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Magnetization, specific heat, magneto-thermal effect and thermoelectric power of two-dimensional electron gas in a quantizing magnetic field
Surface Science 142 (1984) 225-235 North-Holland, Amsterdam
225
MAGNETIZATION, SPECIFIC HEAT, MAGNETO-THERMAL AND THERMOELECTRIC POWER OF TWO-DIMENSIONAL ELECTRON GAS IN A QUANTIZING MAGNETIC FIELD Wlodek
EFFECT
ZAWADZKI
Institute of Physics, Polish Academy of Sciences, 02 668 Warsaw, Poland and Institute for Experimental Physics, University of Innsbruck, Innsbruck, Austria
and Rudolf
LASSNIG
Institute for Experimental
Physics, University of Innsbruck,
Innsbruck,
Au~trla
Received 12 July 1983; accepted for publication 6 September 1983
Oscillatory magnetization, specific heat, magneto-thermal effect and thermoelectric power of the two-dimensional gas of noninteracting electrons in the presence of a quantizing magnetic field and a finite temperature are considered theoretically. The magnetization due to Landau quantization oscillates around the zero value as a function of magnetic field and vanishes in the low-field limit. The specific heat consists of an intralevel and an interlevel contributions, both strongly oscillating. Magneto-thermal oscillations of a thermally isolated superlattice are shown to be of the order of 1 K at low temperatures, which is 10 3 times larger than in 3D semimetals. The thermoelectric power at high magnetic fields does not depend on electron scattering and it is directly proportional to the entropy. Its oscillations in a magnetic field as well as its dependence on the temperature and the level broadening are discussed.
1. Introduction
A two-dimensional electron gas in the presence of a quantizing magnetic field is known to possess strongly variable properties depending on the relative position of the Fermi level with respect to the Landau levels. In transport effects the mobility edges are of crucial meaning, whereas in equilibrium phenomena extended and localized states play comparable roles. The purpose of this paper is to consider equilibrium thermodynamics of the 2D electron gas in strong magnetic fields, when the electron scattering is of secondary importance, in order to provide independent information on the density of states, pinning of the Fermi energy and related problems. 0039-6028/84/$03.00
(North-Holland
0 Elsevier Science Publishers Physics Publishing Division)
B.V.
W. Zawadzki,
226
R. hssnrg
/ ZDEG
rn yuuntirtng
mugnerrc f&d
2. Electron density, Fermi energy, magnetization We consider thermodynamic properties of a 2D gas of noninteracting electrons in a parabolic, spherical energy band at finite temperature T in the presence of a quantizing magnetic field H. We include the spin degeneracy but assume the spin-splitting factor g * = 0. Inversion layers and superlattices based on GaAs satisfy quite well these assumptions, if the g*-value enhancement is neglected. An incorporation of the spin splitting into the theory is straightforward. We assume further that only one electric subband is populated. The energetic density of states is taken in the form of a sum of Gaussian peaks
(1) where L = (fic/eH) ‘I2 , X = Aw (n + 5) and r is the broadening parameter (the level width AC = 2r). i depindence of broadening on the magnetic field is neglected, cf. ref. [l]. The electron density in 1 cm2 is
exp(-2y,f)
(2)
dz,
whereA = (l/n)(eH/hc),y, = (Z - e,,)/yandz =~/kT,q = [/kT,e,, =X,,/kT, y = T/kT are the reduced quantities. The filling factor of the system is defined as v = N/A, denoting the number of occupied Landau levels. The condition of a constant electron density in the sample leads to an integral equation for the Fermi energy [( H). Fig. 1 shows this dependence calculated form* = O.O665m,, N, = 8 X 10” cmP2, r= 0.5 meV and T= 6 K. The free energy of the system is F= NC-
kT/p(r)
ln[l
+exp($$j]
dc
(3)
/ 25
0
20
LO
60
80
H[kGI
IM3
Fig. 1. The Fermi energy versus magnetic field calculated for the 2D electron gas in GaAs constant electron density and the temperature of 6 K. The Landau levels are also indicated.
at a
W. Zawadzki,
It is convenient
R. Lmsnig
/ 2DEG in quantizing
magnetic field
221
to write P(C) in the form
p(~, H) = aH R(E, H) where a = (l/a)(e/Ac). After some manipulation
(4) The magnetization one obtains
of the system
is M = -d F/dH.
In order to calculate a contribution to magnetization from one completely filled Landau level, one should put n - z B 0 and the integration limits + cc. The integral can then be calculated to give M, = kTa( v - 20,). Fig. 2 shows the magnetization calculated according to eq. (5) for the above m*, N,,, r and T = 4.2 K. It can be seen that the diamagnetism of the 2D electron gas oscillates symmetrically around the zero value, vanishing in the limit of H + 0. The inclusion of the spin splitting does not change the situation - it simply doubles the number of levels. As follows from figs. 1 and 2, the magnetization oscillations follow quite closely those of the Fermi level. On the other hand, the ShubnikovvDe Haas oscillations of magnetoresistance have maxima when the Landau levels cross the Fermi energy. It follows then that there should be a phase shift between the magnetization and the magneto-resistance oscillations. In fact, this has been observed recently in the magnetization measurements on GaAs-GaAlAs superlattices [3]. The first theory of magnetization for the 20 electron gas has been carried out using the delta-like density of states, with the results quite similar to those presented in fig. 2 [2]. It is also of interest to note that in the first explanation of the dHvA effect in 3D metals Peierls used a two-dimensional model of a metal at T = 0. Already this early consideration exhibited certain features presented above (cf. ref. [4]). 30 M/M,
I
0
20
LO
60
80
HIkGl
Fig. 2. Magnetization of the 2D electron gas versus conditions as in fig. 1 and T= 4.2 K. MO = kT(2e/hc).
magnetic
field
calculated
for the same
228
W. Zawadzkr,
R. Lmsnig
3. Specific heat, magneto-thermal
/ 2DEG
rn quantizrng magnetrc field
effect
The specific heat is given in general
as
in which
The dependence ag/aT at a constant concentration is determined ating eq. (2) with respect to T and using eq. (7). This leads to
aSlaT=
by differenti-
- L,/L,,
(8)
c,,=k&,-L:&),
(9)
where
(10) in which x = z - 17 and the other quantities are defined above. The Fermi energy is first calculated from the condition N = const. and then all the integrals computed as functions of the magnetic field for the corresponding [ values. Fig. 3 shows the specific heat calculated for the above parameters at T=6K. The specific heat of a 2D electron gas is seen to consist of two contributions. At high magnetic fields, where Aw, > kT, only the intralevel thermal
15 cv W 10
20
0 Fig. 3. Specific
HIkGl
gas versus magnetic
from the intralevel
seen. A, = (l/n)(eH/hc)
80
60
heat of the 2D electron
as in fig. 1. A transition clearly
10
100
field calculated
regime to the interlevel
for H = 1 kG.
for the same conditions
regime of thermal
excitations
is
W. Zawadrki, R. Lassnig / 2DEG in quantizing magnetic field
229
excitations contribute to C,. When the Fermi energy is between two Landau levels, the lower levels are completely filled, the upper ones completely empty, and C, vanishes. At such a magnetic field the system cannot absorb low energy excitations. At weaker magnetic fields the interlevel excitations begin to come into play if the temperature is not too low. they are of importance when the Fermi energy lies between two Landau levels. The interlevel contribution to C,, is seen in fig. 3 in the form of sharp spikes since, as follows from fig. 1, the Fermi energy “jumps” between two Landau levels within a narrow range of magnetic field strength. The rise of this contribution with decreasing magnetic field follows the general behavior of C, for two-level systems [4]. Thus, fig. 3 illustrates a continuous transition of the specific heat from the intralevel to the interlevel behavior, which is characterized, among others, by a change of phase of the magneto-oscillations. At very high magnetic fields the Fermi energy is forced below the lowest Landau level and the electron statistics becomes non-degenerate. In this limit only the lowest level is occupied and the specific heat as well as the electron concentration can be calculated analytically, to give at a finite temperature C,. = +y=kN.
(11)
It can be seen that the intralevel part of C, depends crucially on the level broadening, going to zero at vanishing r. Fig. 4 shows the peak value of C, for the filling factor v = 2.5 (at H = 66 kG) as a function of kT/r. Consider the regime Aw, s kT and ho, >> r, and the Fermi energy in the vicinity of the n th Landau level. The completely filled levels contain A . n electrons, which do not contribute to the specific heat. One can show both analytically and numerically that in this situation at a fixed magnetic field the contribution of the n th level to C, is determined only by 20
cv
v=2.5
0.8 Fig. 4. Intralevel contribution function of kT/r.
kT/r
to the specific heat calculated for the filling factor Y = 2.5 as a
kT/T. This means that the dependence shown in fig. 4 is universal in the sense that it can be used for any given r in order to determine the behavior of C,, with temperature. It is of interest to examine the validity of the general formula for the specific heat of a strongly degenerate electron gas: C,. = (7r2/3) k’Tp({). It can easily be seen that this formula does not account for the interlevel contribution to C,,, describing only the linear range of the intralevel part, as seen in fig. 4. The above expression is usually derived using the Sommerfeld expansion of the statistical integrals with respect to the small parameter ( kT/[)2, in which the Fermi energy { is counted from the band edge [5]. In our case. however, the completely filled Landau levels below the Fermi energy are statistically inactive and within one partly occupied level the expansion parameter becomes (kT/r)2. In good samples this parameter is small with respect to one only at very low temperatures (below 1 K). Thus, for the 2D electron gas in a magnetic field the whole concept of strong degeneracy and the validity of corresponding approximations is limited to the very narrow range of extremely low temperatures. If the above effects are to be observable, the specific heat of the electron gas should be comparable to that of the lattice. A typical period of a superlattice is 200 A. For this value we calculate, using the coefficient A in eq. (2). the electronic specific heat C,./k = 2.42 X lOI [. . .] (cmP3), where the values in brackets are plotted in figs. 3 and 4. At low temperatures the specific heat of the lattice, due to three acoustic phonon branches, is C,/ = 234k( Nz,/2)(T/8,,)‘, where N, is the number of atoms and 8, is the Debye temperature [4]. The factor l/2 accounts for two atoms per unit cell in GaAs-type materials. Taking for GaAs N, = 2.21 X 10” crne3 and 8, = 426 K we obtain C,!/k = 3.34 X lOI T’ (cm-‘), so that at T = 1 K the electronic and the lattice specific heats are, in fact, comparable. Next we consider magnetothermal oscillations, which denote temperature variations of a thermally isolated system as a function of magnetic field. This effect has been measured in three-dimensional semimetals [6], where the temperature changes are of the order of 10-j K. In a thermally isolated sample the processes occur adiabatically, i.e. the entropy must remain constant. Since the entropy is an oscillatory function of a magnetic field and an increasing function of temperature, the latter must oscillate in order to keep the entropy constant. The same principle is used in magnetic cooling. The entropy is S,, = - (dF/aT),., which becomes in the previous notation
(12) It can easily be seen from the above expression that the completely filled levels (x = z - n ==z0) give vanishing contribution to the entropy. Together with the condition of a constant electron concentration, cf. eq. (2) we have two integral
W. Zawndzki,
501
100
200
Fig. 5. Magneto-thermal calculated for a constant is not included.
R. Lassnig
/ 2DEG
60
rn quanmrng
LO
magnerrc field
231
HlkGI 7
oscillations of the 2D electron gas in GaAs versus the filling factor electron concentration and a constant entropy. The entropy of the lattice
equations for two unknowns l(H) and T(H), the second being of experimental interest. Fig. 5 shows the temperature of the two-dimensional electron gas as a function of magnetic field calculated for a GaAs sample, as characterized above. The constant value of entropy is chosen to be S, = 9.68 x 10’hk
100
2-
60
LO
20
H[kGI
I'
TIKI
16-
12-
Fig. 6. Magneto-thermal oscillations versus the filling superlattice including the entropy of the lattice.
factor
calculated
for a GaAs-GaAIAs
232
W. Zawadzkr,
R. Lassnig / 2DEG
tn quantizing magnetic field
(corresponding to T = 1.25 K at H = 100 kG). The very strong oscillations of temperature result from vanishing values of the intralevel specific heat at the minima (cf. fig. 3). On the other hand, the phallic shapes of the peaks and the strong damping at lower magnetic fields are due to the fact that, as the temperature rises, the interlevel contribution to C,, comes into play hampering a further temperature increase. The magneto-thermal oscillations can be calculated for the temperature region, where C,r = aT3, the entropy of the lattice is simply S, = C,!/3. This can be included in the constant-entropy condition: S,, = S, = const. The result is shown in fig. 6, with S,, + S, = 9.68 X 10lhk. Since the lattice prevents a strong temperature rise, the interlevel contribution to the electronic specific heat is negligible and the magneto-thermal oscillations have a spike-like behavior. This should allow one to use the effect as a spectroscopic tool in 2D systems. At low temperatures the amplitude of oscillation is of the order of 1 K, which is 10’ times larger than in the three-dimensional case. The effect is so large that it can be used for magnetic cooling.
4. Thermoelectric
power
A quantum theory of thermo-magnetic transport phenomena offers some serious difficulties, since in the presence of a temperature gradient the system is not homogeneous. As a consequence, the automatic application of the Kubo method led in the past to results, which did not satisfy the Onsager symmetry relations, violated the third law of thermodynamics, etc. [7-91. These paradoxes and puzzles were resolved by Obraztsov [lO,ll], who showed that, in order to obtain a correct description of the off-diagonal components of thermomagnetic tensors, one should explicitly include in the theory a contribution of the magnetization. This is related to the fact that the microscopic surface currents, which determine the Landau magnetization of conduction electrons, make a significant contribution to the macroscopic current density when a temperature gradient is present. At high magnetic fields, i.e. for w,r > 1, the diagonal components of the transport tensor may be neglected with respect to the off-diagonal ones. The latter do not depend on electron scattering the high-field limit. Taking into account the contribution of magnetization M one obtains for the off-diagonal component of the macroscopic thermoelectric tensor (13) where /?,“, determines the microscopic current density. When /?,“, is calculated using the standard methods of the density matrix, the equality (13) is obtained,
W. Zawadzki, R. Lassnig / 2DEG in quantizing magnetic field
Fig. 7. The thermoelectric power of the 2D electron gas versus magnetic parameters and T = 6 K. The dashed line indicates values of ( - e/k)(ln
in which becomes
a(
S is the entropy
w ) = &/u,,
of the electron
gas. The
233
field calculated 2)/v.
for GaAs
thermoelectric
power
= - S/eN.
(14)
N is given in eq. (2) and S in eq. (12) so that LY(H) can be readily calculated in the no-scattering limit. By measuring cx at a constant electron concentration one determines directly the entropy of the electron gas. Fig. 7 shows the thermo-power of the 2D gas in a strong transverse magnetic field calculated for the above parameters and T = 6 K. At high fields, as long as the interlevel contribution to C, is negligible (cf. fig. 3) the entropy (and consequently cx) vanishes when the Fermi energy is between two Landau 1
%I 0.8 -
0.6 -
0.4 -
0.2 I
0
0.2
OL
0.6
08
kTlr
Fig. 8. The thermoelectric power at the filling factor Y = n + 5 calculated as a function the range t2wC s kT and Aw, B r. We denote g, = a( - e/k)( n + f)/(ln 2).
of kT/T
in
234
W. Zmvadrkl,
R. Lmsnig / 2DEG in quontiring magnem
field
levels. At lower fields the interlevel contribution to C,. becomes of importance and the entropy, as well as (Y, does not reach zero values. These general predictions agree quite well with the first experimental observations of the thermo-power oscillations in GaAs-GaAlAs heterostructures [12]. In fig. 8 we show the maximal values of a(H) calculated from eqs. (2) and (12) as functions of kT/T in the limit of Aw, B kT and AU, >> r. This dependence is universal in the same sense as the one presented in fig. 4. It is of interest to compare the above results with the published calculations of magneto-thermo-power in 2D systems [13-151. All the above authors used the delta-like density of states, which is equivalent to taking r = 0. In this limit there is fi/r(l/y) exp( - 2~~~) ---) 6(z - d,,). The integrals in eqs. (2) and (12) are then equal to their integrands with z replaced by 6,,.A maximum of the nth peak occurs for 77= d,,, i.e. for x = 0, at which the filling factor is Y = n + 5. Hence, we have N = A( n - 4) and the thermo-power at the maximum becomes an
- --__k In2 e i7++’
(15)
This is the result quoted in refs. [14,15] (the authors of ref. (131 erroneously estimated v = n). It should be kept in mind that the assumptions r = 0 and T f 0 correspond to the limit kT/T = cc. It can be seen from fig. 8 that, in fact, we obtain the above value of cx in the limit of kT/T= co. However, the experiments are usually performed in the range of kT/T < 0.5, where the maximal values of (Yare distinctly lower, as seen from fig. 8. We conclude: (1) The “universal values” of thermopower, as claimed in the quoted papers, are not really universal; (2) the temperature dependence of the thermopower can be used to determine the value of r.
5. Summary The magnetization, the specific heat, the magneto-thermal effect and the thermoelectric power of the two-dimensional electron gas have been calculated as functions of a magnetic field. The oscillatory character of these effects is much more dramatic than that for the three-dimensional system, due to more pronounced oscillations of the energetic density of states. A concept of strong degeneracy of the 2D gas in a magnetic field is limited to very low temperatures (below 1 K). A broadening of the Landau levels r is not essential for the behavior of magnetization but it is of great importance for the thermal effects, as it allows intralevel, low-energy excitations. At high fields, when the interlevel thermal excitations are negligible, the thermodynamic properties become universal functions of kT/T. These theoretical predictions are in good agreement with preliminary observations of the magnetization and of the thermoelectric power in GaAs-GaAlAs heterostructures.
W. Zawadzki,
R. Lmsnrg / 2DEG in quantizing magnetic field
235
Acknowledgments We are grateful to Dr. H.L. Stormer and to Professor K. von Klitzing for information on their experimental results prior to publication. One of us (W.Z.) acknowledges the generous hospitality of Professor Erich Gornik and of the University of Innsbruck during his stay in Austria, where most of this work has been done.
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15]
T. Ando, A.B. Fowler and F. Stern, Rev. Mod. Phys. 54 (1982) 537. W. Zawadzki, Solid State Commun. 47 (1983) 317. H.L. Stormer et al., to be published. C. Kittel, Introduction to Solid State Physics, 3rd ed. (Wiley, New York, 1967). A.H. Wilson, The Theory of Metals (Cambridge Univ. Press, 1965) p. 144. J.E. Kunzler, F.S.L. Hsu and W.S. Boyle, Phys. Rev. 128 (1962) 1084. AI. Anselm and B.M. Askerov, Soviet Phys.-Solid State 3 (1961) 2665. M.I. Klinger, Soviet Phys-Solid State 3 (1961) 974. L.E. Gurevich and G. Nedlin, Soviet Phys.-Solid State 3 (161) 2029. Yu.N. Obraztsov, Soviet Phys-Solid State 6 (1964) 331. Yu.N. Obraztsov, Soviet Phys.-Solid State 7 (1965) 455. K. von Klitzing, private communication. S.P. Zelenin, A.S. Kondrat’ev and A.E. Kuchma, Soviet Phys.-Semicond. 16 (1982) 355 S.M. Girvin and M. Jonson, J. Phys. C. (Solid State Phys.) 15 (1982) L1147. P. Streda, J. Phys. C (Solid State Phys.) 16 (1983) L369.
225
MAGNETIZATION, SPECIFIC HEAT, MAGNETO-THERMAL AND THERMOELECTRIC POWER OF TWO-DIMENSIONAL ELECTRON GAS IN A QUANTIZING MAGNETIC FIELD Wlodek
EFFECT
ZAWADZKI
Institute of Physics, Polish Academy of Sciences, 02 668 Warsaw, Poland and Institute for Experimental Physics, University of Innsbruck, Innsbruck, Austria
and Rudolf
LASSNIG
Institute for Experimental
Physics, University of Innsbruck,
Innsbruck,
Au~trla
Received 12 July 1983; accepted for publication 6 September 1983
Oscillatory magnetization, specific heat, magneto-thermal effect and thermoelectric power of the two-dimensional gas of noninteracting electrons in the presence of a quantizing magnetic field and a finite temperature are considered theoretically. The magnetization due to Landau quantization oscillates around the zero value as a function of magnetic field and vanishes in the low-field limit. The specific heat consists of an intralevel and an interlevel contributions, both strongly oscillating. Magneto-thermal oscillations of a thermally isolated superlattice are shown to be of the order of 1 K at low temperatures, which is 10 3 times larger than in 3D semimetals. The thermoelectric power at high magnetic fields does not depend on electron scattering and it is directly proportional to the entropy. Its oscillations in a magnetic field as well as its dependence on the temperature and the level broadening are discussed.
1. Introduction
A two-dimensional electron gas in the presence of a quantizing magnetic field is known to possess strongly variable properties depending on the relative position of the Fermi level with respect to the Landau levels. In transport effects the mobility edges are of crucial meaning, whereas in equilibrium phenomena extended and localized states play comparable roles. The purpose of this paper is to consider equilibrium thermodynamics of the 2D electron gas in strong magnetic fields, when the electron scattering is of secondary importance, in order to provide independent information on the density of states, pinning of the Fermi energy and related problems. 0039-6028/84/$03.00
(North-Holland
0 Elsevier Science Publishers Physics Publishing Division)
B.V.
W. Zawadzki,
226
R. hssnrg
/ ZDEG
rn yuuntirtng
mugnerrc f&d
2. Electron density, Fermi energy, magnetization We consider thermodynamic properties of a 2D gas of noninteracting electrons in a parabolic, spherical energy band at finite temperature T in the presence of a quantizing magnetic field H. We include the spin degeneracy but assume the spin-splitting factor g * = 0. Inversion layers and superlattices based on GaAs satisfy quite well these assumptions, if the g*-value enhancement is neglected. An incorporation of the spin splitting into the theory is straightforward. We assume further that only one electric subband is populated. The energetic density of states is taken in the form of a sum of Gaussian peaks
(1) where L = (fic/eH) ‘I2 , X = Aw (n + 5) and r is the broadening parameter (the level width AC = 2r). i depindence of broadening on the magnetic field is neglected, cf. ref. [l]. The electron density in 1 cm2 is
exp(-2y,f)
(2)
dz,
whereA = (l/n)(eH/hc),y, = (Z - e,,)/yandz =~/kT,q = [/kT,e,, =X,,/kT, y = T/kT are the reduced quantities. The filling factor of the system is defined as v = N/A, denoting the number of occupied Landau levels. The condition of a constant electron density in the sample leads to an integral equation for the Fermi energy [( H). Fig. 1 shows this dependence calculated form* = O.O665m,, N, = 8 X 10” cmP2, r= 0.5 meV and T= 6 K. The free energy of the system is F= NC-
kT/p(r)
ln[l
+exp($$j]
dc
(3)
/ 25
0
20
LO
60
80
H[kGI
IM3
Fig. 1. The Fermi energy versus magnetic field calculated for the 2D electron gas in GaAs constant electron density and the temperature of 6 K. The Landau levels are also indicated.
at a
W. Zawadzki,
It is convenient
R. Lmsnig
/ 2DEG in quantizing
magnetic field
221
to write P(C) in the form
p(~, H) = aH R(E, H) where a = (l/a)(e/Ac). After some manipulation
(4) The magnetization one obtains
of the system
is M = -d F/dH.
In order to calculate a contribution to magnetization from one completely filled Landau level, one should put n - z B 0 and the integration limits + cc. The integral can then be calculated to give M, = kTa( v - 20,). Fig. 2 shows the magnetization calculated according to eq. (5) for the above m*, N,,, r and T = 4.2 K. It can be seen that the diamagnetism of the 2D electron gas oscillates symmetrically around the zero value, vanishing in the limit of H + 0. The inclusion of the spin splitting does not change the situation - it simply doubles the number of levels. As follows from figs. 1 and 2, the magnetization oscillations follow quite closely those of the Fermi level. On the other hand, the ShubnikovvDe Haas oscillations of magnetoresistance have maxima when the Landau levels cross the Fermi energy. It follows then that there should be a phase shift between the magnetization and the magneto-resistance oscillations. In fact, this has been observed recently in the magnetization measurements on GaAs-GaAlAs superlattices [3]. The first theory of magnetization for the 20 electron gas has been carried out using the delta-like density of states, with the results quite similar to those presented in fig. 2 [2]. It is also of interest to note that in the first explanation of the dHvA effect in 3D metals Peierls used a two-dimensional model of a metal at T = 0. Already this early consideration exhibited certain features presented above (cf. ref. [4]). 30 M/M,
I
0
20
LO
60
80
HIkGl
Fig. 2. Magnetization of the 2D electron gas versus conditions as in fig. 1 and T= 4.2 K. MO = kT(2e/hc).
magnetic
field
calculated
for the same
228
W. Zawadzkr,
R. Lmsnig
3. Specific heat, magneto-thermal
/ 2DEG
rn quantizrng magnetrc field
effect
The specific heat is given in general
as
in which
The dependence ag/aT at a constant concentration is determined ating eq. (2) with respect to T and using eq. (7). This leads to
aSlaT=
by differenti-
- L,/L,,
(8)
c,,=k&,-L:&),
(9)
where
(10) in which x = z - 17 and the other quantities are defined above. The Fermi energy is first calculated from the condition N = const. and then all the integrals computed as functions of the magnetic field for the corresponding [ values. Fig. 3 shows the specific heat calculated for the above parameters at T=6K. The specific heat of a 2D electron gas is seen to consist of two contributions. At high magnetic fields, where Aw, > kT, only the intralevel thermal
15 cv W 10
20
0 Fig. 3. Specific
HIkGl
gas versus magnetic
from the intralevel
seen. A, = (l/n)(eH/hc)
80
60
heat of the 2D electron
as in fig. 1. A transition clearly
10
100
field calculated
regime to the interlevel
for H = 1 kG.
for the same conditions
regime of thermal
excitations
is
W. Zawadrki, R. Lassnig / 2DEG in quantizing magnetic field
229
excitations contribute to C,. When the Fermi energy is between two Landau levels, the lower levels are completely filled, the upper ones completely empty, and C, vanishes. At such a magnetic field the system cannot absorb low energy excitations. At weaker magnetic fields the interlevel excitations begin to come into play if the temperature is not too low. they are of importance when the Fermi energy lies between two Landau levels. The interlevel contribution to C,, is seen in fig. 3 in the form of sharp spikes since, as follows from fig. 1, the Fermi energy “jumps” between two Landau levels within a narrow range of magnetic field strength. The rise of this contribution with decreasing magnetic field follows the general behavior of C, for two-level systems [4]. Thus, fig. 3 illustrates a continuous transition of the specific heat from the intralevel to the interlevel behavior, which is characterized, among others, by a change of phase of the magneto-oscillations. At very high magnetic fields the Fermi energy is forced below the lowest Landau level and the electron statistics becomes non-degenerate. In this limit only the lowest level is occupied and the specific heat as well as the electron concentration can be calculated analytically, to give at a finite temperature C,. = +y=kN.
(11)
It can be seen that the intralevel part of C, depends crucially on the level broadening, going to zero at vanishing r. Fig. 4 shows the peak value of C, for the filling factor v = 2.5 (at H = 66 kG) as a function of kT/r. Consider the regime Aw, s kT and ho, >> r, and the Fermi energy in the vicinity of the n th Landau level. The completely filled levels contain A . n electrons, which do not contribute to the specific heat. One can show both analytically and numerically that in this situation at a fixed magnetic field the contribution of the n th level to C, is determined only by 20
cv
v=2.5
0.8 Fig. 4. Intralevel contribution function of kT/r.
kT/r
to the specific heat calculated for the filling factor Y = 2.5 as a
kT/T. This means that the dependence shown in fig. 4 is universal in the sense that it can be used for any given r in order to determine the behavior of C,, with temperature. It is of interest to examine the validity of the general formula for the specific heat of a strongly degenerate electron gas: C,. = (7r2/3) k’Tp({). It can easily be seen that this formula does not account for the interlevel contribution to C,,, describing only the linear range of the intralevel part, as seen in fig. 4. The above expression is usually derived using the Sommerfeld expansion of the statistical integrals with respect to the small parameter ( kT/[)2, in which the Fermi energy { is counted from the band edge [5]. In our case. however, the completely filled Landau levels below the Fermi energy are statistically inactive and within one partly occupied level the expansion parameter becomes (kT/r)2. In good samples this parameter is small with respect to one only at very low temperatures (below 1 K). Thus, for the 2D electron gas in a magnetic field the whole concept of strong degeneracy and the validity of corresponding approximations is limited to the very narrow range of extremely low temperatures. If the above effects are to be observable, the specific heat of the electron gas should be comparable to that of the lattice. A typical period of a superlattice is 200 A. For this value we calculate, using the coefficient A in eq. (2). the electronic specific heat C,./k = 2.42 X lOI [. . .] (cmP3), where the values in brackets are plotted in figs. 3 and 4. At low temperatures the specific heat of the lattice, due to three acoustic phonon branches, is C,/ = 234k( Nz,/2)(T/8,,)‘, where N, is the number of atoms and 8, is the Debye temperature [4]. The factor l/2 accounts for two atoms per unit cell in GaAs-type materials. Taking for GaAs N, = 2.21 X 10” crne3 and 8, = 426 K we obtain C,!/k = 3.34 X lOI T’ (cm-‘), so that at T = 1 K the electronic and the lattice specific heats are, in fact, comparable. Next we consider magnetothermal oscillations, which denote temperature variations of a thermally isolated system as a function of magnetic field. This effect has been measured in three-dimensional semimetals [6], where the temperature changes are of the order of 10-j K. In a thermally isolated sample the processes occur adiabatically, i.e. the entropy must remain constant. Since the entropy is an oscillatory function of a magnetic field and an increasing function of temperature, the latter must oscillate in order to keep the entropy constant. The same principle is used in magnetic cooling. The entropy is S,, = - (dF/aT),., which becomes in the previous notation
(12) It can easily be seen from the above expression that the completely filled levels (x = z - n ==z0) give vanishing contribution to the entropy. Together with the condition of a constant electron concentration, cf. eq. (2) we have two integral
W. Zawndzki,
501
100
200
Fig. 5. Magneto-thermal calculated for a constant is not included.
R. Lassnig
/ 2DEG
60
rn quanmrng
LO
magnerrc field
231
HlkGI 7
oscillations of the 2D electron gas in GaAs versus the filling factor electron concentration and a constant entropy. The entropy of the lattice
equations for two unknowns l(H) and T(H), the second being of experimental interest. Fig. 5 shows the temperature of the two-dimensional electron gas as a function of magnetic field calculated for a GaAs sample, as characterized above. The constant value of entropy is chosen to be S, = 9.68 x 10’hk
100
2-
60
LO
20
H[kGI
I'
TIKI
16-
12-
Fig. 6. Magneto-thermal oscillations versus the filling superlattice including the entropy of the lattice.
factor
calculated
for a GaAs-GaAIAs
232
W. Zawadzkr,
R. Lassnig / 2DEG
tn quantizing magnetic field
(corresponding to T = 1.25 K at H = 100 kG). The very strong oscillations of temperature result from vanishing values of the intralevel specific heat at the minima (cf. fig. 3). On the other hand, the phallic shapes of the peaks and the strong damping at lower magnetic fields are due to the fact that, as the temperature rises, the interlevel contribution to C,, comes into play hampering a further temperature increase. The magneto-thermal oscillations can be calculated for the temperature region, where C,r = aT3, the entropy of the lattice is simply S, = C,!/3. This can be included in the constant-entropy condition: S,, = S, = const. The result is shown in fig. 6, with S,, + S, = 9.68 X 10lhk. Since the lattice prevents a strong temperature rise, the interlevel contribution to the electronic specific heat is negligible and the magneto-thermal oscillations have a spike-like behavior. This should allow one to use the effect as a spectroscopic tool in 2D systems. At low temperatures the amplitude of oscillation is of the order of 1 K, which is 10’ times larger than in the three-dimensional case. The effect is so large that it can be used for magnetic cooling.
4. Thermoelectric
power
A quantum theory of thermo-magnetic transport phenomena offers some serious difficulties, since in the presence of a temperature gradient the system is not homogeneous. As a consequence, the automatic application of the Kubo method led in the past to results, which did not satisfy the Onsager symmetry relations, violated the third law of thermodynamics, etc. [7-91. These paradoxes and puzzles were resolved by Obraztsov [lO,ll], who showed that, in order to obtain a correct description of the off-diagonal components of thermomagnetic tensors, one should explicitly include in the theory a contribution of the magnetization. This is related to the fact that the microscopic surface currents, which determine the Landau magnetization of conduction electrons, make a significant contribution to the macroscopic current density when a temperature gradient is present. At high magnetic fields, i.e. for w,r > 1, the diagonal components of the transport tensor may be neglected with respect to the off-diagonal ones. The latter do not depend on electron scattering the high-field limit. Taking into account the contribution of magnetization M one obtains for the off-diagonal component of the macroscopic thermoelectric tensor (13) where /?,“, determines the microscopic current density. When /?,“, is calculated using the standard methods of the density matrix, the equality (13) is obtained,
W. Zawadzki, R. Lassnig / 2DEG in quantizing magnetic field
Fig. 7. The thermoelectric power of the 2D electron gas versus magnetic parameters and T = 6 K. The dashed line indicates values of ( - e/k)(ln
in which becomes
a(
S is the entropy
w ) = &/u,,
of the electron
gas. The
233
field calculated 2)/v.
for GaAs
thermoelectric
power
= - S/eN.
(14)
N is given in eq. (2) and S in eq. (12) so that LY(H) can be readily calculated in the no-scattering limit. By measuring cx at a constant electron concentration one determines directly the entropy of the electron gas. Fig. 7 shows the thermo-power of the 2D gas in a strong transverse magnetic field calculated for the above parameters and T = 6 K. At high fields, as long as the interlevel contribution to C, is negligible (cf. fig. 3) the entropy (and consequently cx) vanishes when the Fermi energy is between two Landau 1
%I 0.8 -
0.6 -
0.4 -
0.2 I
0
0.2
OL
0.6
08
kTlr
Fig. 8. The thermoelectric power at the filling factor Y = n + 5 calculated as a function the range t2wC s kT and Aw, B r. We denote g, = a( - e/k)( n + f)/(ln 2).
of kT/T
in
234
W. Zmvadrkl,
R. Lmsnig / 2DEG in quontiring magnem
field
levels. At lower fields the interlevel contribution to C,. becomes of importance and the entropy, as well as (Y, does not reach zero values. These general predictions agree quite well with the first experimental observations of the thermo-power oscillations in GaAs-GaAlAs heterostructures [12]. In fig. 8 we show the maximal values of a(H) calculated from eqs. (2) and (12) as functions of kT/T in the limit of Aw, B kT and AU, >> r. This dependence is universal in the same sense as the one presented in fig. 4. It is of interest to compare the above results with the published calculations of magneto-thermo-power in 2D systems [13-151. All the above authors used the delta-like density of states, which is equivalent to taking r = 0. In this limit there is fi/r(l/y) exp( - 2~~~) ---) 6(z - d,,). The integrals in eqs. (2) and (12) are then equal to their integrands with z replaced by 6,,.A maximum of the nth peak occurs for 77= d,,, i.e. for x = 0, at which the filling factor is Y = n + 5. Hence, we have N = A( n - 4) and the thermo-power at the maximum becomes an
- --__k In2 e i7++’
(15)
This is the result quoted in refs. [14,15] (the authors of ref. (131 erroneously estimated v = n). It should be kept in mind that the assumptions r = 0 and T f 0 correspond to the limit kT/T = cc. It can be seen from fig. 8 that, in fact, we obtain the above value of cx in the limit of kT/T= co. However, the experiments are usually performed in the range of kT/T < 0.5, where the maximal values of (Yare distinctly lower, as seen from fig. 8. We conclude: (1) The “universal values” of thermopower, as claimed in the quoted papers, are not really universal; (2) the temperature dependence of the thermopower can be used to determine the value of r.
5. Summary The magnetization, the specific heat, the magneto-thermal effect and the thermoelectric power of the two-dimensional electron gas have been calculated as functions of a magnetic field. The oscillatory character of these effects is much more dramatic than that for the three-dimensional system, due to more pronounced oscillations of the energetic density of states. A concept of strong degeneracy of the 2D gas in a magnetic field is limited to very low temperatures (below 1 K). A broadening of the Landau levels r is not essential for the behavior of magnetization but it is of great importance for the thermal effects, as it allows intralevel, low-energy excitations. At high fields, when the interlevel thermal excitations are negligible, the thermodynamic properties become universal functions of kT/T. These theoretical predictions are in good agreement with preliminary observations of the magnetization and of the thermoelectric power in GaAs-GaAlAs heterostructures.
W. Zawadzki,
R. Lmsnrg / 2DEG in quantizing magnetic field
235
Acknowledgments We are grateful to Dr. H.L. Stormer and to Professor K. von Klitzing for information on their experimental results prior to publication. One of us (W.Z.) acknowledges the generous hospitality of Professor Erich Gornik and of the University of Innsbruck during his stay in Austria, where most of this work has been done.
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15]
T. Ando, A.B. Fowler and F. Stern, Rev. Mod. Phys. 54 (1982) 537. W. Zawadzki, Solid State Commun. 47 (1983) 317. H.L. Stormer et al., to be published. C. Kittel, Introduction to Solid State Physics, 3rd ed. (Wiley, New York, 1967). A.H. Wilson, The Theory of Metals (Cambridge Univ. Press, 1965) p. 144. J.E. Kunzler, F.S.L. Hsu and W.S. Boyle, Phys. Rev. 128 (1962) 1084. AI. Anselm and B.M. Askerov, Soviet Phys.-Solid State 3 (1961) 2665. M.I. Klinger, Soviet Phys-Solid State 3 (1961) 974. L.E. Gurevich and G. Nedlin, Soviet Phys.-Solid State 3 (161) 2029. Yu.N. Obraztsov, Soviet Phys-Solid State 6 (1964) 331. Yu.N. Obraztsov, Soviet Phys.-Solid State 7 (1965) 455. K. von Klitzing, private communication. S.P. Zelenin, A.S. Kondrat’ev and A.E. Kuchma, Soviet Phys.-Semicond. 16 (1982) 355 S.M. Girvin and M. Jonson, J. Phys. C. (Solid State Phys.) 15 (1982) L1147. P. Streda, J. Phys. C (Solid State Phys.) 16 (1983) L369.