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Broadening effect on magnetothermal oscillations in 2-D electron systems
Solid State Communications, Printed in Great Britain.
BROADENING
Vol. 50, No. 1, pp. 35-37,
0038-1098/84 $3.00 + .OO Pergamon Press Ltd.
1984.
EFFECT ON MAGNETOTHERMAL
OSCILLATIONS
IN 2-D ELECTRON SYSTEMS
Y. Shiwa and A. Isihara Statistical
Physics Laboratory,
Department of Physics, State University Buffalo, NY, 14260, U.S.A.
of New York at Buffalo,
(Received 15 October 1983; in revised form 19 December 1983 by R.H. Silsbee)
Landau level broadening reduces considerably the amplitude of the temperature oscillations in 2-D electron systems when the magnetic field is changed adiabatically, but the oscillations are still much larger than the 3-D case. the delta functions as follows:
1. INTRODUCTION IN A RECENT ARTICLE [ 11, we have pointed out that adiabatic changes of strong magnetic field cause significant and peculiar temperature oscillations in a 2-D ideal electron system. The amplitude of these oscillations was found to be an order of magnitude larger than the case in bulk [2]. This largeness is due to the lack of electron motion in the direction of magnetic field and also to the sharp changes of the chemical potential at low temperatures as a function of l/ye = eO/pBH which is a crucial dimensionless parameter. Here, e. is the ideal Fermi energy in the absence of magnetic field, i.e., (h2/2m)(2nn), n and pBH being electron density and the field energy respectively. While the former cause exists at all temperatures, the latter is a low temperature characteristic of a 2-D electron system and is the primary factor for the large amplitude. The sharp changes of the chemical potential, which is the actual Fermi energy in the presence of magnetic field, are understandable because at low temperatures the Landau levels are filled one by one from the bottom. However, in real 2-D systems, the Landau levels may be broadened due to impurity scattering. Such broadening is expected to reduce the magnetothermal oscillations as in the case of the de Haas-van Alphen oscillations. In view of the anomalously large effect in the ideal case, it is then very important to investigate the broadening effect on the temperature oscillations. The present paper has been written for this investigation.
8(e-eE,)+i (E 7T
6(e - E,) by Lorentzian
functions
l/T
(1)
en)2 + (l/7)2’
where l/r is the half width. Here, and henceforth, we have set Ii = 1 and 2m = 1. We introduce a broadening parameter by r = l/(2&),
(2)
where a2 = pBH.
(3)
The Landau levels are given by efl = (2n+l)/~~H+p~H,
(n=0,1,2
,...)
(4)
where the Lande’s g-factor is chosen to be 2 and no effective mass correction is made. Broadening in the dHvA effect is often characterized by the temperature defined by (5) It is called the Dingle temperature. It turns out that the direct use of the above replace.ment in the grand potential leads to undesirable divergent terms. Similar divergences appear in the 3-D case which Dingle eliminated by a somewhat obscure argument that the energy could be minus infinity. By integrating the grand potential twice before applying the above replacement, we have succeeded to make a satisfactory approach in which obscure steps are eliminated. From the grand potential, we have derived the entropy as follows:
2. METHOD OF APPROACH Effects of Landau level broadening appear in some other quantum oscillations. Based on a phenomenological consideration of level broadening, Dingle found that the amplitude of the dHvA oscillations in 3-D is reduced [3]. We have adopted a similar but improved method. Our improvement consists of integrating by parts the grand partition function and then replacing
cm (WY) .smh (n21/a)
1’ Here, A is the surface area, /I = l/kT, L(x) is the Langevin function 35
(6)
36
BROADENING
EFFECT ON MAGNETOTHERMAL
OSCILLATIONS
Vol. 50, ho. 1
Fig. 1. Magnetothermal oscillations in 2-D electron systems with broadening parameter I’ = 0.3. Solid curve: n’/cr = 1.5 and dotted curve: rr2/o = 0. The former corresponds roughly to 1 K and 10 T in Si inversion layers. The latter represents the case of absolute zero. The arrows indicate the ordinate to be used.
exponentially. Since the derivative in the denominator of equation (10) represents the specific heat, this difference affects the magnitude of the left side significantly. In order to obtain explicit results, we have to estimate the broadening parameter. Following Brailsford [4], we interpret 7/2 as scattering time which can be estimated from mobility. For Si inversion layer electrons with effective mass 0.19mo, m, being the bare mass, and mobility lo4 cm2 V-’ s-’ we find i? = 27rr 0.3 for 10 T. This corresponds to To of order 0.2 K. For 3-D, McCombe and Seidel [5] got a similar value. Figure 1 represents our main results obtained without the effective mass and for the case n2/a = 1S. This choice of the parameter corresponds roughly to 1 K and 10 T. In comparison with Fig. 1 of our previous article [ 11, the amplitude is much less and the oscillations somewhat distorted from the sinusoidal curve. The dotted curve corresponds to the case of absolute zero. In both cases, the broadening parameter F = 0.3. Although the oscillations look different from the ideal case, the period of oscillation stays the same:
L(X) = coth x - 1/x,
A(l/ro)
(Y = prrH/kTand
Also, the sharp increase in dT/T takes place near odd integral values of l/y, as in the ideal case. Therefore, by making use of these properties, it seems to be possible to determine the Bohr magneton rather accurately. The magnetothermal effect is very sensitive to the broadening parameter F. The dramatic changes in the oscillating pattern from the ideal case suggest that the effect of level broadening could be assessed by making use of the magnetothermal effect. On the other hand, for the estimate we shall have to calculate dT/T for several different values of F. For large F, the numerical evaluation is relatively easy because of the fast convergence. However, we have found that f; values of order 0.3 require inclusion of many terms in equations such as
W = exp [-rr/(m2)].
(7)
The dimensionless parameter y represents the chemical potential I-1as follows: I/y
= p/a2.
(8)
From the number density relation, this parameter is implicitly determined in the following way: l/y,
sin (nZ/y)
= l/T+++
Finally, the temperature dimensionless form: dT -=_ T
(as/a%,. T@sla
OH,
(9)
smh (n21/cu)’
1
changes are obtained
dH. n
in a
(10)
Since the entropy has been given as a function of 1/y, the two derivatives in the above relation have to be evaluated by making use of equation (9). As a result, the ratio dT/Tis evaluated as a function of I/ye = eel/-@ 3. RESULTS AND DISCUSSION Before trying to get explicit results, we remark on the role played by the broadening parameter r. In the presence of this parameter, we have found that the chemical potential changes much more gradually than the case without it and the familiar linear specific heat reappears. In its absence, the specific heat decreases
= 2.
(11)
(6) or (9). In Fig. 1 note that the curve for absolute zero has amplitude which is approximately twice as large as the case n2/o = 1S. No effective mass has been used in these results. By decreasing the temperature and increasing the magnetic field, we will have significant increase in the amplitude because the effect is sensitive to the dimensionless parameter o. We remark that if the effective mass m* and effective Lande’s factor g* are used, they appear in the grand partition function as a phase factor in a combined form g*m*/(gm). For Si inversion layers, this factor is around 0.4. Since the magnetothermal oscillations are primarily due to electron’s orbital motion, we expect that no significant changes in the amplitude will be caused by the effective quantities. However, we shall investigate more closely realistic cases in a later article.
Vol. 50, No. 1
BROADENING
EFFECT ON MAGNETOTHERMAL
We conclude this article by emphasizing that the 2-D magnetothermal effect is considerably larger than the 3-D case. In view of the fact that 3-D oscillations have been observed [S], it is hoped that 2-D systems be explored experimentally in the near future.
31
REFERENCES 1 2.
3. Acknowledgement - This work was supported by the ONR under Contract NO001 4-79X045 1.
OSCILLATIONS
A. Isihara & Y. Shiwa, Solid State Commun. 48, 1081 (1983). A.V. Gold, in Solid State Physics, (Edited by J.F. Cochran & R.R. Haering) p. 39. Gordon and Breach, New York (1968); B.D. McCombe & G. Seidel, Phys. Rev. 155,633 (1967). R.B. Dingle, Proc. Roy. Sot. (London) A21 1,5 17 (1952). A.D. Brailsford, Phys. Rev. 149,456 (1966). B.D. McCombe & G. Seidel, Zoc cit.
BROADENING
Vol. 50, No. 1, pp. 35-37,
0038-1098/84 $3.00 + .OO Pergamon Press Ltd.
1984.
EFFECT ON MAGNETOTHERMAL
OSCILLATIONS
IN 2-D ELECTRON SYSTEMS
Y. Shiwa and A. Isihara Statistical
Physics Laboratory,
Department of Physics, State University Buffalo, NY, 14260, U.S.A.
of New York at Buffalo,
(Received 15 October 1983; in revised form 19 December 1983 by R.H. Silsbee)
Landau level broadening reduces considerably the amplitude of the temperature oscillations in 2-D electron systems when the magnetic field is changed adiabatically, but the oscillations are still much larger than the 3-D case. the delta functions as follows:
1. INTRODUCTION IN A RECENT ARTICLE [ 11, we have pointed out that adiabatic changes of strong magnetic field cause significant and peculiar temperature oscillations in a 2-D ideal electron system. The amplitude of these oscillations was found to be an order of magnitude larger than the case in bulk [2]. This largeness is due to the lack of electron motion in the direction of magnetic field and also to the sharp changes of the chemical potential at low temperatures as a function of l/ye = eO/pBH which is a crucial dimensionless parameter. Here, e. is the ideal Fermi energy in the absence of magnetic field, i.e., (h2/2m)(2nn), n and pBH being electron density and the field energy respectively. While the former cause exists at all temperatures, the latter is a low temperature characteristic of a 2-D electron system and is the primary factor for the large amplitude. The sharp changes of the chemical potential, which is the actual Fermi energy in the presence of magnetic field, are understandable because at low temperatures the Landau levels are filled one by one from the bottom. However, in real 2-D systems, the Landau levels may be broadened due to impurity scattering. Such broadening is expected to reduce the magnetothermal oscillations as in the case of the de Haas-van Alphen oscillations. In view of the anomalously large effect in the ideal case, it is then very important to investigate the broadening effect on the temperature oscillations. The present paper has been written for this investigation.
8(e-eE,)+i (E 7T
6(e - E,) by Lorentzian
functions
l/T
(1)
en)2 + (l/7)2’
where l/r is the half width. Here, and henceforth, we have set Ii = 1 and 2m = 1. We introduce a broadening parameter by r = l/(2&),
(2)
where a2 = pBH.
(3)
The Landau levels are given by efl = (2n+l)/~~H+p~H,
(n=0,1,2
,...)
(4)
where the Lande’s g-factor is chosen to be 2 and no effective mass correction is made. Broadening in the dHvA effect is often characterized by the temperature defined by (5) It is called the Dingle temperature. It turns out that the direct use of the above replace.ment in the grand potential leads to undesirable divergent terms. Similar divergences appear in the 3-D case which Dingle eliminated by a somewhat obscure argument that the energy could be minus infinity. By integrating the grand potential twice before applying the above replacement, we have succeeded to make a satisfactory approach in which obscure steps are eliminated. From the grand potential, we have derived the entropy as follows:
2. METHOD OF APPROACH Effects of Landau level broadening appear in some other quantum oscillations. Based on a phenomenological consideration of level broadening, Dingle found that the amplitude of the dHvA oscillations in 3-D is reduced [3]. We have adopted a similar but improved method. Our improvement consists of integrating by parts the grand partition function and then replacing
cm (WY) .smh (n21/a)
1’ Here, A is the surface area, /I = l/kT, L(x) is the Langevin function 35
(6)
36
BROADENING
EFFECT ON MAGNETOTHERMAL
OSCILLATIONS
Vol. 50, ho. 1
Fig. 1. Magnetothermal oscillations in 2-D electron systems with broadening parameter I’ = 0.3. Solid curve: n’/cr = 1.5 and dotted curve: rr2/o = 0. The former corresponds roughly to 1 K and 10 T in Si inversion layers. The latter represents the case of absolute zero. The arrows indicate the ordinate to be used.
exponentially. Since the derivative in the denominator of equation (10) represents the specific heat, this difference affects the magnitude of the left side significantly. In order to obtain explicit results, we have to estimate the broadening parameter. Following Brailsford [4], we interpret 7/2 as scattering time which can be estimated from mobility. For Si inversion layer electrons with effective mass 0.19mo, m, being the bare mass, and mobility lo4 cm2 V-’ s-’ we find i? = 27rr 0.3 for 10 T. This corresponds to To of order 0.2 K. For 3-D, McCombe and Seidel [5] got a similar value. Figure 1 represents our main results obtained without the effective mass and for the case n2/a = 1S. This choice of the parameter corresponds roughly to 1 K and 10 T. In comparison with Fig. 1 of our previous article [ 11, the amplitude is much less and the oscillations somewhat distorted from the sinusoidal curve. The dotted curve corresponds to the case of absolute zero. In both cases, the broadening parameter F = 0.3. Although the oscillations look different from the ideal case, the period of oscillation stays the same:
L(X) = coth x - 1/x,
A(l/ro)
(Y = prrH/kTand
Also, the sharp increase in dT/T takes place near odd integral values of l/y, as in the ideal case. Therefore, by making use of these properties, it seems to be possible to determine the Bohr magneton rather accurately. The magnetothermal effect is very sensitive to the broadening parameter F. The dramatic changes in the oscillating pattern from the ideal case suggest that the effect of level broadening could be assessed by making use of the magnetothermal effect. On the other hand, for the estimate we shall have to calculate dT/T for several different values of F. For large F, the numerical evaluation is relatively easy because of the fast convergence. However, we have found that f; values of order 0.3 require inclusion of many terms in equations such as
W = exp [-rr/(m2)].
(7)
The dimensionless parameter y represents the chemical potential I-1as follows: I/y
= p/a2.
(8)
From the number density relation, this parameter is implicitly determined in the following way: l/y,
sin (nZ/y)
= l/T+++
Finally, the temperature dimensionless form: dT -=_ T
(as/a%,. T@sla
OH,
(9)
smh (n21/cu)’
1
changes are obtained
dH. n
in a
(10)
Since the entropy has been given as a function of 1/y, the two derivatives in the above relation have to be evaluated by making use of equation (9). As a result, the ratio dT/Tis evaluated as a function of I/ye = eel/-@ 3. RESULTS AND DISCUSSION Before trying to get explicit results, we remark on the role played by the broadening parameter r. In the presence of this parameter, we have found that the chemical potential changes much more gradually than the case without it and the familiar linear specific heat reappears. In its absence, the specific heat decreases
= 2.
(11)
(6) or (9). In Fig. 1 note that the curve for absolute zero has amplitude which is approximately twice as large as the case n2/o = 1S. No effective mass has been used in these results. By decreasing the temperature and increasing the magnetic field, we will have significant increase in the amplitude because the effect is sensitive to the dimensionless parameter o. We remark that if the effective mass m* and effective Lande’s factor g* are used, they appear in the grand partition function as a phase factor in a combined form g*m*/(gm). For Si inversion layers, this factor is around 0.4. Since the magnetothermal oscillations are primarily due to electron’s orbital motion, we expect that no significant changes in the amplitude will be caused by the effective quantities. However, we shall investigate more closely realistic cases in a later article.
Vol. 50, No. 1
BROADENING
EFFECT ON MAGNETOTHERMAL
We conclude this article by emphasizing that the 2-D magnetothermal effect is considerably larger than the 3-D case. In view of the fact that 3-D oscillations have been observed [S], it is hoped that 2-D systems be explored experimentally in the near future.
31
REFERENCES 1 2.
3. Acknowledgement - This work was supported by the ONR under Contract NO001 4-79X045 1.
OSCILLATIONS
A. Isihara & Y. Shiwa, Solid State Commun. 48, 1081 (1983). A.V. Gold, in Solid State Physics, (Edited by J.F. Cochran & R.R. Haering) p. 39. Gordon and Breach, New York (1968); B.D. McCombe & G. Seidel, Phys. Rev. 155,633 (1967). R.B. Dingle, Proc. Roy. Sot. (London) A21 1,5 17 (1952). A.D. Brailsford, Phys. Rev. 149,456 (1966). B.D. McCombe & G. Seidel, Zoc cit.