Effect of intervalley electron-electron coupling on the cyclotron resonance line shape for electrons in the Si surface inversion layers

Surface Science 98 (1980) 437-441 0 North-Hol~nd Pub~s~m~ Company and Yamada Science Foundation

EFFECT OF INTERVALLEY ELECTRON-ELECTRON COUPLING ON THE CYCLOTRON RESONANCE LINE SHAPE FOR ELECTRONS IN THE Si

SURFACE INVERSION

LAYERS

C.S. TING Uttiversity of Houston, Houston, Texas 77004, USA

Received 20 July 1979

The cyclotron resonance for two types of carriers is studied by using the method of kinetic equations. The memory function approach is applied to evaluate the electron-electron relaxations time T&W). We show that it is the imaginary part of 7;’ (w) which contributes significantly to the cyclotron resonance tine shape for electrons in a Si surface inversion layer at low temperatures.

There are two sets of nonequivalent subbands on the Si(100) surface [I]. One with two degenerate valleys at the center of the two-dimensional Brilloum Zone and one with four degenerate valleys centered near the corners along the (110) directions. The subbands of the center valleys are energetically lower than those of the corner valleys. Application of a uniaxial stress along (001) direction causes two of the four corner valleys to be lowered in energy with respect to the center valleys [2]. Therefore it is possible for electrons to occupy the subbands in both the center valleys and the corner valleys by either applying a uniaxial stress or by rising the temperature. Since the mass of the carriers are different for these two nonequivalent valleys, one would expect two distinct resonances in cyclotron resonance experiments. Instead, experiments [3,4] have indicated that there is only a single resonance. These experimental results had been qualitatively explained by a theory of Kelly and Falicov [5] who proposed a model in which an extremely strong intervalley coupling is assumed and that causes the formation of charge density waves. Similar problems were studied by Takada and Ando [6] using Fermi liquid theory at the temperature T= 0 K, and also by Appel and Overhauser [7] using the method of kinetic equations at finite temperature. A careful study of these works shows that both of the previous approaches are valid only in the limits of o + EF and T< EF, where o is the angular frequency of the applied ac electric field, EI; can be regarded as the Fermi energy of the carriers in either the center valleys or the corner valleys. However, the experiments [4,5] were performed at carrier density ~2= 5 X 10” c~ll-~ and w = 3.5 meV. Assuming the equal occupancy of these two nonequivalent set of valleys. We fined EF 2: 1.0 meV at T = 0 K. At 437

higher temperatures Er; becomes large and negative, the statistics of these carriers transform from Fermi-Dirac to Boltzmann distribution. Therefore none of the limits w < EF and 7’< EF were satisfied under the experimental condition. In the present manuscript we shall use the method of kinetic equations to study the cyclotron resonance for carriers of two different types, one with mass IMPand the other with mass m2. The inverse relaxation time T,‘(W) between carriers will be evaluated to the lowest order in the intervalley electron-electron interaction using the memory function approach. Assuming that the external magnetic field N is ~e~endicular to the Si(lO0) surface and a circular polarized ac electric field is applied along the surface, the conductivity cf.. = o,, .- io,, can be obtained from the method of kinetic equations [7,8],

with

where nj and Tj are respectively the density and the single particle scattering time of carriers with mass mi. And r,(w) is the relaxation time between carriers with mass is evaluated only to the lowest order in the intervalley elecml and m2. If ril(o) tron-electron interaction v12(q), it is straight forward to show that ri’(~> = 44(u), and the memory function M(w) is given by ]9]

sR(q,

w) contains a real part and an imaginary respectively given by

Re sR($, W) = i’ 2’ 0 + Re S(‘)(q, X [Re S(‘)(q,

part, at finite temperature

they are

coth iW’ 5 ) i’ Im S(‘){q, w’) [Re S(*)(q, w’ ~ w)

w’ + w)] + Im ,S(*)(q, 0’) w’ - o) + Re S(r)(q,

W’ + o)]}

I

(3)

439

lm Sa(q, o) = 7 $ -co X Im S(‘)(q,

[coth($)

~ coth(*)]

w’) Im St2)(q, 0 - 0’) ,

(4)

where S(‘)(q. w) and SC2)(q, w) are respectively the density-density correlations in RPA approximation [I l] for carriers of type 1 and 2. The explicitly expression for S(‘)(q, w) or S(*)(q, w) at finite temperature can be obtained from the work of Ganguly and Ting [ I I]. At this stage it would be interesting to compare our approach with those of refs. [6] and [7]. The authors of ref. [i’] considered only the real part of r;‘(w) and neglected its imaginary part. This approximation may be valid in the limit of o -+ 0. It is more difficult to compare our work with ref. [6]. The authors there used the Fermi liquid theory to calculate the magnetic-conductivity. However, the comparison is still possible to be made if we expand the conductivity a_(w) of ref. [6] to the lowest order in the Fermi liquid interacting parameter A ,a, and identify it to be the corresponding term in the conductivity defined in eq. (1). A relation between Al2 and r,‘(w) can thus be obtained, and it depends on the magnetic field Hand the single particle scattering time ii. Since both of A,, and r,‘(o) are calculated for zero nlagnetic field and infinite long 3, this relation is mea~ngful only if U and 7,:’ are set equal to zero, A-

i I2

-2w

fE1FI2

( F7l,rn2

l/2

1

cm2 mlnl

-

md2 +m2n2

7-yo) e

The right hand side of the above equation depends on w, and r,l(w) has both a real part and an imaginary part. A 12 obtained in ref. [6] is independent of w and is real and positive. Therefore the above equation is valid only when o + 0, where Re M(w) 0~w and Im M(o) a u2. The numerical computation is performed under the following ideal situation: we assume that a suitable uniaxial stress has been applied on Si in the (001) direction, which lowers the ground state subbands of two of the four degenerate corner valleys and coincides them exactly with those of the center valleys. Under this condition the Fermi energies of carriers in these two nonequivalent sets of subbands are identical. Using the density of the carriers n = 0.5 X 1012 cms2, o = 3.5 meV (which is equivalent to the frequency v = 890.7 GHz). ml = 0.195 m, and m2 = 0.417 m,, where m, is the free electron mass. The computation for eqs. (3) and (4) can be done numerically and it involves triple integrations and is quite time consuming especially for Re M(o). At this stage the results are accurately obtained only for T = 10 K, we have Re M(u) =L 1.9 meV and Im M(w) = 0.15 meV. The imaginary part of M(o) obtained here corresponds to the inverse relaxation time ril of ref. [?I, and they seem to be in the same order of magnitude. Neglecting the imaginary part on the right hand side of eq. (5) the Fermi liquid interaction parameter Al2 “estimated” in the present calculation is Ai2 % 0.064, while the value of Al2 obtained from ref. [6] is 0.12. One of the causes for this difference might be

440

C.S. Ting f h-2rervaEe.v electron-electron

40

80

120

coupling

160

200

MAGNETIC FIELD (kG1

Fig. 1. The real part of u_(w) is calculated as a function of the magnetic field at T= 10 K and w = 3.5 meV. Here we have set M(w) = 0 for curve (a). In M(w) = 0.13 meV and Re M(u) = 0 for curve (b) and ImM(w) = 0.15 meV and Re Mu) = 1.9 meV for curve (c).

that we did the computation at w = 3.5 meV and the authors of ref. ]6] calculated A 12 at w = 0. substituting the values of M(w) to the conductivity formula defined by eq. (I), the real part of the conductivity has been calculated as a function of magnetic field H by taking err = wr2 = 3.5. Th e results are shown in fig. 1, where we have set M(o) = 0 for curve (a), Im M(w) # 0 and Re M(o) = 0 for curve (b), Im M(o) # 0 and Re M(o) $I 0 for curve Cc). It is clear that the real part of ~,r(w) or Im M(w) has little influence on the cyclotron resonance line shape at low temperature [7]. The reason we almost have a single resonance line (curve (c)) shape is entirely due to the large imag~ary part of T;~(w) which was compieteIy neglected in ref. [7]. At higher temperatures our numerical results show that Im M(W) u 1.4 and 3.1 meV respectively for 'T=30 and 90 K. We therefore expect Im Mfo) should have signi~cant influence on the line shape of the cyclotron resonance at higher temperature range. At present moment we are not able to obtain the high temperature results for Re M(w) because of the convergency problem. However, our preliminary calculation shows that the value of Re M(u) becomes smaller as the temperature is rising. The work of the temperature dependent line shape is in progress and the results will be presented elsewhere.

C.S. Ting / Intervalley electron-electron

coupling

441

Acknowledgement I would like to thank Mr. L.Y. Liou and Mr. T.H. Lin for computational This work is supported in part by AR0 contract DAAG 29-79-COO39.

help.

References [l] F. Stern and W.E. Howard, Phys. Rev. 163 (1967) 861. [2] G. Dorda, J. Eisele and H. Gesch, Phys. Rev. B17 (1978) 1785; and see the reference therein. [3] H. Kuhlbeck and J.P. Kotthaus, Phys. Rev. Letters 35 (1975) 1019. [4] P. Stallhofer, J.P. Kotthaus and J.F. Koch, Solid State Commun. 20 (1976) 519. [S] M.J. Kelly and L.M. Falicov, Phys. Rev. Letters 37 (1976) 1021; Phys. Rev. B15 (1977) 1983. [6] Y. Takada and T. Ando, Phys. Sot. of Japan 44 (1978) 905. [7] J. Appel and A.W. Overhauser, Phys. Rev. B18 (1978) 758. [8] Eq. (1) is slightly different from that in ref. [7], where we have not set 71 = 72 = 7. [9] Similar but not identical result was also obtained by J.J. Quinn (private communication). [lo] A.K. Ganguly and C.S. Ting, Phys. Rev. B16 (1976) 3541.