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Quantum theory of the cyclotron resonance lineshape for a two-dimensional electron-phonon system
Volume 70A, number 2
PHYSICS LETTERS
19 February 1979
QUANTUM THEORY OF THE CYCLOTRON RESONANCE LINESHAPE FOR A TWO-DIMENSIONAL ELECTRON-PHONON SYSTEM 1
Mahendra PRASAD
Department of Physics and Astronomy, State University of New York at Buffalo, Amherst, NY 14260, USA
Received 12 July 1978 Revised manuscript received 22 November 1978
Based on the super resolvent operator representation of Kubo’s current correlation formula [1] and its proper connected diagram expansion [2] a theory of cyclotron resonance lineshape is presented. Effects of the average phonon field are taken into account when treating the scattering of a given electron with the reference phonon. This gives rise to a self-consistent equation for the irreducible collision operator, which possesses a built in structure of the “gain” and “loss” like terms occurring in the kinetic description of the relaxation process. Equations for the cyclotron resonance linewidths rN and the frequency shifts ~N associated with the electronic transitions between the Landau subbands N and N+ 1 are derived for the case when the electron—phonon interaction induces the scatteringof the electron in the same Landau subband. Temperature dependent cyclotron resonance linewidth data of Kueblbeck and Kotthaus [3] compare very well with the theoretical prediction made here and one parameter fitting yields the deformation potential in reasonable agreement with that found for silicon. Kubo’s formula for the dynamic magnetoconductivity tensor is given by
à~_(w)= i lim urn UmL2 tr [TR{(a~i/au) [‘to~‘ph
+ 2’-11e—ph
—
z]1J~}]
(1)
a-.O u—~O
where “tr” and “TR” stand for the single electron and many phonon traces, respectively, h 0, single electron hamiltonian under a magnetic field perpendicular to the surface in the Landau gauge, HPh, many phonon hamiltonian and He_ph stands for the electron—phonon coupling hamiltonian. A “hat” () on the letters denotes the Liouville operator corresponding to them. 1e—ph j~~u~)]}—1 (2) = {1 + exp [I3(12~ + HPh + ~‘ 1+ andj_ are the current components; ~ is the Fermi energy, L2 the normalization area, u isa c-number and z stands for w + ia. The cyclotron resonance lineshape arises mainly from the super resolvent operator occurring in eq. (1) and the interaction term from ff can be dropped, then ,i-~ii~. Proper connected diagram analysis of eq. (1) then yields the following result: —
—
—
o÷_(w)= i lim lim LimL2 tr [(a/au) ~O>ph~’I
(3)
‘
a-GO u—*O
1He_ph)~~, (4,5) 0~‘ph ~_z]~hj~, b A2
[1z
—
—
means many phonon average.
Finding the traces occurring in eq. (3) and the u-derivative we obtain the following result: Present address: Clarendon Laboratory and Department of Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK.
127
Volume 70A, number 2
PHYSICS LETTERS
19 February 1979
i E[f(~)_f(EN÷1)1(N+l)
e2
N
(6)
(wo—w+z~N)+iFN
where EN = (N+~)hw0,f(E), the Fermi distribution function and
rN, ~N
satisfy the following equations:
1_2fd2q(l+nq)IcqI2[{K1(N+l,N+l;q)_K2(N,N;q)}{wq2+r~}_1 + {K1(N, N; q) — K2(N, N; q)}{w~+
1 +h_2fd2qnqIcqI2[{K1(N,N;q)_K2(N,N;q)}{w~ +F~} +{K
(7)
1],
1(N+ 1,N+ 1; q) —K2(N,N; q)}{o.~~~ + F~}— ~
=
fd2q(1 +flq)ICqI2[{Ki(N+ 1,N+ 1; q)—K
2 +
2(N,N; q)}(wq ~~N)’((~’q ~~N) + {K
2 + F~)]
1(N, N; q)
—
K2(N, N; q)}(’~N Wq)/((wq —
—
~N)
+ 7j_2 fd2qnqlCql2[{K
2 + F~)
1(N, N; q)
—
K2(N,N; q)}(wq + L\N)/((Wq + ~N)
+ {K
2+ I’~)] (8) 1(N+ 1, N+ 1; q) K2(N, N, q)} (~N “q)’((~N (.~.)q) The terms occurring in the above equations can be physically interpreted as arising from the processes involving emission (1 + flq terms) and absorption (flq terms) of phonons. The functions K 1 and K2 are given by .
—
—
—
~
(9)
~
~
(10)
f
(11)
J~,~’(X,q~,X’)-dx~(XX)exp(iq~x)~N(~X).
112 is the radius of the ground Landau orbit and X the centre coordinate the Landau cyclotron orbit. flq stands the occupancy of the phonons in the mode q, given by ØN(x) areofthe wave functions, r0 =for(h/eB) flq
=
[exp(i371wq)
—
1]_l.
(12)
Eq. (8) implies that the frequency shift no longer vanishes as opposed to the electron impurity case [4] and is rather complicated. For electron acoustic phonon interactions we have ICqI2Aq
and
wq—sq,
(13,14)
s being the sound wave velocity. A is constant and contains the mass density and the deformation potential corresponding to the Si (100) surface. When the phonon energies are small compared to the thermal energy ( kB T = 1 /i3) we can write flq kB T/hqs and then the solutions of eq. (7) for a few lowest Landau transitions are summarized below. (i) N 0 * N 1 transition. 1
r[1 —7~+7~ exp(’y~)Ei(’y~)] .
This result has been reported earlier [5]. 128
(15)
Volume 70A, number 2
19 February 1979
PHYSICS LETTERS
(ii) N =1 * N =2 transition. 1 =~r[2_2’y~ —3’y~—’y~ ~
(16)
.
(iii) N 2* N =3 transition. 1
= ~
r [4
—
{4’y~+
14’y~+ ~
+ i~1~
+~ ~
164
4O} + {~‘y~+ 4~4 +
+
24y~+ 1 2} ~ exp (~) Ei (ij)].
(17)
(iv) N=3* N 4 transition. ~
(18)
4}]
—{36y~+ 2104 + 318~4+q2.y~ +2ji~y~O +~ 7~2+~‘y~ where r = 2IrAkBT/s3h3, “N = rNrO/s.~/~(N= 0, 1, 2, ), and Ei(x) stands for an exponential integral given by ...
Ei(x)f~-_—du.
(19)
Reduced resonance linewidths 7N given by eqs. (15), (16), (17) and (18) are plotted against the reduced temperature r in fig. 1. Some interesting features of these curves are enumerated below. (1) Reduced widths ‘IN monotonously grow with the increase of the temperature. (2) At any fixed temperature differences among the linewidths due to various transitions are almost constant. Also this constancy is maintained throughout the temperature region of interest. (3) The linewidth arising due to the N = 0* N = 1 transition is maximum and reduces in magnitude for the transitions of higher Landau index N. This reduction of the resonance width for higher Landau levels can be explained as below. The de Broglie wavelength of any Landau level is of the order of r 0/(2N + 1)1/2, thus decreases as N increases and the electron—phonon interaction weakens as N increases thereby reducing the linewidths due to the transitions between higher Landau levels. Temperature dependent cyclotron resonance linewidth data of Kueblbeck and Kotthaus are fitted to the elec5cmAec. rs-5.9~jlO 0100A — Theoretical curve e Exp~riment~l an- points From reF3 •
I
i
/
/
-
/
/
/
/
2.5
2.0
N-0--N- 1 N-i ~N-2
15
N-2—-N-3 I -3-=N-4 to ________________________________ 0 2.0 4 6.0 8.0 10.0 12.0
—T---. Fig. 1.
Reduced widths
31t3.
“N = rNro/s%J~are plotted against the reduced temperature r = 2irAk8T/s
129
Volume 70A, number 2 tronic transitions between two lowest
PHYSICS LETTERS
Landau levels, characterized by eq. (15). One
19 February 1979 experimental point correspond-
ing to T 65 K is fitted on the theoretical curve and thus the constant A is found (A = 2.8 X l0~50cgs). This in turn gives rise to a deformation potential 10 eV. This value is in reasonable agreement with that found for the inversion layers in Si. Other experimental points are close to the theoretical curve and thus a good agreement is
found to result. It should be remarked that the treatment based on the extension of Adams and Holstein’s [6] theory leads to a divergence at w = w0, which is circumvented in the present paper. The two-dimensional phonon model used here is also justified based on the momentum balance in the emission and absorption scattering rates. More detailed results will be reported later. I am grateful to Professor R.J. Elliott and Dr. R.A. Stradling for encouragement and fruitful discussions. Initial guidance of Professor S. Fujita is gratefully acknowledged. References [1] R. Kubo,J. Phys. Soc. Japan 12(1957) 570. 121 S. Fujita and C.C. Chen, Intern. J. Theor. Phys. 2(1969) 59; J.R. Barker, J. Phys. C 6 (1973) 2663; A. Lodder and S. Fujita, J. Phys. Soc. Japan 25 (1968) 775. [3] H. KueblbeckandJ.P. Kotthaus, Phys. Rev. Lett. 35(1975)1019. [4] M. Prasad and S. Fujita, Solid State Commun. 23 (1977) 551; S. Fujita and M. Prasad, J. Phys. Chem. Solids, to be published. [51 M. Prasad, T.K. Srinivas and S. Fujita, Solid State Commun. 24 (1977) 439. [6] E.N. Adams and T.D. Holstein, J. Phys. (Them. Solids 10 (1959) 254.
130
PHYSICS LETTERS
19 February 1979
QUANTUM THEORY OF THE CYCLOTRON RESONANCE LINESHAPE FOR A TWO-DIMENSIONAL ELECTRON-PHONON SYSTEM 1
Mahendra PRASAD
Department of Physics and Astronomy, State University of New York at Buffalo, Amherst, NY 14260, USA
Received 12 July 1978 Revised manuscript received 22 November 1978
Based on the super resolvent operator representation of Kubo’s current correlation formula [1] and its proper connected diagram expansion [2] a theory of cyclotron resonance lineshape is presented. Effects of the average phonon field are taken into account when treating the scattering of a given electron with the reference phonon. This gives rise to a self-consistent equation for the irreducible collision operator, which possesses a built in structure of the “gain” and “loss” like terms occurring in the kinetic description of the relaxation process. Equations for the cyclotron resonance linewidths rN and the frequency shifts ~N associated with the electronic transitions between the Landau subbands N and N+ 1 are derived for the case when the electron—phonon interaction induces the scatteringof the electron in the same Landau subband. Temperature dependent cyclotron resonance linewidth data of Kueblbeck and Kotthaus [3] compare very well with the theoretical prediction made here and one parameter fitting yields the deformation potential in reasonable agreement with that found for silicon. Kubo’s formula for the dynamic magnetoconductivity tensor is given by
à~_(w)= i lim urn UmL2 tr [TR{(a~i/au) [‘to~‘ph
+ 2’-11e—ph
—
z]1J~}]
(1)
a-.O u—~O
where “tr” and “TR” stand for the single electron and many phonon traces, respectively, h 0, single electron hamiltonian under a magnetic field perpendicular to the surface in the Landau gauge, HPh, many phonon hamiltonian and He_ph stands for the electron—phonon coupling hamiltonian. A “hat” () on the letters denotes the Liouville operator corresponding to them. 1e—ph j~~u~)]}—1 (2) = {1 + exp [I3(12~ + HPh + ~‘ 1+ andj_ are the current components; ~ is the Fermi energy, L2 the normalization area, u isa c-number and z stands for w + ia. The cyclotron resonance lineshape arises mainly from the super resolvent operator occurring in eq. (1) and the interaction term from ff can be dropped, then ,i-~ii~. Proper connected diagram analysis of eq. (1) then yields the following result: —
—
—
o÷_(w)= i lim lim LimL2 tr [(a/au) ~O>ph~’I
(3)
‘
a-GO u—*O
1He_ph)~~, (4,5) 0~‘ph ~_z]~hj~, b A2
[1z
—
—
means many phonon average.
Finding the traces occurring in eq. (3) and the u-derivative we obtain the following result: Present address: Clarendon Laboratory and Department of Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK.
127
Volume 70A, number 2
PHYSICS LETTERS
19 February 1979
i E[f(~)_f(EN÷1)1(N+l)
e2
N
(6)
(wo—w+z~N)+iFN
where EN = (N+~)hw0,f(E), the Fermi distribution function and
rN, ~N
satisfy the following equations:
1_2fd2q(l+nq)IcqI2[{K1(N+l,N+l;q)_K2(N,N;q)}{wq2+r~}_1 + {K1(N, N; q) — K2(N, N; q)}{w~+
1 +h_2fd2qnqIcqI2[{K1(N,N;q)_K2(N,N;q)}{w~ +F~} +{K
(7)
1],
1(N+ 1,N+ 1; q) —K2(N,N; q)}{o.~~~ + F~}— ~
=
fd2q(1 +flq)ICqI2[{Ki(N+ 1,N+ 1; q)—K
2 +
2(N,N; q)}(wq ~~N)’((~’q ~~N) + {K
2 + F~)]
1(N, N; q)
—
K2(N, N; q)}(’~N Wq)/((wq —
—
~N)
+ 7j_2 fd2qnqlCql2[{K
2 + F~)
1(N, N; q)
—
K2(N,N; q)}(wq + L\N)/((Wq + ~N)
+ {K
2+ I’~)] (8) 1(N+ 1, N+ 1; q) K2(N, N, q)} (~N “q)’((~N (.~.)q) The terms occurring in the above equations can be physically interpreted as arising from the processes involving emission (1 + flq terms) and absorption (flq terms) of phonons. The functions K 1 and K2 are given by .
—
—
—
~
(9)
~
~
(10)
f
(11)
J~,~’(X,q~,X’)-dx~(XX)exp(iq~x)~N(~X).
112 is the radius of the ground Landau orbit and X the centre coordinate the Landau cyclotron orbit. flq stands the occupancy of the phonons in the mode q, given by ØN(x) areofthe wave functions, r0 =for(h/eB) flq
=
[exp(i371wq)
—
1]_l.
(12)
Eq. (8) implies that the frequency shift no longer vanishes as opposed to the electron impurity case [4] and is rather complicated. For electron acoustic phonon interactions we have ICqI2Aq
and
wq—sq,
(13,14)
s being the sound wave velocity. A is constant and contains the mass density and the deformation potential corresponding to the Si (100) surface. When the phonon energies are small compared to the thermal energy ( kB T = 1 /i3) we can write flq kB T/hqs and then the solutions of eq. (7) for a few lowest Landau transitions are summarized below. (i) N 0 * N 1 transition. 1
r[1 —7~+7~ exp(’y~)Ei(’y~)] .
This result has been reported earlier [5]. 128
(15)
Volume 70A, number 2
19 February 1979
PHYSICS LETTERS
(ii) N =1 * N =2 transition. 1 =~r[2_2’y~ —3’y~—’y~ ~
(16)
.
(iii) N 2* N =3 transition. 1
= ~
r [4
—
{4’y~+
14’y~+ ~
+ i~1~
+~ ~
164
4O} + {~‘y~+ 4~4 +
+
24y~+ 1 2} ~ exp (~) Ei (ij)].
(17)
(iv) N=3* N 4 transition. ~
(18)
4}]
—{36y~+ 2104 + 318~4+q2.y~ +2ji~y~O +~ 7~2+~‘y~ where r = 2IrAkBT/s3h3, “N = rNrO/s.~/~(N= 0, 1, 2, ), and Ei(x) stands for an exponential integral given by ...
Ei(x)f~-_—du.
(19)
Reduced resonance linewidths 7N given by eqs. (15), (16), (17) and (18) are plotted against the reduced temperature r in fig. 1. Some interesting features of these curves are enumerated below. (1) Reduced widths ‘IN monotonously grow with the increase of the temperature. (2) At any fixed temperature differences among the linewidths due to various transitions are almost constant. Also this constancy is maintained throughout the temperature region of interest. (3) The linewidth arising due to the N = 0* N = 1 transition is maximum and reduces in magnitude for the transitions of higher Landau index N. This reduction of the resonance width for higher Landau levels can be explained as below. The de Broglie wavelength of any Landau level is of the order of r 0/(2N + 1)1/2, thus decreases as N increases and the electron—phonon interaction weakens as N increases thereby reducing the linewidths due to the transitions between higher Landau levels. Temperature dependent cyclotron resonance linewidth data of Kueblbeck and Kotthaus are fitted to the elec5cmAec. rs-5.9~jlO 0100A — Theoretical curve e Exp~riment~l an- points From reF3 •
I
i
/
/
-
/
/
/
/
2.5
2.0
N-0--N- 1 N-i ~N-2
15
N-2—-N-3 I -3-=N-4 to ________________________________ 0 2.0 4 6.0 8.0 10.0 12.0
—T---. Fig. 1.
Reduced widths
31t3.
“N = rNro/s%J~are plotted against the reduced temperature r = 2irAk8T/s
129
Volume 70A, number 2 tronic transitions between two lowest
PHYSICS LETTERS
Landau levels, characterized by eq. (15). One
19 February 1979 experimental point correspond-
ing to T 65 K is fitted on the theoretical curve and thus the constant A is found (A = 2.8 X l0~50cgs). This in turn gives rise to a deformation potential 10 eV. This value is in reasonable agreement with that found for the inversion layers in Si. Other experimental points are close to the theoretical curve and thus a good agreement is
found to result. It should be remarked that the treatment based on the extension of Adams and Holstein’s [6] theory leads to a divergence at w = w0, which is circumvented in the present paper. The two-dimensional phonon model used here is also justified based on the momentum balance in the emission and absorption scattering rates. More detailed results will be reported later. I am grateful to Professor R.J. Elliott and Dr. R.A. Stradling for encouragement and fruitful discussions. Initial guidance of Professor S. Fujita is gratefully acknowledged. References [1] R. Kubo,J. Phys. Soc. Japan 12(1957) 570. 121 S. Fujita and C.C. Chen, Intern. J. Theor. Phys. 2(1969) 59; J.R. Barker, J. Phys. C 6 (1973) 2663; A. Lodder and S. Fujita, J. Phys. Soc. Japan 25 (1968) 775. [3] H. KueblbeckandJ.P. Kotthaus, Phys. Rev. Lett. 35(1975)1019. [4] M. Prasad and S. Fujita, Solid State Commun. 23 (1977) 551; S. Fujita and M. Prasad, J. Phys. Chem. Solids, to be published. [51 M. Prasad, T.K. Srinivas and S. Fujita, Solid State Commun. 24 (1977) 439. [6] E.N. Adams and T.D. Holstein, J. Phys. (Them. Solids 10 (1959) 254.
130