THE QUANTUM HALL EFFECT : TRANSFER OF ELECTRONS AND LOCALIZATION

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THE QUANTUM HALL EFFECT : TRANSFER OF ELECTRONS AND LOCALIZATION

A. Raymond, K. Karrai

To cite this version:

A. Raymond, K. Karrai. THE QUANTUM HALL EFFECT : TRANSFER OF ELECTRONS AND LOCALIZATION. Journal de Physique Colloques, 1987, 48 (C5), pp.C5-491-C5-494.

�10.1051/jphyscol:19875104�. �jpa-00226686�

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Colloque C 5 , supplgment au noll, Tome 48, novembre 1987

THE QUANTUM HALL EFFECT : TRANSFER OF ELECTRONS AND LOCALIZATION

A. RAYMOND' ) and K. KARRAI

SNCI-CNRS, 166x, F-38042 Grenoble Cedex, France

Nous proposons un modele dans lequel la densitk totale Ns d'klectrons mobiles 2D doit varier pour repondre A la statistique. En supposant que pxy = B/Nse, le modele permet de rendre compte de la largeur et de la forme des marches de l'effet Hall quantique pour toutes les hktkrojonctions GaAs-GaAlAs ktudikes et ce ?I toutes tempkratures.

A model is proposed in which the total density Ns of mobile 2D electrons must change in order to obey the statistics. Assuming that p = B/NSe, we account for the width and shape of the Hall steps for all theXYinvestigated GaAs-GaAlAs heterostructures whatever the temperature may be.

Immediately after the original paper on the quantum Hall effect (QHE) by Von Klitzing et a1 (l), two models have been proposed in order to account for the quantization of px in plateaus. The first one 2 - 4 assumes that the two dimensional electron gas (2DEGY is isolated (which means that the total electron density Ns has a constant value when B varies), thatpxy does not represent the total density of mobile electrons (carrying current electrons) and that the disoder, i.e. localized states, are responsible for the formation of the plateaus. The second model (5) proposes for GaAs-GaAlAs heterojunction (GaAs H), when B varies, a tunneling of electrons between the ionized donors of the doped layer and the inversion layer, i.e. a magnetic field dependence of Ns. The first-model is now well established but it leads to some inconsistancies between the conclusions of the experimental analysis and the intrinsic characteristics of the structures and it does not give at present a good understanding of the width and shape of the Hall steps. On the other hand, it seems now that nobody believes in the second model.

In this paper, we propose a new model which takes into account both the transfer of electrons and the role of localized states ; two kinds of QHE can be observed depending only on the experimental temperature (T) : one at higher T involving a transfer and the other one at lower T involving localization. We have analyzed experimental results in the litterature as well as our own results under hydrostatic pressure. We observed that this model allows to account for the flatness and the width of the plateaus of pTy without any fitting parameter, and the quantized values h/ie2 are exactly obtained. In the model of an isolated 2DEG (Ns = cte), the explanation of a finite width of the pxy~plateaus requires the existence of localized states. However, we observe experimentally that the value of pxy on those plateaus is exactly h/ie2, as if the total current was independent of the relative density of localized states. Prange (3) as well as Aoki and Ando(2) have shown that this was possible because the remaining delocalized states carry exactly enough extra current to compensate for its loss ; on the other hand Laughlin (4) uses a very elegant gedanken experiment to argue that gauge invariance requires the quantization of p x to h/ie2, in spite of localization.

There have been several approacges to understand the origins of the Quantized Hall plateaus in the concept of an isolated 2 DEG but he has never been, up to now, any quantitative account for the Hall step widths and shapes (6). Several experiments have been performed in order to determine the density of states (DOS) but all of them have been analysed in the isolated 2 DEG approximation, i e. with Ns magnetic field independent (6). The main result is that all the previous experiments reveal that the relative density of extended states only represents a small percentage of the total DOS and that a substantial DOS must exist between Landau levels (LLS) even in high mobility and high homogeneity samples. The large broadening parameter of the LLS obtained in all these studies is much larger than predicted from either

(')permanent address : GES-USTL, Place EugGne Bataillon. F-34060 Montpellier Cedex, France

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19875104

C5-492 JOURNAL DE PHYSIQUE

the scattering-time approximation or the self-consistent Born approximation for the case of short-range scatterers.

On the other hand a comparative analysis of the plateau widths A performed outside the spin splitting and Fractional Quantum Hall Effect regimes (FQHE) (low B) for samples with similar values of Ns but different mobility, leads us to a relevant remark : it seems that for very low T (T << 1K) A is larger for samples with lower mobility when for higher T (T > 1K) A is larger for samples having a higher mobility, which is inconsistent with the model. These inconsistencies seem to indicate that, in the model of the isolated quantum well (QW) previously summarized, some hypotheses are incorrect.

Let us now assume that the density Ns of charges involved in pxy is the density of delocalized electrons, and that, as in Ref. (5), a transfer of electrons may be possible as the 2 DEG is not an isolated system (i.e. there must be a reservoir to supply electrons to the 2 DEG). If the magnetic field dependence of the DOS of the 2 DEG is different from the one of the reservoir, then, at the thermodynamical equilibrium, the density Ns of the 2 DEG must depend on B.

As the charge density can now be modified by the magnetic field, the electrostatic equilibrium of the heterojunction is then B dependent. Two cases will be considered here :

i) the reservoir is not in the QW : the electrons may transfer, for instance, by tunneling on ionized donors or on residual impurities or defects in the spacer layer. In this case the equilibrium of the heterojunction must be recalculated.

ii) the reservoir is in the QW : the electrons do not leave the QW. They can be localized for example on interface states. In this case, the accumulated charge due to electrons in the well does not change and the equilibrium does not change either.

In order to verify this model we have calculated the B dependence of N, and Pxy for GaAs H. In such a real structure, we have assumed that the Fermi energy EF, which is a constant in the whole structure, does not depend on B since in the undepleted regions of GaAs and of GaAlAS, EF is in first approximation B independent (EF - E0 can very because the lowest electrical subband energy E, can vary). As we previously explained, in case i, when Ns varies, the electric field F varies and consequently E, varies too, when, in case it, the variation of Ns does not change F and Eo. The calculations have been performed in the triangular well approximation (TWA) assuming that E, is the only populated subband. As shown by Stern and Das Sarma ( 7 ) , the TWA gives reasonably correct values, if F is the mean electric field. We have considered non interacting electrons in a spherical parabolic energy band. We used a DOS in form of a sum of Gaussian peaks. In the calculations we make the hypothesis that all the 2 DEG states are non localized states, which means that all the localized states involved in this model in case ii are additional states.

Ns is then given by :

r

^'

Ns = 3 1 r

'sn-o h ?r r - m ^E - EF

l dE(1) l + exp [ ^kT 1

where I'

-

is the broadening parameter and p the zero field mobility, s = + 1 and 8=g*mx/2m0.

In equation (1) Ns is a function of B and in case i (E - E,) depends also on B. The value of the g factor, g* is calculated taking into account the spin splitting enhancement by the exchange interaction of electrons (8). We write g*

-

go* + ^{A g with} :

N t (L) represents the density of mobile electrons having t (1) spin. 1,

-

is the magnetic length. We have performed an iterative numerical calculation of Ns for each value of B by using equation (1) : pxyl corresponds to pXy in case i with an external reservoir, the equilibrium of the GaAs H being recalculated for each

value of B ; pxy2 correspond to case ii with localization in the QW.

The results of this calculation are reported Fig. l(a) for sample 1. The following parameters have been used : m*/mo

-

-

On Fig.l(b) the experimental curves of p,, and pxy for sample 1 are reported. We can notice forpxy the good agreement obtained without fittlng parameter for the width of the plateaus even when the spin splitting occurs. A comparitive analysis of Fig.la and lb clearly shows the competition bet- ween IQHE and FQHE for the filling factor

-

We have investigated numerous cases found in the litterature, and our own results, with N var ing between 0.6 X 1011 and 7.8 x lofl cm- S . In all the case as for sample 1 , we observed a good agreement between the experimental curve of and the theoretical one in case i (pxy:yYfor I larger than approximately 1K.

This means that, above such a temperature, the QHE is a consequence of a transfer of electrons due to the statistics between the 2 DEG and for example ionized donors, outside the QW (case i).

In a recent publication Nizhankovskii et a1 (9) have investigated the effect of magnetic field on the chemical potential of electrons in GaAs H. They conclude that the oscillatory dependence of Ns on the magnetic field strength they deduced al- 0 2 4 6 8 B(T) lows to explain the QHE they observe at 4.2K without invoking the localization concept. This analysis corresponds to case i of our model.

When the temperature is very much lower Fig.la) Theoretical curves of than lK, the width of the plateaus pXy : pxyl corresponds to the increases and our model indicates that the case of an external reservoir situation is now well described by the (higher T). pxy2 corresponds to case ii: the QHE is a consequence of the the localization of non mobile transfer of electrons from mobile states electrbns in the well at lower T in the broadened LLS, into the additional (internal reservoir). The insert localized states. These localized states represents the theoretical B can be due to interface states or other dependence of the g factor for disorder effects in the structure. As an sample 1. example, on Fig. 2 we have reported b) Experimental curves for pxx theoretical (pxy2-dashed line) and and pxy. Dashed lines correspond experimental curves of pxy given in Fig. 3 to the theoretical curve pxyl. of Ref. 10.

As for Fig. 1, we can observe an excellent agreement without fitting parameter between theoretical and experimental results outside the spin splitting regime and a qualitative good agreement inside it.

We have also investigated QtiE experiments performed under hydrostatic pressure P for

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T higher than 1K. The experimental and theoretical magnetic field dopendences of

are respectively reported Fig. 3(a)

23

Fig. 3 shows clelrly that the proposed model correctly accounts for the width and shape of the Hall steps (the disagreement observed under pressure of 1 bar and 3.2 Kbar, for V

-

, , , , , , ) and FQHE).

C 2 4 6 In conclusion, we have proposed a model in

B(T) which we assumed that the total density of Fig.2. pxy as a function of B

for very low T : full line cor- responds to the experimental results (10) ; the dashed line corresponds to the theoretical results in the case of localiza- tion ( ~ ~ ~ 2 ) .

Fig.3. Experimental (a) and theoretical (b) QHE for sample 1 at 1.3K under different values of the pressure P given in Kbar.

P = 0 ; Ns = 3 . 7 x 1 0 ~ ~ ,

p = 4.19~105 c m 2 / ~ s - ~ = 11.3 ; N,

-

p

-

mobile electrons must change in order to obey the statistics. This model shows that two regimes exist in the QHE : above a temperature of the order of X , the plateaus are due to a transfer of

electrons between the 2DEG and an external reservoir ; below this temperature the process is similar but the reservoir is inside the QW, i.e. the non mobile electrons are localized in the QW. This may correspond to a DOS in form of a sum of Gaussian peaks which contain the mobile (extended) states, superimposed on a background of localized states. In spite of simplified hypotheses, this model seems to account for all the experimental results found in the litterature, as well as our own results, without any fitting parameter .

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