helpful discussions. This work was supported by the Russian Foundation for the Basic Research, Grant No. 99-02-17019.
References [1] L. Braginsky, Phys. Rev. B 57 (1998) R6870; cond-mat=9705004. [2] R.J. Elliot, Phys. Rev. 108 (1957) 1384. [3] T. Ando, S. Wakahara, H. Akera, Phys. Rev. B 40 (1989) 11 609. [4] V.A. Volkov, T.N. Pinsker, Sov. Phys. JETP 72 (1977) XXX [Z. Exp. i Teor. Fiz. 72 (1977) 1087]. [5] F.G. Bass, I.M. Fuks, Wave Scattering from Statistically Rough Surfaces, Pergamon Press, Oxford, 1979. [6] D.J. Lockwood, Solid State Commun. 92 (1994) 101. [7] D. Schwall, F. Otter, J. Galligan, Philos. Mag. B 75 (1997) 887. [8] K. Glassford, J. Chelikowsky, Phys. Rev. B 46 (1992) 1284. [9] Shang-i Mo, W.Y. Ching, Phys. Rev. B 51 (1995) 13 023. [10] K. Yamada, S. Kohiki, Physica E 4 (1999) 228. [11] E.L. Ivchenko, G.E. Pikus, Superlattices and other Heterostructures. Symmetry and Optical Phenomena, 2nd Edition, Springer, Berlin, 1997. [12] N. Beck et al., J. Non-Cryst. Solids 198–200 (1996) 903.
www.elsevier.nl/locate/physe
Interband light absorption at a rough interface L. Braginsky ∗ Institute of Semiconductor Physics, 630090 Novosibirsk, Russia Received 30 August 1999; accepted 26 October 1999
Abstract Light absorption at the boundary of indirect-band-gap and direct-forbidden gap semiconductors is analyzed. It is found that the possibility of the electron momentum nonconservation at the interface leads to essential enhancement of absorption in porous and microcrystalline semiconductors. The e ect is more pronounced at a rough boundary due to enlargement of the share of the interface atoms. ? 2000 Elsevier Science B.V. All rights reserved. PACS: 78.66.−w; 78.55.Mb Keywords: Porous and microcrystalline semiconductors; Quantum dots; Optical properties
1. Introduction It is well known that the direct interband electron transitions are the main mechanism of light absorption in pure semiconductors. These transitions are direct owing to the momentum conservation law for the excited electron. The momentum, which this electron obtains from the light wave (∼ ˜=0 , where 0 is the wavelength of the light), is small in comparison with the electron momentum in the crystal (∼ ˜=, where is the electron wavelength). It is clear, however, that the momentum is not conserved if the absorption takes place at the crystal boundary or at the interface between two crystals. The possibility of indirect electron transitions at the interface results in enhancement of the absorption. This means that considerable enhance∗ Tel.: +7-3832-333264; fax: +7-3832-332771. E-mail address: [email protected] (L. Braginsky)
ment of light absorption has to be expected in porous and microcrystalline semiconductors where the share of the interface atoms is suciently large. This interface mechanism of light absorption becomes most important if indirect electron transitions are more preferable than the direct ones. This happens, rst, in indirect-band-gap semiconductors where the electron transition to the side valley in the bulk should be accompanied by the electron–phonon or electron–impurity interaction. Second, this happens in direct-forbidden-band-gap semiconductors where the direct electron transitions between the top of the valence band and the bottom of the conduction band are prohibited. Both possibilities are considered in this paper. We investigate the frequency dependence of the absorption at the fundamental absorption edge and estimate the relative value of the interface absorption. In addition to the light absorption at a plane interface of
1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 2 5 6 - 8
L. Braginsky / Physica E 5 (2000) 142–152
indirect-band-gap semiconductors considered in Ref. [1], we also consider the interband light absorption at a rough interface. Such an interface is characteristic of the intercrystallite boundary in porous and microcrystalline semiconductors. 2. The model We consider the light absorption at the surface of the semiconductor quantum dot or the crystallite embedded in an insulator media. Suppose that each size of the crystallite considerably exceeds the lattice constant. On this assumption the envelope function approximation is valid. Moreover, this assumption allows us to consider the light absorption as inelastic scattering of the electron at the interface if the electron wavelength is suciently small. We introduce the Cartesian coordinate system, where the z-axis is normal to the interface, and assume that semiconductor occupies the region z ¡ 0. The probability for the photon to be absorbed in the crystallite is 2 9 (2˜)2 e2 P f i (c − v − ˜!); (1) = 2 me c!nS p;q 9z where ˜; e; me ; and c are the fundamental constants, ! is the photon frequency, n is the refraction index, S is the area of the crystallite side z = 0 where the absorption is considered, p and q are the electron momenta, p and q are their z components, c (q) and v (p) are the energies of the electron in the conduction and valence bands. The electric eld of the light is directed along the z-axis; ii and fi are the wave functions of the electron before the excitation (in the valence band) and after it (in the conduction band) correspondingly. Wave functions ii and fi are determined by the band structure of the crystallite and the boundary conditions for the envelope wave functions at the interface between the crystallite and the insulator. To consider the Coulomb interaction, which occurs between the excited electron and the hole in the valence band, we have to input the weight factor ( ) = exp =(sinh ) [where = (qaB =˜)−1 , and aB is the e ective Bohr radius] into sum (1) [2]. The Coulomb interaction becomes important if the electron and the hole are too close to the band extrema, so that their wavelengths are large. In this case the
143
electron density at the place where the hole is situated is not small. It is possible to change summation in Eq. (1) by integration over the electron energy and parallel-to-interface components of the momentum q|| . We obtain =
e2 mc mh a2 (2)2 ˜4 m2e !cn Z ( )|Pvc |2 dc d 2 q|| q × q ; 2mh (˜! − c ) − q||2 2mc (c − Eg ) − q||2 (2)
where Pvc = −i
N Ns
9 f i ; 9z
mc and mh are the e ective masses of the electron in the conduction and valence bands respectively, N and Ns are the numbers of atoms in the crystallite and at the interface, a is the lattice constant, and Eg is the gap. The limits of integration are determined by the region where the expressions under the square roots in the integrand are positive. 3. Boundary conditions for the envelope wave function 3.1. Boundary condition at the plane interface 3.1.1. Interface of semiconductors with nondegenerate band structure Let z = 0 be the interface between semiconductor (z ¡ 0) and an insulator (z ¿ 0). For a simple nondegenerate electron spectrum the boundary conditions for the envelope wave function at the plane interface can be written in the form [1] ‘ ( 0‘ ) = b11 r ( 11 ); b21 ‘ ( 21 ) = r ( 0r );
(3)
where ‘ and r are the envelopes corresponding to the di erent, left (‘) or right (r), sides of the interface; |bik | ∼ 1 and | ik | ∼ a are the parameters of the boundary conditions, which relate the envelopes at the di erent sites ik at the interface. It is important that these sites are close to the interface. The generally accepted form of the boundary conditions [3] can be obtained from Eq. (3) if we expand ( ) at the interface
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z = 0 assuming | 0 |.1 (where 0 = 9 =9z). Then we nd ‘ r ˆ = T where Tˆ = ||tik ||; r0 ‘0 t11 =
b11 b21 11 − 0r ; b11 ( 11 − 0r )
t21 = −
b11 b21 − 1 ; b11 ( 11 − 0r )
t12 =
b11 b21 11 21 − 0r 0‘ ; b11 ( 11 − 0r )
t22 = −
b11 b21 21 − 0‘ : b11 ( 11 − 0r )
(4)
The values of bik , ik , and tik are independent of the electron energy and can be used as the phenomenological parameters. The determinant of the tik matrix should be equal to mr =m‘ (where mr and m‘ are e ective masses of the electron on each side of the interface) in order to ensure the conservation of the electron probability ux at the interface. It follows from Eq. (4) that, in general, t11 ∼ 1, t12 ∼ a, t21 ∼ a−1 , and t22 ∼ 1. To understand the physical meaning of this estimation, let us consider the electron scattering at the interface. The envelope wave function of the electron is ( ipz [e + Re−ipz ]eip|| ; z ¡ 0; = T e− z+ip|| ; z ¿ 0; where p|| is the parallel-to-interface component of the electron momentum, and is the coordinate vector in this (XY ) plane; (p. .a−1 ) is the decay exponent of the electron wave function in the insulator, R and T are the re ection and transmission coecients. We can use the boundary conditions (4) to obtain the re ection coecient 2ipt22 : (5) 1+R=− t21 + t11 − ipt22 The pole of this expression determines the energy E of the electron level, which separates from the band at the interface. We can rewrite Eq. (5) as follows: 2ip 1+R=− ; (6) − ip √ where = 2mE = (t21 + t11 )=t22 ' t21 =t22 is the decay exponent of the electron wave function of the interface state. Eq. (6) allows to relate the value of t21 with the energy of the interface level. We see that the parameter t21 indicates the value of separation of the interface level from the band extremum. Indeed, for
Fig. 1. The two ways of the light absorption at the indirect-band-gap semiconductor interface: the immediate electron transition to the side valley Ps , and the vertical transition Pc followed by the conversion to the side valley (dotted arrow). Ev and Ec are the interface levels.
t21 ∼ a−1 E ∼ ˜2 =(2ma2 ), i.e., the separation is of the order of the bandwidth. However, the interface level becomes close to the band extremum when t21 → 0. 3.1.2. Interface of semiconductors with degenerate band structure The boundary conditions for the envelope wave function becomes more complicated if the intervalley or interband degeneracy occurs in the band structure of the semiconductor. For simplicity, let us consider the two-valley conduction band presented in Fig. 1. The boundary conditions for the electron in this band can be written as follows [1]: 1 ( 01 ) = c11 r ( 11 ); 2 ( 02 ) = c22 r ( 22 ); c31 1 ( 31 ) + c32 2 ( 32 ) = r ( 03 );
(7)
where 1 and 2 are the envelopes corresponding to degenerate conduction band in the left semiconductor, and parameters cik ∼ 1 and | | ∼ a are independent of the electron energy. The probability ux conservation holds at the interface for arbitrary electron energies. To ensure this, the parameters of the boundary conditions
L. Braginsky / Physica E 5 (2000) 142–152
145
(7) must satisfy the following relations: c11 ( 11 − 03 ) c31 ( 31 − 01 ) ; =− m1 m c22 ( 11 − 03 ) c32 ( 31 − 02 ) ; =− m2 m 11 = 22 ;
(8)
where m1 and m2 are the e ective masses of the electron in the central and side valleys of the conduction band of the semiconductor, and m is the e ective mass in the insulator. An important simpli cation arises when the large band o set occurs at the contact of a two-valley semiconductor and an insulator. If so, then r ˙ exp(− r z), where the r value can be considered as independent of the electron energy. Eliminating r from Eqs. (7), we nd 1 ( ˜11 ) + c˜12 2 ( ˜12 ) = 0; c˜21 1 ( ˜21 ) + 2 ( ˜22 ) = 0;
(9)
where c˜ij ∼ cij and ˜ij ∼ ij are known functions of cij , ij , and r . The boundary conditions (9) look quite similar to Eq. (3). Therefore, Eq. (4) holds in this case too. This allows us to use Eqs. (3) and (4) to consider the light absorption also at the interface of indirect-band-gap semiconductor. The more general boundary conditions that are applicable at an interface with a large band o set has been proposed in Ref. [4]. Our Eq. (9) holds in the e ective mass approximation. This approximation has been used in Ref. [1] to obtain Eqs. (3) and (7) and estimate parameters cij and ij . 3.2. Boundary conditions at a rough interface Nonlocal form of the boundary conditions (3) can be used to obtain the boundary conditions for the envelope wave function at a rough interface. We consider the special form of a rough interface that is presented in Fig. 2. The interface looks like an array of the plane areas of the same crystallographic orientation. The random function z = (r) of the coordinates in XY plane determines the positions of these areas relative to z = 0. We assume (r) = 0, so that z = 0 is the average interface plane. We suppose the average height of roughnesses to be small in comparison with the electron wave-
Fig. 2. The model of the rough interface: side view.
length. Then it is possible to describe the rough interface by means of the correlation function W (r 0 ; r00 ) = (r0 )(r00 ). For the homogeneous rough interface W (r 0 ; r00 ) = W (r 0 − r00 ), i.e., the correlation function is the function of one variable: = r 0 − r00 . There are two parameters that are most important when the statistical properties of a rough interface are considered: 2 = W (0) and the correlation length l – the mean attenuation length of the correlation function. In our model the correlation length l can be associated with the mean size of the plane area (Fig. 2). We shall analyze the relation between the light absorption and the parameters and l. The special form of the rough interface (Fig. 2) allows us to apply the boundary conditions (3), which are applicable at a plane interface, at each plane z = . Note that these boundary conditions are not applicable at the vicinity of the corner points (like point 1 in Fig. 2). We assume the mean size of the plane areas to be large in comparison with the lattice constant, so that the relative number of the corner points is small. To obtain the boundary conditions at the rough interface, it is necessary to expand the envelopes in Eq. (3) at z = (r), instead of z = 0. This means that values in Eq. (4) should be replaced by + , so that the matrix of the boundary conditions Tˆ becomes of the form t − t21 t12 + (t11 − t22 ) − t21 2 ; (10) Tˆ = 11 t21 t22 + t21 where tik are the components of the boundary conditions matrix (4) at the plane interface. It is important that now the boundary condition matrix depends on r. This results in the di use components of the wave functions. We write the envelopes
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L. Braginsky / Physica E 5 (2000) 142–152
as the sum of their average ‘; r and di use ’‘; r components [5]: ‘; r
= ‘; r + ’‘; r
where
‘; r
= ‘; r ;
’‘; r = 0: (11)
To obtain the boundary conditions for ‘; r and ’‘; r , we substitute the envelopes (11) into Eq. (4) using the boundary conditions matrix (10). We also have to average these equations and subtract the average equations from the initial ones. Then we obtain r = t11 ‘ + t12 ‘0 − t21 2 ‘0 −t21 ’‘ + (t11 − t22 )’0‘ ; r0 = t21 ‘ + t22 ‘0 + t21 ’0‘ ; ’r = t11 ’‘ + t12 ’0‘ − t21 ‘ +(t11 − t22 )‘0 − t21 2 ’0‘ ; ’0r = t21 ’‘ + t22 ’0‘ + t21 =‘0 :
(12)
The values 2 − 2 , ’‘ − ’‘ , and ’0‘ − ’0‘ are omitted; they are small if =l.1. We use Eqs. (12) to consider the electron scattering at the rough interface. The envelope wave function can be written in the form ‘
‘ r
‘
‘
‘
= ei(p|| +pz z) + Rei(p|| −pz z) ; = Te
r i(p|| +pzr z)
’r (r) = kz =
p
P k
P k
;
where W () = (r + )(r) is the correlation function of the rough interface, and W˜ is its Fourier transform Z W˜ (k) = W ()e−ik d 2 : For the di use component ’‘ (r) we obtain Z 2pz‘ A(kz )(k − p|| )ei(k−kz z) d 2 k; ’‘ (r) = (2)2 where
kz t21 (t˜22 − ipzr t˜12 ) + (t˜21 − ipzr t˜11 )[kz (t11 − t22 ) + it21 ] : [t21 − ikz (t11 + t22 ) − kz2 t12 ][t˜21 − ipzr t˜11 − ipz‘ t˜22 − pz‘ pzr t˜12 ]
’˜ ‘ (k)ei(k−kz z) ; ’˜ r (k)ei(k+kz z) :
2m − k2 ;
t˜11 = t11 + it21 Z kz [t21 + ikz (t11 − t22 )] ˜ × W (k − p|| ) d 2 k; t21 − ikz (t11 + t22 ) − kz2 t12 t˜12 = t12 − 2 t21 Z 2 + kz2 (t11 − t22 )2 ] [t21 + W˜ (k − p|| ) d 2 k; t21 − ikz (t11 + t22 ) − kz2 t12 2 t˜21 = t21 − t21 Z kz2 W˜ (k − p|| ) d 2 k; × t21 − ikz (t11 + t22 ) − kz2 t12 t˜22 = t22 + it21 Z kz [t21 − ikz (t11 − t22 )] ˜ × W (k − p|| ) d 2 k; t21 − ikz (t11 + t22 ) − kz2 t12 (14)
(15)
A(kz ) = − ’‘ (r) =
then from rst two equations (12). These equations accept the form of Eq. (4) if we introduce the e ective parameters t˜ik of the boundary conditions as follows:
(13)
where is the electron energy, pz;‘;||r and kz and k are z and XY components of the electron momenta. It follows from Eq. (13) that p||‘ = p||r = p|| . We use last two Eqs. (12) to express the di use components ’‘ and ’r as functions of , R, and T . The re ection (R) and transmission (T ) coecients can be obtained
We should emphasize the two-fold in uence of the interface roughness. First, the roughness of the interface results in the di use component of the scattered wave. This component is always small for the long wavelength electrons and vanish when → ∞. Second, the e ective parameters of the boundary conditions depend on the interface roughness. This e ect depends on the relation between the electron wavelength and the correlate length of the rough interface. Indeed, we can assume W˜ (k − p|| ) ∼ 2 (k − k0 ) if .l. Then the corrections to the e ective parameters of the boundary conditions (14), caused by the roughnesses, become proportional to 2 =(l) and vanish when → ∞.
L. Braginsky / Physica E 5 (2000) 142–152
Corrections to tik become independent of when /l. This is clear from Eq. (14) because in this case |p|| |.|k|. It is easy to understand the reason. Formally, the roughnesses, the size of which is less then the electron wavelength, could be taken into account when the parameters bik and ik of the boundary conditions are obtained. In this consideration the roughnesses a ect the value of these parameters but do not make them dependent on [1]. It follows from Eq. (14) that -dependent corrections to t˜ik are of the order of (l=)2 ; this is in agreement with estimations [1]. It should be noted that values of t˜ik are not real at a rough interface. The reason for this is the diffuse component of the scattered wave that spoils the time-inversion symmetry.
4. Interband light absorption at a plane interface 4.1. Interface light absorption in direct-forbidden gap semiconductors Let us consider the light absorption at the interface between semiconductor (z ¡ 0) and an insulator (z ¿ 0). The band structure of the semiconductor is presented in Fig. 3. We can write the electron wave
147
functions as follows: 1 ii = √ N ( [vp (r)eipz + Rv vp∗ (r)e−ipz ]eip|| ; z ¡ 0; z ¿ 0; Tv e− v z+ip|| ; 1 fi = √ N ( ∗ [uq (r)e−iqz + Rc uq (r)eiqz ]eiq|| ; z ¡ 0; Tc e− c z+iq|| ;
z ¿ 0;
(16)
where uq (r) and vp (r) are the Bloch amplitudes in the conduction and valence band respectively; c:v are decay exponents of the wave functions apart o the interface; Rv ; Rc ; Tv , and Tc are the re ection and transmission coecients. It is possible to relate these values with positions of the interface levels (see Eq. (6)) 2ip 2iq ; 1 + Rc = ; (17) v − ip c + iq p √ where v = 2mh Ev and c = 2mc (Eg − Ec ). In direct-forbidden gap semiconductors u0 (r) = 0. We assume uq (r) = 2i sin(qa=2) wq (r), where wq (r) is the periodic function of r that does not vanish at q = 0. Then for the interband matrix element Pvc we obtain
1 + Rv = −
Ns (1 + Rv )(1 + R∗c )U p|| ;q|| ; N
Pvc = where Z U=
0
w0
9v0 3 d r; 9z
(18)
so that |Pvc |2 =
Fig. 3. Schematic band structure of the direct-forbidden gap semiconductor. Here Ev and Ec are the interface levels. Arrows indicate the two ways in which light absorption takes place: direct electron transition 1 → 2, and indirect 3 → 4 that occurs at the interface.
4U 2 Ns2 (c − Eg )(! − c ) p ;q : N 2 (c − Ec )(! − c + Ev ) || ||
We see that energy dependence of |Pvc |2 is sensitive to the position of the interface levels Ev and Ec , whether or not they are close to the corresponding band extrema. If we assume the Coulomb exponent ( ) as independent of q, then the result of integration (2) can be written as follows: ˙ (˜! − Eg ) ;
(19)
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where 3 if ˜! − Eg .min(Ev ; Eg − Ec ); 2 if min(Ev ; Eg − Ec ).˜! − Eg = .max(Ev ; Ec ); 1 if ˜! − Eg /max(Ev ; Eg − Ec ): 4.2. Interface light absorption in indirect-band-gap semiconductors The band structure of the semiconductor is presented in Fig. 1. For simplicity, we assume the nondegenerate valence band. There are two valleys in the conduction band: the central valley with the minimum at the center of Brillouin zone, and the side valley at the edge of it. It is important that the side valley of the conduction band is situated in the direction of the normal to the interface. We write the electron wave functions as follows: 1 (20) ii = √ [v(r)eipz + Rv v∗ (r)e−ipz ]eip|| ; N 1 fi = √ [Rc uc (r)ez + Rs us∗ (r)ei(q−=a)z N + us (r)e−i(q−=a)z ]eiq|| ; where v; uc , and us are the Bloch amplitudes in the valence band, the central and side valleys of the conduction band; p, , and q are z components of the wave numbers in these valleys. The coecients Rv ; Rc , and Rs are determined by the boundary conditions for the envelopes. For the interband matrix element Pvc we nd [1] 4pq 1 CPc + v + Ps ; Pvc = (c + iq)(v − ip) a 2 + p2 2 Z Pc = −i˜
0
Z Ps = −i˜
0
u0∗
9v 3 d r; 9z
uq∗
9v 3 d r; 9z
The energy dependence of |Pvc |2 (for p.) is the same as Eq. (18); therefore, |Pc |2 ˙ (˜! − Eg ) |Ps |2 + ; (22) (a)2 where is determined by Eq. (19), and is the dimensionless parameter, which has arisen from the rst term of expression (21). Two terms in the square brackets of Eq. (22) can be interpreted as follows: the rst term corresponds to the immediate transition of the electron to the side valley at the interface, while the second term corresponds to the excitation of the electron to the virtual interface state of the central valley followed by conversion to the side valley; it prevails if the valleys; minima are close (a.1). The Coulomb interaction between the electron in the conduction band and hole in the valence bands can a ect the frequency dependence of the absorption. Indeed, ( ) is independent of q only if qaB /˜. Otherwise, if qaB .˜, it is proportional to q−1 ; in this case the exponent changes, so that → − 1=2 when ˜! − Eg . e4 =(22 ˜2 ) (where −1 2 −1 = m−1 c + mh ; = n ). Thus, the frequency dependence of the absorption at the fundamental absorption edge essentially depends on the conditions at the interface, namely whether or not the interface electron levels are close to the corresponding bands extrema. 5. Light absorption at a rough interface According to Eq. (11), we write i
= i + ’i
where c and v are the electron energies in the conduction and valence bands, respectively, C is the coecient depending on the interface structure, and 0 is the unit cell.
f
= f + ’f ;
(23)
where i; f and ’i; f are the average and di use components of the corresponding wave functions: ( [vp (r)eipz + Rv vp∗ (r)e−ipz ]eip|| ; z ¡ 0; 1 i = √ N Tv e− v z+ip|| ; z ¿ 0; ’i (r) =
(21)
and
2p √ (2)2 N Z × vk (r)Av (kz )(k − p|| )ei(k−kz z) d 2 k;
1 f = √ N
(
[uq∗ (r)e−iqz
iqz
iq||
+ Rc uq (r)e ]e
Tc e− c z+iq|| ;
(24)
; z¡0 z ¿ 0;
L. Braginsky / Physica E 5 (2000) 142–152
’f (r) =
2q √ (2)2 N Z × uk∗ (r)Ac (kz )(k − q|| )ei(k+kz z) d 2 k;
where Av and Ac are the coecients A(kz ) (15) for the valence and conduction bands respectively. By substituting the wave functions (24) into Eq. (16), we express the matrix element Pvc as follows: (1) (2) (3) + Pvc + Pvc ; Pvc = Pvc
where (1) Pvc
∗ ˆ f H int i
=
(2) = Pvc
and (3) Pvc
Z
Z [ Z
=
dV;
∗ ˆ f H int ’i
(25)
+ ’∗f Hˆ int i ] dV;
’∗f Hˆ int ’i dV:
We have to obtain the square module of Eq. (25) and then to average it over the realizations of the random function (). For |Pvc |2 we have (1) 2 |Pvc |2 = |Pvc | +
(2) 2 |Pvc |
+
(1) (3)∗ Pvc Pvc
+
∗
(3) (1) Pvc Pvc :
From (25) and (25) we obtain Ns (1) Pvc = (1 + Rv )(1 + R∗c )U p|| ;q|| ; N (2) 2 |Pvc | =
(26a)
4Ns2 |U |2 W (p|| − q|| ) N 2S |p(1 + R∗c )Av (kz ) − q(1 + Rv )A∗c (kz )|2 (26b)
and Ns (3) Pvc = 4ipqU p|| ;q|| Z N × Av (kz )A∗c (kz )W (k|| − q|| ) d 2 k:
(26c)
Expressions (26) allow to estimate in uence of the roughness on the value and the frequency dependence (1) value is of the of the absorption. Note that the Pvc same form as that for the plane interface. The main di erence concerns parameters of the boundary conditions, Eq. (14), wherein the in uence of the interface roughness has been taken into account. This results
149
in the shift of the interface levels that manifests itself in the frequency dependence of the absorption. This e ect is most important if the interface electron levels are close to both, valence and conduction, bands or if the interface is smooth; the values of 1 + Rv and 1 + Rc are not small in these cases. Indeed, expressions (26b) and (26c) determine the in uence of the di use components of the scattered waves on the absorption; the values of these components are as small as =. On the contrary, this small parameter is absent in Eq. (26a) that determines the in uence of the average component (i.e., position of the interface electron level) on the absorption. However, Eq. (26a) is proportional to (1 + Rv )(1 + R∗c ). This value is small if the interface level is not close to any band. Then the values of Eqs. (26b) and (26c) may be of the same order or even exceed that of Eq. (26a). Comparing Eqs. (26b) and (26c) with Eq. (26a), we nd that the roughness in uence on the absorption is determined by the value =
2 pq2 ∼ |1 + Rc ||1 + Rv | 4 ∼
q
(v2 + p2 )(c2 + q2 )
2 q mc mh (˜! + Ev − Eg )(˜! − Ec ): 2
(27)
The correlation length l of the rough interface is also important for the light absorption. We can assume W˜ (k) = 2 (k) if l/. It can be shown that (2) 2 (1) (3)∗ (1)∗ (3) |Pvc | + Pvc Pvc + Pvc Pvc = 0 in this case. It is easy to understand the reason. Roughnesses, the mean length of which (Fig. 2) essentially exceeds the electron wavelength, could not a ect the electron properties of the interface. In the opposite limiting case l. the roughness in uence is determined by the term P(3) . Enhancement of the absorption in this case arises due to an increase of the number of the interface atoms Ns → Ns =a, in the vicinity of which the interband absorption with the momentum nonconservation occurs. The interesting situation arises when l ∼ . In this case the di use scattering that is described by the term (2) 2 (1) 2 |Pvc | ∼ c2 2 l2 [2mc|| (˜! − Eg )]|Pvc | leads to the change in the frequency dependence of the absorption. This means the rapid increase of absorption at the high frequencies where 2mc|| (˜! − Eg )l2 ∼ 1, so that exponent changes its value from → + 1.
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L. Braginsky / Physica E 5 (2000) 142–152
6. Discussion We considered the interband light absorption at the interface and found that possibility of the electron momentum nonconservation leads to enhancement of the absorption in the small microcrystallite. To estimate the value of the interface absorption, we have to compare the absorption of the microcrystalline solid obtained from the crystallites under consideration, = =L ˙ w=L , with the light absorption in the bulk semiconductor. Here L is length of the crystallite under consideration, w is the main size of the region at the interface where the interband electron transitions with the momentum nonconservation are possible. To estimate this value, we can use Eqs. (17) and (18): w ∼ if the interface electron levels are close to √ both (conduction and valence) bands extrema, w ∼ a if the interface level is close to any (conduction or valence) band extremum, and w ∼ a if both interface levels separate too far o the band extrema. The small value w=L characterizes the ratio of the number of atoms at this interface region to the number of atoms in the whole of solid. In order that the interface absorption becomes important in Ge and the indirect-band-gap semiconductors of the AIII BV group this small value should exceed the small parameter of electron–phonon interaction g (this value is about 10−3 –10−2 ). In these semiconductors the interface absorption essentially increases due to the intervalley conversion, which is determined by the second term in Eq. (22). That is possible if w=L(a)−2 ¿g. The main mechanism of light absorption in bulk Si is the impurity absorption. In order that the interface absorption becomes signi cant in this material, the number of the interface atoms should exceed the number of impurities. It seems possible that the indirect interband electron transitions at the interface considered here are responsible for an increase of luminescence of the porous Si (see Refs. [6,7] and references therein). Enhancement of the light absorption at the fundamental absorption edge has to be expected in direct-forbidden gap semiconductors, wherein the direct electron transitions between the band extrema are prohibited. The essential di erence in the e ective masses of the valence and conduction bands is favourable for the e ect. Such situation is character-
istic for transition metal oxides semiconductors [8,9]. The conduction band in these materials is composed mainly of d orbitals of the metal. These orbitals are strongly localized. For this reason the conduction band is narrow and even becomes at in a certain direction ( -M for TiO2 ). It is important that direct interband electron transition is dipole-forbidden in this material. There is two-fold advantage from the indirect electron transitions. First, they are allowed, i.e., the dipole matrix element of the indirect transitions is not small. Second advantage arises due to the large density of states for the electron in the conduction band. Let us compare the interband transitions in the bulk of semiconductor 1 → 2 and at the interface 3 → 4 (Fig. 3). The absorption is proportional to the density of electron states in each band. The density of states in the conduction band of TiO2 essentially exceeds that in the valence band. Nevertheless, it is the density of electron states in the valence band which determines the absorption in the bulk of crystal. This happens because of the law of conservation of momentum, which makes the electron states with the large momentums inaccessible for the excited electrons. On the contrary, indirect electron transitions make such states accessible. We can use Eq. (19) to estimate the value of absorption coecient of the microcrystalline direct-forbidden gap semiconductor: 2 2 w mc w Wv ∼ ∼ ; 0 L mh L Wc where 0 is the absorption coecient of the bulk monocrystal, Wv is width of the valence band (Wc ∼ 10 meV and Wv ' 1 eV for TiO2 [8,9]). Thus, the interface mechanism of the absorption becomes comparable with the bulk one for the microcrystalline TiO2 the mean size of the crystallite in which is L6w(Wv =Wc )2 ∼ 100 nm. Signi cant increase of optical absorption has been observed in the microcrystalline BaTiO3 [10]. We found that the absorption value and its frequency dependence at the fundamental absorption edge are sensitive to the structure of the interface. Thus, essential increase of the absorption has to be expected at a smooth interface. For such an interface, the main size of the interface region (i.e., the region at the interface where one crystal structure changes into another) considerably exceeds the lattice constant. It
L. Braginsky / Physica E 5 (2000) 142–152
is found that t21 = 0 at a smooth interface [11], so that c = v = 0, and the interband matrix element Pvc increases, because it no longer proportional to the small ratio (a=)2 [see Eqs. (17) and (18)]. Indeed, signi cant (in 100 times) enhancement of the light absorption has been observed at the interface a-Si=mc-Si [12]. The possible existence of the interface electron levels is essential for the optical properties of a sharp interface. Energy position of these levels depends not only on the bordering materials, but also on the interface itself. Structure of the interface as well as impurities and defects on it a ect positions of these levels. Their positions can be measured in optical experiments as the singularities in the absorption spectrum at the energies below the gap value. It seems that the electron interface level should be close to the valence band at least in wide-gap semiconductors. The interface level becomes empty when it is shifted too far from the top of the valence band. This results in a large surface charge and a strong band bending that is not favourable from the energetical point of view. Nevertheless, the interface level can be shifted as a result of structure reconstruction of the interface. This reconstruction does not essentially a ect the interatomic spacings or angles, but it makes the interface level closer to the top of the valence band. The roughness of the interface is one of the possible ways of such reconstruction. It follows from our consideration (Eq. (14)) that signi cant shift of the interface level occurs if the correlation length of rough interface is small (v l.˜). This causes the interface level to be closer to the band extremum. This is the particular case where the structure reconstruction of the interface decreases the interface energy. If so, then the rough interface becomes more favourable than the plane one. It seems strange that the interface roughnesses result only in the shift of the interface levels, but cannot be the cause for such levels. In other words, t˜21 in Eq. (14) vanishes for an appropriate correlation function W˜ (k), but it always equal to zero if t21 = 0. This is the result of our assumption that the interface is smooth (l/a). Perhaps, the interface level can be separated from the band extremum due to the interface roughnesses if l ∼ a. In this case the interface level arises as a result of the rearrangement of the chemical bonds at the rough interface.
Fig. 4. Interband sorption edge in the in uence of ˜2 (2mc|| l2 )−1 ; Ew
151
light absorption at the fundamental abthe crystallite. The dotted line indicates the interface roughness. Here El = Eg + = Eg + W c .
The value of (Eq. (27)) determines the roughnesses in uence on the absorption. This value is small, ∼ 2 =2 , if the interface electron levels are close to the extremum of both the bands. In this case the main size w of the region at the interface where the momentum nonconservation is possible is of the order of the electron wavelength , so that the small roughnesses cannot a ect the absorption. This is not the case if both interface levels shift far from the bands extrema; then w ∼ a and value is of about or even exceeds unity. The roughnesses in uence on the absorption is signi cant at the low frequencies when ˜!6Eg + ˜2 (2mc l2 )−1 ; enhancement of absorption in this case is due to an e ective increase of the number of the interface atoms (Ns → Ns =a) in the vicinity of which the interface light absorption occurs. The possible deviation of the absorption vs. energy curve at the fundamental absorption edge caused by the interface roughnesses is schematically presented in Fig. 4. In conclusion, we show that the possibility of the momentum nonconservation at the interface leads to enhancement of the interband light absorption in small crystallites, the size of which is about a few 10 nm. The interband absorption is sensitive to the interface electron levels, namely, whether or not they are close to the bands extrema. The in uence of the interface
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L. Braginsky / Physica E 5 (2000) 142–152
roughnesses is essential if these levels are not close to any or both conduction and valence bands. The e ect of the roughnesses is two-fold. First, they result in the di use electron scattering at the interface. This leads to enhancement of the absorption due to the e ective increase of the share of the interface atoms. Second, roughnesses result in e ective shift of the electron interface levels. Acknowledgements The author wishes to thank Prof. E. Ivchenko, who introduced him to the problem of in uence of the interface roughnesses on the absorption, Prof. V. Volkov and Prof. M. Entin for the helpful discussions. This work was supported by the Russian Foundation for the Basic Research, Grant No. 99-02-17019.
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