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Theory of the temperature-dependent cyclotron resonance lineshape of inversion-layer electrons
Surface Science 73 (1978) 505-508 0 North-Holland Publishing Company
THEORY OF THE TEMPERATURE-DEPENDENT LINESHAPE OF INVERSION-LAYER
CYCLOTRON
RESONANCE
ELECTRONS
M. PRASAD, T.K. SRINIVAS and S. FUJITA Department of Physics and Astronomy, New York 14260, USA
State University of New York at Buffalo, Amherst,
Unlike the electron-impurity interation, the electron-accoustic phonon interaction generates a temperature-dependent cyclotron resonance lineshape. The difference is due to the temperature-dependent phonon distribution and the quasi-inelastic nature of the electron phonon scattering. Calculations based on the proper connected diagram expansion are in qualitative agreement with experiments on inversion-layer electrons in Si in the temperature range 8-65 K.
Recent measurements by Kueblbeck and Kotthaus [l] of the cyclotron resonance of inversion layer electrons in Si show that (a) the resonance width r approximately varies linearly with temperature Tin the domain 8-6.5 K, and (b) the resonance maximum shifts toward the low frequency side with T. Theories by Ando and Uemura [2] and by two of the present authors (M.P. and S.F.) [3,4] indicate that the impurity contribution to width P and shift A are temperature-independent and that A = 0, both of which are in qualitative agreement with data [ 1,5] at liquid helium temperatures and at surface carrier densities higher than 1 X 10” cm-*. The temperature-dependent lineshape at higher temperatures may therefore be thought to arise from the electron-phonon interaction. In the present paper we report on calculations of the resonance lineshape due to the electron-accousticphonon interaction in the extreme quantum region, based on the general theory developed earlier [6,7], the proper connected diagram expansion of Kubo’s formula [8] for the dynamic conductivtity. The average power p delivered by a circularly polarized microwave of frequency o and field strength E is given by p = $E* Re{a+_(o)}
,
(1)
where u+_ is the current correlation integral (Kubo’s formula). We assume that (a) the magnetic field B is perpendicular to the plane (layer) of the electron motion, (b) field B is so high that the resonance lineshape is mainly determined by the contribution associated with transitions between states of lowest Landau quantum numbers n = 0 and 1, (c) phonons act only as scatterers, and the phonon distribution is characterized by the Planck distribution function nq 2 [exp(@w,) - 11-l with p =(knT)-’ and w, being the frequency of a phonon with wave vector Q, (d) the 505
506
M. Prasad et al. /Temperature-dependent
cyclotron resonance lineshape
virtual and real transition processes are described to the second order in the electron-phonon interaction strength. Under these conditions, we found that
where no is the density of electrons, and wg E eBfm* is the cyclotron Width l7 and shift A satisfy the follo~ng equation
frequency.
-iK’-(mot-A-o) = 2nh-2~d*q(;
+ 2nFi-2
tnq)lCq12t2e-’
s
d2qnqlCq12t2e-’
w +a _ 1 _ z 0 4 1
wO+Atwq-w+ilT’
where C, represents the electron-phonon t = ;rgq=
interaction
)
and (4)
with r. 5 (~/eB) ‘12 being the radius of the cyclotron orbit corresponding to the states with n = 0. The first integral with the factor (1 t n,) represents the contribution of a relaxation process involving emission of a phonon while the second term with the factor n4 arises from a process with absorption of a phonon. We note that eq. (3) reduces to eq. (2.4) of ref, [3] for the case of the electron-impurity interaction if the energy of phonon, AL+,, is neglected. At the resonance maxnium: w = w. + A, comparison of the imaginary part of eq. (3) yields 1 = 27rRm2 d*q(l t 2nq)IC412r2e-‘(~~ 1 Comparison -A=
276-*
t 271Fl-*
+ r2)-l.
(51
of the real parts at w = o. leads to s
A - w4
d2q(l t nq)ICq/2t2e-’ (ws
-
A)2 + r2
A+u9
s
d2q%$~~2f2e-t
(A + w9)2 + r~ .
(6)
Eq. (5) indicates that the width I? monotonously grows with (1 + 2n9) as temperature is raised. Eq. (6) means that shift A is non-zero and depends on temperature. For the electron-accoustic phonon interaction, we may assume that wq=sq
(7)
where s is the speed of sound, and that [9] Cq = i~q(~/2~#9)~12
,
(8)
M. Prasad et al. / Temperature-dependent
lCq12 =Aq
A =DQz(2ps)-’
,
cyclotron resonance lineshape
)
(9)
where D is the deformation potential constant, density. At relatively high temperatures, nq 1 kBT/hq
mass
(10)
may be assumed. Under the assumptions
(7)-(IO),
1 - y2 + y4 exp(y2) Ei(r’)]
where y2 = r2ri/2s2, Ei(x) = J
and p the two-dimensional
9 1
1 = (ksTA/s3h3)[
507
eq. (5) can be reduced to
,
(11)
and Ei(x) is the exponential
integral:
eC”UP1du.
Fig. 1. shows the curve of the dimensionless width y against the dimensionless temperature r E (27r)‘A kBT/s3h3. Experimental data taken from ref. [ 11, indicated by circles, are fitted by adjusting the constant A = 1.76 X 1O-49 (cgs) at T= 6.5 K,
-
THEORETICAL 0
CURVE
EXPERIMENTAL FROM
REF
POINTS I
3.0 -
y.g
2.5-
t
i 2.0 -
1.5-
0
l.OO.0
2.0
4.0
6.0
---C--O
Fig. 1. The solid curve of the reduced width y against the reduced temperature t represents the solution of eq. (11). Experimental points correspond to the observed ratios wg/r = 6, 3.2, 2.5, 2 and 1.8 at temperatures T= 7.8, 25.5, 41.5, 56 and 65 K, respectively, as reported in ref. [ 11.
508
M. Prasad et al. / Temperature-dependent
cyclotron resonance lineshape
where s = 5.9 X 10’ cm set-’ is used. This value of A yields D = 9.0 X lo-” erg for the deformation potential constant, which appears to be reasonable [lo]. The temperature dependence of width r appears to be in good order. The frequency shift A calculated from (6) using the same data, vanishes above 20 K. More quantitative analysis, however, must come after the theory is improved, in particular by taking account of electron transition involving a few lowest Landau levels, not just the lowest two. It is also important to solve eqs. (5) and (6) without using the high temperature approximation (9). At extreme low temperatures so that nq < 1, eq. (5) shows that width r approaches a constant independent of temperature. In conclusion the electron-phonon interaction, because of its quasi-inelastic nature (w4 # 0), gnerates a lineshape quite different from that due to the electronimpurity interaction, which is characterized by A = 0 and by the temperature independence.
References [l] H. Kueblbeck and J.P. Kotthaus, Phys. Rev. Letters 35 (1975) 1019; T.A. Kennedy, B.D. McCombe, D.C. Tsui and R.J. Wagner, in: Proc. 2nd Intern. Conf. EP2DS, pp. 912-916. [2] T. Ando and Y. Uemura, J. Phys. Sot. Japan 36 (1974) 956; T. Ando, J. Phys. Sot. Japan 38 (1975) 989. [3] S. Fujita and M. Prasad, J. Phys. Chem. Solids 38 (1977) 1351. [4] M. Prasad and S. Fuji@ Solid State Commun. 21 (1977) 1105. [5] G. Abstreiter, J.F. Koch, P. Goy and Y. Couder, Phys. Rev. B14 (1976) 2492; T.A. Kennedy, R.J. Wagner, B.D. McCombe and D.C. Tsui, Phys. Rev. Letters 35 (1975) 1031. [6] S. Fujita and C.C. Chen, J. Theor. Phys. 2 (1969) 59. [7] A. Lodder and S. Fujita, J. Phys. Sot. Japan 25 (1968) 774. [8] R. Kubo, J. Phys. Sot. Japan 12 (1957) 570. [9] R. Kubo, S.J. Miyake and N. Hashitsume, in: Solid State Physics, Vol. 17, Eds. F. Seitz and D. Turnbull (Academic Press, New York, 1965) p. 256. [lo] A value of 9 eV for the bulk deformation potential constant was suggested in the work by Ohtsuka et al., private communication.
THEORY OF THE TEMPERATURE-DEPENDENT LINESHAPE OF INVERSION-LAYER
CYCLOTRON
RESONANCE
ELECTRONS
M. PRASAD, T.K. SRINIVAS and S. FUJITA Department of Physics and Astronomy, New York 14260, USA
State University of New York at Buffalo, Amherst,
Unlike the electron-impurity interation, the electron-accoustic phonon interaction generates a temperature-dependent cyclotron resonance lineshape. The difference is due to the temperature-dependent phonon distribution and the quasi-inelastic nature of the electron phonon scattering. Calculations based on the proper connected diagram expansion are in qualitative agreement with experiments on inversion-layer electrons in Si in the temperature range 8-65 K.
Recent measurements by Kueblbeck and Kotthaus [l] of the cyclotron resonance of inversion layer electrons in Si show that (a) the resonance width r approximately varies linearly with temperature Tin the domain 8-6.5 K, and (b) the resonance maximum shifts toward the low frequency side with T. Theories by Ando and Uemura [2] and by two of the present authors (M.P. and S.F.) [3,4] indicate that the impurity contribution to width P and shift A are temperature-independent and that A = 0, both of which are in qualitative agreement with data [ 1,5] at liquid helium temperatures and at surface carrier densities higher than 1 X 10” cm-*. The temperature-dependent lineshape at higher temperatures may therefore be thought to arise from the electron-phonon interaction. In the present paper we report on calculations of the resonance lineshape due to the electron-accousticphonon interaction in the extreme quantum region, based on the general theory developed earlier [6,7], the proper connected diagram expansion of Kubo’s formula [8] for the dynamic conductivtity. The average power p delivered by a circularly polarized microwave of frequency o and field strength E is given by p = $E* Re{a+_(o)}
,
(1)
where u+_ is the current correlation integral (Kubo’s formula). We assume that (a) the magnetic field B is perpendicular to the plane (layer) of the electron motion, (b) field B is so high that the resonance lineshape is mainly determined by the contribution associated with transitions between states of lowest Landau quantum numbers n = 0 and 1, (c) phonons act only as scatterers, and the phonon distribution is characterized by the Planck distribution function nq 2 [exp(@w,) - 11-l with p =(knT)-’ and w, being the frequency of a phonon with wave vector Q, (d) the 505
506
M. Prasad et al. /Temperature-dependent
cyclotron resonance lineshape
virtual and real transition processes are described to the second order in the electron-phonon interaction strength. Under these conditions, we found that
where no is the density of electrons, and wg E eBfm* is the cyclotron Width l7 and shift A satisfy the follo~ng equation
frequency.
-iK’-(mot-A-o) = 2nh-2~d*q(;
+ 2nFi-2
tnq)lCq12t2e-’
s
d2qnqlCq12t2e-’
w +a _ 1 _ z 0 4 1
wO+Atwq-w+ilT’
where C, represents the electron-phonon t = ;rgq=
interaction
)
and (4)
with r. 5 (~/eB) ‘12 being the radius of the cyclotron orbit corresponding to the states with n = 0. The first integral with the factor (1 t n,) represents the contribution of a relaxation process involving emission of a phonon while the second term with the factor n4 arises from a process with absorption of a phonon. We note that eq. (3) reduces to eq. (2.4) of ref, [3] for the case of the electron-impurity interaction if the energy of phonon, AL+,, is neglected. At the resonance maxnium: w = w. + A, comparison of the imaginary part of eq. (3) yields 1 = 27rRm2 d*q(l t 2nq)IC412r2e-‘(~~ 1 Comparison -A=
276-*
t 271Fl-*
+ r2)-l.
(51
of the real parts at w = o. leads to s
A - w4
d2q(l t nq)ICq/2t2e-’ (ws
-
A)2 + r2
A+u9
s
d2q%$~~2f2e-t
(A + w9)2 + r~ .
(6)
Eq. (5) indicates that the width I? monotonously grows with (1 + 2n9) as temperature is raised. Eq. (6) means that shift A is non-zero and depends on temperature. For the electron-accoustic phonon interaction, we may assume that wq=sq
(7)
where s is the speed of sound, and that [9] Cq = i~q(~/2~#9)~12
,
(8)
M. Prasad et al. / Temperature-dependent
lCq12 =Aq
A =DQz(2ps)-’
,
cyclotron resonance lineshape
)
(9)
where D is the deformation potential constant, density. At relatively high temperatures, nq 1 kBT/hq
mass
(10)
may be assumed. Under the assumptions
(7)-(IO),
1 - y2 + y4 exp(y2) Ei(r’)]
where y2 = r2ri/2s2, Ei(x) = J
and p the two-dimensional
9 1
1 = (ksTA/s3h3)[
507
eq. (5) can be reduced to
,
(11)
and Ei(x) is the exponential
integral:
eC”UP1du.
Fig. 1. shows the curve of the dimensionless width y against the dimensionless temperature r E (27r)‘A kBT/s3h3. Experimental data taken from ref. [ 11, indicated by circles, are fitted by adjusting the constant A = 1.76 X 1O-49 (cgs) at T= 6.5 K,
-
THEORETICAL 0
CURVE
EXPERIMENTAL FROM
REF
POINTS I
3.0 -
y.g
2.5-
t
i 2.0 -
1.5-
0
l.OO.0
2.0
4.0
6.0
---C--O
Fig. 1. The solid curve of the reduced width y against the reduced temperature t represents the solution of eq. (11). Experimental points correspond to the observed ratios wg/r = 6, 3.2, 2.5, 2 and 1.8 at temperatures T= 7.8, 25.5, 41.5, 56 and 65 K, respectively, as reported in ref. [ 11.
508
M. Prasad et al. / Temperature-dependent
cyclotron resonance lineshape
where s = 5.9 X 10’ cm set-’ is used. This value of A yields D = 9.0 X lo-” erg for the deformation potential constant, which appears to be reasonable [lo]. The temperature dependence of width r appears to be in good order. The frequency shift A calculated from (6) using the same data, vanishes above 20 K. More quantitative analysis, however, must come after the theory is improved, in particular by taking account of electron transition involving a few lowest Landau levels, not just the lowest two. It is also important to solve eqs. (5) and (6) without using the high temperature approximation (9). At extreme low temperatures so that nq < 1, eq. (5) shows that width r approaches a constant independent of temperature. In conclusion the electron-phonon interaction, because of its quasi-inelastic nature (w4 # 0), gnerates a lineshape quite different from that due to the electronimpurity interaction, which is characterized by A = 0 and by the temperature independence.
References [l] H. Kueblbeck and J.P. Kotthaus, Phys. Rev. Letters 35 (1975) 1019; T.A. Kennedy, B.D. McCombe, D.C. Tsui and R.J. Wagner, in: Proc. 2nd Intern. Conf. EP2DS, pp. 912-916. [2] T. Ando and Y. Uemura, J. Phys. Sot. Japan 36 (1974) 956; T. Ando, J. Phys. Sot. Japan 38 (1975) 989. [3] S. Fujita and M. Prasad, J. Phys. Chem. Solids 38 (1977) 1351. [4] M. Prasad and S. Fuji@ Solid State Commun. 21 (1977) 1105. [5] G. Abstreiter, J.F. Koch, P. Goy and Y. Couder, Phys. Rev. B14 (1976) 2492; T.A. Kennedy, R.J. Wagner, B.D. McCombe and D.C. Tsui, Phys. Rev. Letters 35 (1975) 1031. [6] S. Fujita and C.C. Chen, J. Theor. Phys. 2 (1969) 59. [7] A. Lodder and S. Fujita, J. Phys. Sot. Japan 25 (1968) 774. [8] R. Kubo, J. Phys. Sot. Japan 12 (1957) 570. [9] R. Kubo, S.J. Miyake and N. Hashitsume, in: Solid State Physics, Vol. 17, Eds. F. Seitz and D. Turnbull (Academic Press, New York, 1965) p. 256. [lo] A value of 9 eV for the bulk deformation potential constant was suggested in the work by Ohtsuka et al., private communication.