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Theory of direct interband transitions
OOZZ-3697184 $3.00 + .oO Pergamon Fess Ltd.
, phy~. Chem S&/s Vol. 45. No. 11112.PP. 1243-1247. $984 Printed in the U.S.A.
THEORY
OF DIRECT SANG
DON
INTERBAND
TRANSITIONS
and OK HEE CHUNG
CHOI
Department of Physics, Kyungpook National University, Taegu, South Korea (Received 21 November 1983; accepted 12 March 1984) Abstract--A theory of direct interband optical transitions in electron-phonon systems is presented with
the help of the projection operator method. The caiculated linewidth for ge~~urn at 300 K and 4.66 t&as is approximately 1.3 x 10-s eV, which is in qualitative agreement with the Lorentzian part of the experimental lineshape measured by Burstein et al. and with the theoretical result of Ohta et al. 1. I~ODU~ON
Pb + F+l?+ 1 q’ P9
Y4 =
Interband along with intraband optical transitions have been studied as a tool for investigating transport behaviors of electrons in solids. Many theoretical studies on these topics have appeared. But most of the work has been focused on intraband transitions, especially on cyclotron resonance lineshape. This paper aims at presenting a quantum theory of interband transitions by using the projection operator method[l] introduced by Argyres and Sigel, and comparing the lineshape obtained in germanium with the experimental data of Burstein et. al. [2] and the theoretical result of Ohta et al. [3] in order to justify the theory. The origin of this formalism dates back to the discovery of a theory of cyclotron resonance lineshape by the present authors [4]. The authors also suggested that there exist two coupling schemes in the electron-phonon interactions both for the intraband and the interband optical transitions[5,6]. We will deal with elec~on-phonon interactions in the main text and will discuss the case of impurity scatterings in the conclusion.
vq = C, exp (iq . r),
(2.7)
HP = fi 2 w,,b,+bQ.
(2.8)
P Here C, is the interaction operator, P is the polarization induced in the system, the subscript p on the many body Hamiltonian H and the scattering potential stands for the phonon background, a:(~,) is the creation (annihilation) operator for the electron with momentum p, effective mass 111and energy E, in the eigenstate ltl), and b:(b,) raises (lowers) the q-mode quantum number by 1. The conductivity for this system is then given in the following form: m CT~,(O) = lim
o-o+ s 0
@k/(t)
*k/(t)
=
$
(2.9)
- J *u -
~N~k(tf],
(2.10)
where u is an arbitrary constant vector, 5 the chemical potential, p(H) the equilibrium density operator normal&d to unity, /? = (k,T)-‘, and J(t) denotes the total current operator in the Heisenberg representation, The second equality in eqn (2.10) follows from the following identity:
(2.f)
8s dX exp@Hb)JI exp(-Mb)
s0
= ex~~o)~
(2.3)
- at)&,
(A(-ihA)Jk(t))dX
TR~'(Ho
(2.2)
exp(-iut
(S f = -5 .Y, z);
2. ~ONDU~IVI~ Consider a system of N electrons in interaction with phonons, initially in equilibrium with a temperature T, and subject to an oscillatory external electric field F(t) = F(0) exp(iot). Then in the presence of a staticmagnetic field B = V X A, the total Hamiltonian of the system can be written [5]
(2.6)
I
exp(-M-f&
(2.11)
(2.4)
where HA = Ho - J. u. Note that the total current operator can also be written in terms of the single electron current operator j as
(2.5)
J = ; @lib )+a,.
1243
(2.12)
1244
S. D. CHOI and 0. H. CHLJNG
We assume that the interaction characterized by He,, is so weak that the equilibrium density operator can be factorized as p(Ho-J-u-{N) = P~K + H,,, - J-u - WhW,A
(2.13)
where the subscripts e and p on p respectively imply that the density should be considered over the electron and phonon states. Then using eqns (2.3), (2.12) and (2.13) we obtain the reduction formula: T$‘[p,(H, + Hep-
J . u - cN)J(t)]
= t?‘[j(t)f(h,
+ h, - j-u - 01
(2.14)
where t? denotes the single electron trace andf(x) stands for the Fermi distribution function. Then the conductivity can be written in the single electron formalism:
which can also be obtained from eqn (2.15) in Ref. [9]. For a constant magnetic field B applied along the z-axis and characterized by the vector potential A = (0, Bx, 0), we, neglecting the spins, have h, = [Px*+ (p, + ~w,x)* + ,4*1/(2m 1, (3.3) where o0 = eB/m. It is to be noted that m and w. respectively should be replaced by m&r,) and w,(o,) for the electrons in the conduction (valence) band. The energy eigenvalues and eigenstates are characterized by the Landau index N, the electron wave vector k and the corresponding Bloch functions. In the parabolic band approximation, we have E: = E&k = Et + (N + $hw, + ti*k:/(2m,),
(3.4)
E$ = E&f,kr= -(N’ + $)hw, - h2(k:)*/(2mu),
(3.5)
Ia; c> = Yd+,),
(3.6)
la’; v) = Y~,,K(r)lu,).
(2.15) where $I= -G(w-)j,
(2.16)
G(z) = (hz - L)-‘,
(2.17)
w- = w - iu(a +O+),
(2.18)
L = Lo + L, + Lp.
(2.19)
Here L,,, L, and LP are Liouville operators respectively corresponding to h,, h, and H,, and (..... . .)ph denotes the average over the phonon distribution. We make the following approximation after other authors[7, 81: f(h, + h, -j
. u - [) Xf(h< -j . u - 0. (2.20)
We then obtain
Here E, is the energy gap, the Greek letters c( and CI’ respectively denote the state (N, k) and (N’, k’), 1U,)(lU,)) are the Bloch functions for the conduction (valence) band, and Y,,(r) are the wave functions of the electrons corresponding to N and k: Y,,(r)
= (L,,L1)-“‘4Jx
- X) exp (iyk,, + izk,),
The absorption power for the circularly polarized electromagnetic wave of amplitude F and frequency w, directed along the z-axis in the system, is given by PI P = (F2/2)R,~+-(w),
(3.1)
where the symbol R, means “the real part of’ and the conductivity tensor is defined by eq. (2.21)
(3.8)
q5N(x)= (2NN!r,,,/7t-“2HN(a~) exp ( - cc*x*/2), (3.9)
where HN is the Nth Hermite polynomial, are normalization lengths and
L, and L,
TV= l/r, = (eB/ii)“*,
(3.10)
X = - hk,/(eB).
(3.11)
Let us restrict ourselves to the direct transitions. Since Iu), unlike Y,,, is the rapidly varying factor, we may adopt the following selection rule[lO, 111: (CI; clj+la’; a> = o’+).&.,.
3. THEORY OF INTERBAND TRANSITIONS
(3.7)
(3.12)
Now, following Argyres and Sigel[l], we define the projection operators P and P’ such that & PX =i+ ~‘+)a P’=l-P
(3.13) (3.14)
for any operator X, where X, = (tl; clX(cr; v) has been and will be used for notational convenience. Then keeping in mind the relation (Loj+), = (EZ - EE)(j+),, we easily obtain from eqn (3.2)
1245
Theory of direct interband transitions 1 + hw- - E’, + E; + hw, (3.15)
t++ll=
io’+h i(ho - - E,’ + E,“) + hB,(o)’
X
hf.--E;+E::+hU,
(3.16) 1 ho-
where the lineshape function B,(o) is given by = [i(j+)J’
({(L,
Qcl
1
(
+
hB,(o)
+ n,( c&X
+ L,GoP’LI
with G,, being equal to (hw - - Lo)-‘. The first term of B,(w) may be neglected by assuming that the phonon distribution is random[7]. If the interaction is weak enough, it suffices to take into account the second and third terms only in approximation. Note that the perturbative expansion in eqn (3.17) is valid since there appear no divergent terms for the weak scattering potential h,[12, 131. The second term is obtained as
-
,lTz
+
E;;
-
hwq
’
(3’20)
Now, to make the matters simple, let us consider the direct gap materials and assume that the conduction band is almost vacant and the field is strong. Then although the energy separation between the neighboring Landau levels is not large compared with k,T, we may consider the transition only between the level with (0,O; c) and that with (0,O; u) in approximation. We may also assume that j(iV, 0; cb+lN, 0; u)I((l(O, 0; c~+JO,0; u >I for N # 0 in the strong field condition, since the chance of transitions decreases as the energy difference between the two levels increases. Under these assumptions, we obtain a+-(w)
N(E,
+ hwg/2)[i(hw- - Eg - h&/2) +
hBo(~)l
’
(3.21)
where w: = o, + w, is the reduced cyclotron frequency. The matrices in eqn (3.20) can be calculated as where c?, = (hw- - he)-‘, and (V&o = ((Y; ulV,,lj$ V) = (a; cl V&3; c) and (P’L, j,), = 0 have been used in the calculation. The third term is obtained in a similar way as JNNIX 4X2X’)
= F F 2) hlK(P; &0loJLu; ~>Wq~),, P
X {
Vib,+i,- j+Vib,+ - VqbJ++ j+Fqbq},
E
m s -m
dx&(x
- X)&(x
Some simple calculations
- X’) exp(ixq,).
(3.24)
yield
+ ((a; clGJ@; U))‘{ VqbJ+- j+Vqbq - Pi$b,+j+ + j+~~b~},~(~d,l. (3.19) We now consider the relations (b,+b& = n,, = {exp(@hw,) - 1 1-l and (/3; c(&la; u)[l + hw,@; c/C?&; u)] N (hw- - E$ + EE - hw,)-’ following Ref. [5]. Thus adding up the above two members, we obtain the following formula for the lineshape
K&N, N’; t) = $
W
>.
(0 = T
1 n
t-m
f = ho- - E; + E: - hw P
tm exp(-t)
-$ [z”+~ exp(-t)],
ro2tqx2 + q,2)/2,
AN = IN - N’I,
(3.26)
(3.27) (3.28) (3.29)
1246
D. CHOI and 0. H. CHUNG
S.
N, and N, respectively being the larger and smaller of the two numbers N and N’. Then the lineshape function for a = 0 = (0,O) is obtained as
E:k,T
zzp
m
&rr;s*p)i s 0 (3.30)
t exp( -t)dt
X[C~dq,6($++,+nhw,) h29f 2m - ho, + n’hw” ”
1 ho- - ES + E; - hw 4 +
1 hw- - ES + E;; + hw,
1
Carrying out the t- and q,-integration,
%)
)I
. (4.3)
we obtain
EfkJ(eB)‘~* = (4&),&h7/2
1 + nq
fiw- - E; + El; + hw,
1 + hw- - E; + E;; - hw,
6Lm.
We now apply the formula to the electron-phonon system in Ge. For the interaction between the electrons and long wavelength acoustic phonons in Ge, the deformation potential theory by Bardeen and Schockley[l4] may be applied. In the isotropic coupling approximation, the interaction is defined in terms of the deformation potential constant E,, as
(gj$
(4.1)
where p and s, respectively, are the mass density and the speed of sound defined by wq = sq. We may adopt the high temperature approximation ksT > hw, for T = 300 K at which the experiment by Burstein et al. [2] was performed. Therefore we have nq N ksT/(fisq) > 1. Furthermore, we approximate o,, by the representative value wp defined by m wqnqdq wp= s O m = k,T/h nq dq s0
ll’=’ \
l-1 mv
)
(4.4)
wp_ n’
v wu
(3.31)
4. LINEWIDTH FOR THE ELECTRON-PHONON SYSTEMS
Pq12 =
+5
J
where the cut-off values n, and n, are the maximum integral values of II and n’ respectively defined by n < w,,/w, and n’ < w,,/w,. Note that the linewidth in this approximation is proportional to J B and T. We use the following constants for Ge: S = 5.460 X IO3 m/set, p = 5.330 X lo3 kg/m3, El = 16.50 eV [15], mC = 0.038 M,, m, = 0.360 mp [2]. Then for B = 4.66 teslas and T = 300 K, we obtain from eqn (4.2) up = 4.140 X lOI [l/set], w, = 2.156 X lOI [l/set], and W, = 2.270 X 10” [l/set]. Thus we have hT = 1.3 x 10e3eV,
(4.5)
which is in qualitative agreement with the Lorentzian part of the experimental shape in Ref. [2] and with the theoretical result of Ohta et al. [3, 161. For polar crystals lacking an inversion center, however, the piezoelectric scattering characterized by [ 171
is important, where K is the electromechanical coupling constant. In the same high temperature approximation we obtain
(4.2)
r(w) =
which is a q-independent constant for the sake of mathematical simplicity. Recalling that the linewidth I’(w) is the imaginary part of [iEo(w)] and taking into account the relation lim,+ (x + ia)-’ = p(x-‘) - &5(x), we see that the non-zero terms in eqn (3.31) are the second term of the first part and the first term of the second part at the resonance peak w = ~$12 + EJh:
X
s a
0
1247
Theory of direct interband transitions REFERENCES
The width in this case is proportional to T/JB. It is that eqn (4.7) cannot be checked further, since any experimental data on this scattering mechanism are not available.
2. Burstein E., Picus G. S., Wallis R. F. and Blatt F.,
5. CONCLUDING REMARKS
3. Ohta T., Nagae M. and Miyakawa T., Prog. Theor.
So far we have introduced a quantum theory of interband magneto-optical absorption lineshape due to electron-phonon interactions and have justified the theory by comparing the result obtained for Ge with the experiment of Burstein et al. and the theoretical result of Ohta et al. It should be noted that the present result is based on the assumption that k, = 0, thus the lineshape has been obtained in a Lorentzian form. The result will be improved if, in addition to the inclusion of the k,-dependence and higher order transitions, a non-parabolic band model, the degeneracy of the valence band and impurity scatterings are considered. For the impurity scatterings, the formulae can be obtained by the following replacement:
4. Choi S. D. Chung 0. H., Sug J. Y. and Cho H. M., J. Phys. Chem. Solids MS No PCS 581.
to be regretted
1. Argyres P. N. and Sigel J. L., Phys. Rev. Letf. 31, 1397
(1973). Phys. Rev. 113, 15 (1959).
n, + 1 -
n, (impurity density),
n, -
0,
WQ-
0
in eqns (3.20) and (3.31).
Phys. 23, 229 (1960).
Choi S. D. and Chung 0. H., Solid St. Commun. 46, 717 (1983). Choi S. D. and Chung 0. H.. Left. Nuovo Cimento 38, 221 (1983) and 39, 1 (1984). Kawabata A., J. Phys. Sot. Japan 23, 999 (1967). Lodder A. and Fujita S., J. Phys. Sot. Japan 25, 714 (1968).
9. Argyres P. N. and Sigel J. L., Phys. Rev. BlO,1139 (1974). 10. Madelung O., Introduction to Solid State Theory, D. _ _ 290. Springer-Vqlag, Berlin (1978). II. Palik E. D. and Mitchell D. L.. Phvsics of Solids in Intense Magnetic Field (Edited by E: D. Haidemenakis), p. 115. Plenum Press, New York (1965). 12. Choi S. D. and Chung 0. H., Solid St. Commun. 46, 355 (1983). 13. Choi S. D., Yi S. N. and Chung 0. H., 2. Physik B (in press). 14. Bardeen J. and Schockley W., Phys. Rev. 80, 72 (1950). 15. Dakhovskii I. V., Sov. Phys.-Solid St. 5, 1695 (1964). 16. According to Ref.[3], @F),,,,, = lo-‘eV. But by enlarging the shape (Fig. 12) in Ref. [2], we obtain (hIYB”rm = 2.5 X lo-’ eV. 17. MeijerH. J. G. and Polder D., Physica 19, 255 (1953).
, phy~. Chem S&/s Vol. 45. No. 11112.PP. 1243-1247. $984 Printed in the U.S.A.
THEORY
OF DIRECT SANG
DON
INTERBAND
TRANSITIONS
and OK HEE CHUNG
CHOI
Department of Physics, Kyungpook National University, Taegu, South Korea (Received 21 November 1983; accepted 12 March 1984) Abstract--A theory of direct interband optical transitions in electron-phonon systems is presented with
the help of the projection operator method. The caiculated linewidth for ge~~urn at 300 K and 4.66 t&as is approximately 1.3 x 10-s eV, which is in qualitative agreement with the Lorentzian part of the experimental lineshape measured by Burstein et al. and with the theoretical result of Ohta et al. 1. I~ODU~ON
Pb + F+l?+ 1 q’ P9
Y4 =
Interband along with intraband optical transitions have been studied as a tool for investigating transport behaviors of electrons in solids. Many theoretical studies on these topics have appeared. But most of the work has been focused on intraband transitions, especially on cyclotron resonance lineshape. This paper aims at presenting a quantum theory of interband transitions by using the projection operator method[l] introduced by Argyres and Sigel, and comparing the lineshape obtained in germanium with the experimental data of Burstein et. al. [2] and the theoretical result of Ohta et al. [3] in order to justify the theory. The origin of this formalism dates back to the discovery of a theory of cyclotron resonance lineshape by the present authors [4]. The authors also suggested that there exist two coupling schemes in the electron-phonon interactions both for the intraband and the interband optical transitions[5,6]. We will deal with elec~on-phonon interactions in the main text and will discuss the case of impurity scatterings in the conclusion.
vq = C, exp (iq . r),
(2.7)
HP = fi 2 w,,b,+bQ.
(2.8)
P Here C, is the interaction operator, P is the polarization induced in the system, the subscript p on the many body Hamiltonian H and the scattering potential stands for the phonon background, a:(~,) is the creation (annihilation) operator for the electron with momentum p, effective mass 111and energy E, in the eigenstate ltl), and b:(b,) raises (lowers) the q-mode quantum number by 1. The conductivity for this system is then given in the following form: m CT~,(O) = lim
o-o+ s 0
@k/(t)
*k/(t)
=
$
(2.9)
- J *u -
~N~k(tf],
(2.10)
where u is an arbitrary constant vector, 5 the chemical potential, p(H) the equilibrium density operator normal&d to unity, /? = (k,T)-‘, and J(t) denotes the total current operator in the Heisenberg representation, The second equality in eqn (2.10) follows from the following identity:
(2.f)
8s dX exp@Hb)JI exp(-Mb)
s0
= ex~~o)~
(2.3)
- at)&,
(A(-ihA)Jk(t))dX
TR~'(Ho
(2.2)
exp(-iut
(S f = -5 .Y, z);
2. ~ONDU~IVI~ Consider a system of N electrons in interaction with phonons, initially in equilibrium with a temperature T, and subject to an oscillatory external electric field F(t) = F(0) exp(iot). Then in the presence of a staticmagnetic field B = V X A, the total Hamiltonian of the system can be written [5]
(2.6)
I
exp(-M-f&
(2.11)
(2.4)
where HA = Ho - J. u. Note that the total current operator can also be written in terms of the single electron current operator j as
(2.5)
J = ; @lib )+a,.
1243
(2.12)
1244
S. D. CHOI and 0. H. CHLJNG
We assume that the interaction characterized by He,, is so weak that the equilibrium density operator can be factorized as p(Ho-J-u-{N) = P~K + H,,, - J-u - WhW,A
(2.13)
where the subscripts e and p on p respectively imply that the density should be considered over the electron and phonon states. Then using eqns (2.3), (2.12) and (2.13) we obtain the reduction formula: T$‘[p,(H, + Hep-
J . u - cN)J(t)]
= t?‘[j(t)f(h,
+ h, - j-u - 01
(2.14)
where t? denotes the single electron trace andf(x) stands for the Fermi distribution function. Then the conductivity can be written in the single electron formalism:
which can also be obtained from eqn (2.15) in Ref. [9]. For a constant magnetic field B applied along the z-axis and characterized by the vector potential A = (0, Bx, 0), we, neglecting the spins, have h, = [Px*+ (p, + ~w,x)* + ,4*1/(2m 1, (3.3) where o0 = eB/m. It is to be noted that m and w. respectively should be replaced by m&r,) and w,(o,) for the electrons in the conduction (valence) band. The energy eigenvalues and eigenstates are characterized by the Landau index N, the electron wave vector k and the corresponding Bloch functions. In the parabolic band approximation, we have E: = E&k = Et + (N + $hw, + ti*k:/(2m,),
(3.4)
E$ = E&f,kr= -(N’ + $)hw, - h2(k:)*/(2mu),
(3.5)
Ia; c> = Yd+,),
(3.6)
la’; v) = Y~,,K(r)lu,).
(2.15) where $I= -G(w-)j,
(2.16)
G(z) = (hz - L)-‘,
(2.17)
w- = w - iu(a +O+),
(2.18)
L = Lo + L, + Lp.
(2.19)
Here L,,, L, and LP are Liouville operators respectively corresponding to h,, h, and H,, and (..... . .)ph denotes the average over the phonon distribution. We make the following approximation after other authors[7, 81: f(h, + h, -j
. u - [) Xf(h< -j . u - 0. (2.20)
We then obtain
Here E, is the energy gap, the Greek letters c( and CI’ respectively denote the state (N, k) and (N’, k’), 1U,)(lU,)) are the Bloch functions for the conduction (valence) band, and Y,,(r) are the wave functions of the electrons corresponding to N and k: Y,,(r)
= (L,,L1)-“‘4Jx
- X) exp (iyk,, + izk,),
The absorption power for the circularly polarized electromagnetic wave of amplitude F and frequency w, directed along the z-axis in the system, is given by PI P = (F2/2)R,~+-(w),
(3.1)
where the symbol R, means “the real part of’ and the conductivity tensor is defined by eq. (2.21)
(3.8)
q5N(x)= (2NN!r,,,/7t-“2HN(a~) exp ( - cc*x*/2), (3.9)
where HN is the Nth Hermite polynomial, are normalization lengths and
L, and L,
TV= l/r, = (eB/ii)“*,
(3.10)
X = - hk,/(eB).
(3.11)
Let us restrict ourselves to the direct transitions. Since Iu), unlike Y,,, is the rapidly varying factor, we may adopt the following selection rule[lO, 111: (CI; clj+la’; a> = o’+).&.,.
3. THEORY OF INTERBAND TRANSITIONS
(3.7)
(3.12)
Now, following Argyres and Sigel[l], we define the projection operators P and P’ such that & PX =i+ ~‘+)a P’=l-P
(3.13) (3.14)
for any operator X, where X, = (tl; clX(cr; v) has been and will be used for notational convenience. Then keeping in mind the relation (Loj+), = (EZ - EE)(j+),, we easily obtain from eqn (3.2)
1245
Theory of direct interband transitions 1 + hw- - E’, + E; + hw, (3.15)
t++ll=
io’+h i(ho - - E,’ + E,“) + hB,(o)’
X
hf.--E;+E::+hU,
(3.16) 1 ho-
where the lineshape function B,(o) is given by = [i(j+)J’
({(L,
Qcl
1
(
+
hB,(o)
+ n,( c&X
+ L,GoP’LI
with G,, being equal to (hw - - Lo)-‘. The first term of B,(w) may be neglected by assuming that the phonon distribution is random[7]. If the interaction is weak enough, it suffices to take into account the second and third terms only in approximation. Note that the perturbative expansion in eqn (3.17) is valid since there appear no divergent terms for the weak scattering potential h,[12, 131. The second term is obtained as
-
,lTz
+
E;;
-
hwq
’
(3’20)
Now, to make the matters simple, let us consider the direct gap materials and assume that the conduction band is almost vacant and the field is strong. Then although the energy separation between the neighboring Landau levels is not large compared with k,T, we may consider the transition only between the level with (0,O; c) and that with (0,O; u) in approximation. We may also assume that j(iV, 0; cb+lN, 0; u)I((l(O, 0; c~+JO,0; u >I for N # 0 in the strong field condition, since the chance of transitions decreases as the energy difference between the two levels increases. Under these assumptions, we obtain a+-(w)
N(E,
+ hwg/2)[i(hw- - Eg - h&/2) +
hBo(~)l
’
(3.21)
where w: = o, + w, is the reduced cyclotron frequency. The matrices in eqn (3.20) can be calculated as where c?, = (hw- - he)-‘, and (V&o = ((Y; ulV,,lj$ V) = (a; cl V&3; c) and (P’L, j,), = 0 have been used in the calculation. The third term is obtained in a similar way as JNNIX 4X2X’)
= F F 2) hlK(P; &0loJLu; ~>Wq~),, P
X {
Vib,+i,- j+Vib,+ - VqbJ++ j+Fqbq},
E
m s -m
dx&(x
- X)&(x
Some simple calculations
- X’) exp(ixq,).
(3.24)
yield
+ ((a; clGJ@; U))‘{ VqbJ+- j+Vqbq - Pi$b,+j+ + j+~~b~},~(~d,l. (3.19) We now consider the relations (b,+b& = n,, = {exp(@hw,) - 1 1-l and (/3; c(&la; u)[l + hw,@; c/C?&; u)] N (hw- - E$ + EE - hw,)-’ following Ref. [5]. Thus adding up the above two members, we obtain the following formula for the lineshape
K&N, N’; t) = $
W
>.
(0 = T
1 n
t-m
f = ho- - E; + E: - hw P
tm exp(-t)
-$ [z”+~ exp(-t)],
ro2tqx2 + q,2)/2,
AN = IN - N’I,
(3.26)
(3.27) (3.28) (3.29)
1246
D. CHOI and 0. H. CHUNG
S.
N, and N, respectively being the larger and smaller of the two numbers N and N’. Then the lineshape function for a = 0 = (0,O) is obtained as
E:k,T
zzp
m
&rr;s*p)i s 0 (3.30)
t exp( -t)dt
X[C~dq,6($++,+nhw,) h29f 2m - ho, + n’hw” ”
1 ho- - ES + E; - hw 4 +
1 hw- - ES + E;; + hw,
1
Carrying out the t- and q,-integration,
%)
)I
. (4.3)
we obtain
EfkJ(eB)‘~* = (4&),&h7/2
1 + nq
fiw- - E; + El; + hw,
1 + hw- - E; + E;; - hw,
6Lm.
We now apply the formula to the electron-phonon system in Ge. For the interaction between the electrons and long wavelength acoustic phonons in Ge, the deformation potential theory by Bardeen and Schockley[l4] may be applied. In the isotropic coupling approximation, the interaction is defined in terms of the deformation potential constant E,, as
(gj$
(4.1)
where p and s, respectively, are the mass density and the speed of sound defined by wq = sq. We may adopt the high temperature approximation ksT > hw, for T = 300 K at which the experiment by Burstein et al. [2] was performed. Therefore we have nq N ksT/(fisq) > 1. Furthermore, we approximate o,, by the representative value wp defined by m wqnqdq wp= s O m = k,T/h nq dq s0
ll’=’ \
l-1 mv
)
(4.4)
wp_ n’
v wu
(3.31)
4. LINEWIDTH FOR THE ELECTRON-PHONON SYSTEMS
Pq12 =
+5
J
where the cut-off values n, and n, are the maximum integral values of II and n’ respectively defined by n < w,,/w, and n’ < w,,/w,. Note that the linewidth in this approximation is proportional to J B and T. We use the following constants for Ge: S = 5.460 X IO3 m/set, p = 5.330 X lo3 kg/m3, El = 16.50 eV [15], mC = 0.038 M,, m, = 0.360 mp [2]. Then for B = 4.66 teslas and T = 300 K, we obtain from eqn (4.2) up = 4.140 X lOI [l/set], w, = 2.156 X lOI [l/set], and W, = 2.270 X 10” [l/set]. Thus we have hT = 1.3 x 10e3eV,
(4.5)
which is in qualitative agreement with the Lorentzian part of the experimental shape in Ref. [2] and with the theoretical result of Ohta et al. [3, 161. For polar crystals lacking an inversion center, however, the piezoelectric scattering characterized by [ 171
is important, where K is the electromechanical coupling constant. In the same high temperature approximation we obtain
(4.2)
r(w) =
which is a q-independent constant for the sake of mathematical simplicity. Recalling that the linewidth I’(w) is the imaginary part of [iEo(w)] and taking into account the relation lim,+ (x + ia)-’ = p(x-‘) - &5(x), we see that the non-zero terms in eqn (3.31) are the second term of the first part and the first term of the second part at the resonance peak w = ~$12 + EJh:
X
s a
0
1247
Theory of direct interband transitions REFERENCES
The width in this case is proportional to T/JB. It is that eqn (4.7) cannot be checked further, since any experimental data on this scattering mechanism are not available.
2. Burstein E., Picus G. S., Wallis R. F. and Blatt F.,
5. CONCLUDING REMARKS
3. Ohta T., Nagae M. and Miyakawa T., Prog. Theor.
So far we have introduced a quantum theory of interband magneto-optical absorption lineshape due to electron-phonon interactions and have justified the theory by comparing the result obtained for Ge with the experiment of Burstein et al. and the theoretical result of Ohta et al. It should be noted that the present result is based on the assumption that k, = 0, thus the lineshape has been obtained in a Lorentzian form. The result will be improved if, in addition to the inclusion of the k,-dependence and higher order transitions, a non-parabolic band model, the degeneracy of the valence band and impurity scatterings are considered. For the impurity scatterings, the formulae can be obtained by the following replacement:
4. Choi S. D. Chung 0. H., Sug J. Y. and Cho H. M., J. Phys. Chem. Solids MS No PCS 581.
to be regretted
1. Argyres P. N. and Sigel J. L., Phys. Rev. Letf. 31, 1397
(1973). Phys. Rev. 113, 15 (1959).
n, + 1 -
n, (impurity density),
n, -
0,
WQ-
0
in eqns (3.20) and (3.31).
Phys. 23, 229 (1960).
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9. Argyres P. N. and Sigel J. L., Phys. Rev. BlO,1139 (1974). 10. Madelung O., Introduction to Solid State Theory, D. _ _ 290. Springer-Vqlag, Berlin (1978). II. Palik E. D. and Mitchell D. L.. Phvsics of Solids in Intense Magnetic Field (Edited by E: D. Haidemenakis), p. 115. Plenum Press, New York (1965). 12. Choi S. D. and Chung 0. H., Solid St. Commun. 46, 355 (1983). 13. Choi S. D., Yi S. N. and Chung 0. H., 2. Physik B (in press). 14. Bardeen J. and Schockley W., Phys. Rev. 80, 72 (1950). 15. Dakhovskii I. V., Sov. Phys.-Solid St. 5, 1695 (1964). 16. According to Ref.[3], @F),,,,, = lo-‘eV. But by enlarging the shape (Fig. 12) in Ref. [2], we obtain (hIYB”rm = 2.5 X lo-’ eV. 17. MeijerH. J. G. and Polder D., Physica 19, 255 (1953).