Two-Dimensional Electron Gas in a Magnetic Field

Transcription

1 L. WANC : Two-Dimensional Electron Gas in a Magnetic Field 247 phys. stat. sol. (b) 167, 247 (1990) Subject classification: 71.25; Department of Chemistry, Stanford University1) Two-Dimensional Electron Gas in a Magnetic Field The Non-Degenerate Limit BY L. WANG The thermodynamical properties of a two-dimensional electron gas (2DEG) in a magnetic field with an arbitrary level broadening are discussed in the nondegenerate limit first proposed by Zawadzki. It is concluded that experimental measurements in the non-degenerate regime may open a new avenue for investigations on the density of states of a 2DEG in a magnetic field. Es werden die thermodynamischen Eigenschaften eines zweidimensionalen Elektronengases (2DEG) im Magnetfeld bei einer beliebigen Niveauverbreiterung im nicht,entarteten Grenzfall nach Zuwadzki untersucht. Es wird geschlossen, daij experimentelle Messungen im nichtentartetem Regime eine neue Moglichkeit fur Unt,ersuchungen der Zustandsdichte fur 2DEG in einem Magnetfeld eroffnet. 1. Introduction The density of states (DOS) of a two-dimensional electron gas (2DEG) systems in the presence of a strong magnetic field have been a subject of intensive studies, both theoretically and experimentally [l to 131. The DOS of an ideal 2DEG in a magnetic field consists of a series of sharp Landau levels. But in reality, these Landau levels are broadened due to impurities, phonons, etc.. The experimental investigations 011 the DOS [2 to 61 are performed at low temperatures and by using high electron densities. Hence the electron statistics is always degenerate. Different DOS are found in different kinds of samples, which can be explained qualitatively through various scattering mechanisms [7 to 101. But to achieve a better agreement between theory and experiment, and to completely understand the nature of Landau level broadenings, more work is needed. An interesting question was raised recently by Zawadzki and Lassnig [13]. They considered the case where the magnetic field is so strong that the Fermi energy is forced below the lowest Landau level and the electron statistics becomes non-degenerate. An analytic result for the specific heat was obtained in the special case of Gaussian broadenings [8]. In the present paper we wish to carry out some detailed discussions on this question. We shall study the requirements on the magnetic field, the temperature, and the Landau level width for this non-degenerate situation to occur. A thermodynamical formulation will be developed for an arbitrary shape of Landau level broadening. As revealed by our results, experiments in the non-degenerate regime can bring about more information on the DOS of such a system. 2. Condition for Non-Degeneracy Fermi statistics reduces to non-degenerate Boltzmann statistics when E - p > kt for all E, l) Stanford, CA 94305, USA.

2 248 L. WANG where E is energy, p is the Fermi energy, and T is the temperature. Similar to the threedimensional case, (1) is equivalent to mkt n n<- nh2 for a free 2DEG in the absence of a magnetic field. In (2), n is the number of electrons per unit area, m and m, are the effective and free electron masses, respectively. This equivalence is not evident when there exists a strong magnetic field. In the following we shall estimate the ratio (E - p)/kt for the case where the magnetic field is so strong that only the lowest Landau level is occupied and the Ferrni energy is in the vicinity of the lowest level. One has n = D{esp [(E, -,u)/ktj + l}-l, (3) where E, = hwc/2 represents the lowest Landau level, w, = eb/mc is the cyclotron frequency, B is the magnetic field, and D = eb/nhc = 4.8 x 10lO(B/T) cm-2 is the degeneracy per unit area. Solving (3) we obtain the Fermi energy p=eo- kt ln - - (: Jf we require that E, - p 2 5kT, which is equivalent to, according to (4), x lo9 cm-2 with a niagnetic field B = 20 T. 3. Thermodynamical Formulations (4) n x In the following we derive thermodynamical formulations in the nondegenerate limit, with an arbitrary level broadening. Suppose the density of states of the system is [ll] where the Landau level spect,rum $)+q is given by El, = hfi>c[(z + ; I = 0,1,2,..., CT = jl. Here g is the Lande factor multiplied by (mlm,) and the shape function is normalized, +m j R(x) dz = 1. (7) -w Equation (5) implies that broadening is independent of the Landau level indices (I and a). For instance, the shape functions derived from the self-consistent Born approximation (the semi-elliptic broadening) [7j and the lowest order cumulant method (the Gaussian broadening) [8j are indeed of this type. It is straightforward to calculate the Boltzmann partition function as follows : Z(P) = Z,(B) ZdP) ZdP) Y (8) where,9 = l/kt and the subscripts 0, S, and r represent the orbital, the spin and the broadening contributions. Explicitly one has +w Z,@) = J e-prz R(x) dx. --m

3 Two-Dimensional Electron Gas in a Magnetic Field 249 All thermodynamical properties can be calculated from the partition function in the non-degenerate limit. For example, the internal energy is Hence the specific heat is au cv=---= at a -n - [In Z(B)] = ag ctanh (PhwJZ) ghw - 2 tanh (~g?2wc/4) - 4 = (GI0 + ( G ) S + (Cv)r. The magnetic moment can be evaluated as follows : i a M = n - - [In Z(fl)] =.S ab where pb = eh/zmc is the Bohr magneton. Now we consider an explicit example, e.g., the Gaussian broadening, [8] where the shape function takes the following form: R(z) = exp (--xz). (15) From (ll), we obtain ZI@) = exp (p2r2/4). (16) Substituting (16) into (13) we obtain the specific heat in the case of Gaussian broadening. In particular, the broadening correction is where the temperature dependence of the Landau level width is not quite clear at present. In order to obtain the magnetic moment, we need to know the magnetic field dependence of the level width. In the case of long range scattering, [6 to 81 Pis independent of B, thus the broadening effect on the magnetic moment vanishes, Mr = 0 for long range scatterings, (18) since a,(/?) depends on the magnetic field only through I'. If short range scatterings dominate, we have [Z to 5, 7, 81

4 250 L. WANG where z is the scattering life time. Combination of (14), (16), and (19) gives Mr=- VBBh ntz which is independent of the magnetic field. Other physical properties can be evaluated in a similar fashion. We may make the following observation. Since all thermodynamical quantities depend on the level broadening explicitly, measurements of these effects may bring about more information on the density of states of the system, e.g., the temperature and the magnetic field dependences of the level width. 4. Strong Magnetic Field Limit We now turn to a discussion on the strong magnetic field limit, e.g., Bhwc > 1. The first two terms of the internal energy given by (12) represent the energy in the absence of level broadenings. When the magnetic field is very strong, they reduce to n( 1/2 - - g/4) hoe, which means that all electrons stay in the lowest Landau level. In this case, the first two terms of the specific heat [see (13)] are exponentially small. In other words, the specific heat does not vanish only if there exists level broadenings. For Gaussian broadening, (13) reduces to (17), where the first term of the r.h.s. agrees with a result obtained earlier by Zawadzki and Lassnig [13] (a factor of 2 is due to different definitions of T). But the second term in the r.h.s. of (17) does not necessarily vanish, since, for example, the level width may depend on temperature through the scattering lifetime in the case of short range scattering [see (19)]. Also, the magnetic moment given by (14) saturates at high magnetic fields. In other words, - npb (-1 + $) for short range scatterings, for long range scatterings. (21) We note that the broadening contribution to the magnetic moment is always paramagnetic. 6. Conclusions In summary, we discuss the thermodynamics of a 2DEG in the non-degenerate limit, with an arbitrary Landau level broadening. In particular, the specific heat and the magnetic moment are calculated explicitly. The special case of Gaussian broadening and the strong magnetic field limit are investigated in detail, where our result for the specific heat agrees with that derived earlier by Zawadzki [13] if the level width does not depend on the temperature. The condition to achieve this non-degenerate limit is also considered. We conclude that experimental measurements in the non-degenerate regime may contribute valuable information on the nature of Landau level broadening mechanisms in a 2DEG. Acknowledgement It is a pleasure to thank Prof. R. F. O Connell for many helpful conversations.

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