- Email: [email protected]
Low-field magnetoresistance anomaly in two-dimensional electron gas
Solid State Communications, Vol. 101, No. 4, pp. 243-247, 1997 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/97 $17.00+.00
PII: ~~38-1~8~94)~593-5
LOW-FIELD MAGNETORESISTANCE ANOMALY IN TWO-DIMENSIONAL ELECTRON GAS J.J. Mares, J. KriStofik and P. Hub& institute of Physics, Academy of Sciences of the Czech Republic, Cukrovamicka 10, Praha 6, Czech Republic (Received 1 July 1996; accepted 20 ~e~t~~ber 1996 by A.L. Eji-5s) A strong correlation between the low-field magnetoresistance anomaly and the temperature dependence of the quantum correction to the conductance in a disordered 2D system was experimentally observed in the mT range. The behavior contradicting the generally accepted logarithmic scaling was accounted for by a phase-brea~ng process due to the Aharonov-Bohm effect. Copyright 0 1996 Elsevier Science Ltd Keywords: A. quantum wells, A. semiconductors, D. electron-eIectron interactions, D. electronic transport, D. quantum localization.
Since the eighties, when the notion of weak localization (WL) [l-3] was worked out, the suppression of electron bank-pattering by an external magnetic field in structures with two-dimensional electron gas (2DEG) and its consequence - the giant negative magnetoresistan~ (NMR) of these systems [4-61 - have been well established and, generally speaking, well understood effects. However, some peculiar features of these phenomena, e.g. weak dependence of ma~etoresistan~e on or even its independence from the magnetic field in the miillitesla range, were studied to a considerably lesser extent and mainly theoretically and were interpreted by various authors in a rather controversial way. This effect, called a little bit inapprop~ately “low-field ma~etoresistance anomaly”, bears in fact quite a universal character in 2D systems. It can be observed in disordered metals [7- 1l] as well as in structures with hopping transport [5,12] and from this point of view it is, as we believe, worth looking for a universal explanation of the phenomenon, in case it exists. The present paper is devoted to the investigation of the temperature dependence of NMR and its low-field anomaly in a highly disordered multi-~-layer semiconductor system in the vicinity of the metal-insulator transition. This system was proved [13, 141 to be very suitable for such a study because the effect is well pronounced in the temperature range above 1 K and very easily measurable with high reliability. The aim of this paper is to provide an additional experimental material in this field and assess the validity of some theoretical approaches. For our experiments we have chosen, as a typical
disordered 2D system, an InP-based sulphur-doped MOVPE-grown multi-&layer structure showing at helium temperatures characteristic features of the presence of 2DEG, such as strongly anisotropical high field magnetotransport with QHE [14]. The 2D subsystem consists of ten &layers of an intended donor concentration of 4 x 101’ rns2 per sheet and 10008, apart embedded into the semiinsulating material. Measurements were performed on usual Hall bar samples [14] provided with AuGe alloyed contacts. For the resistance measurements the four probe d.c. method was used employing a Keithley 220 current source and a Keithley 619 electrometer [15]. To ensure the linearity and to avoid the self-heating and high electric field effects it was proved to be sufficient to keep the measuring current at 1 fi. Temperature in the range 1.2-4.2 K inside the sample space of the cryostat was controlled by keeping constant He vapor pressure and measured at zero magnetic field by a carbon resistor calibrated with respect to a Ge normal thermometer. Magnetic fiefd generated by a small superconducting magnet was determined from the current flowing through the series resistance. A special care was taken to eliminate transient effects, so that the experimental points were read under the steady state conditions. This ensured the reproducibility and identity of magnetoresistance curves taken in both sweeping directions. High temperature transport measurements enabled us to determine the actual electron mobility and donor concentration in our samples which were found to be respectively -0.5 m* Vs.-’ and -4.5 x 10” m-’ per one
243
ANOMALY
244 1.6 _--
--
IN TWO-DIMENSIONAL __-.--
ELECTRON
Fig. 1. Temperature dependence of the sheet conductance per layer GO. The insert shows the same dependence plotted in coordinates corresponding to the logarithmic scaling law [24]. B-layer, which means that the samples belonged to the metallic side of the metal-insulator transition of those systems [ 131. Sheet conductance per layer G” (see Fig. 1) at low temperatures and zero magnetic field is, however, linearly increasing with temperature. Suppression of transport by lowering the temperature which is the very symptom of localization is, in our case of metal, the sign of the so-called weak localization. The central role in weak localization phenomena is played by the phase coherence length L, [16]. This length characterizes the extent of the region in which the partial electron waves preserve their coherence, i.e. the ability to interfere. Within this region the amplitudes and not the intensities of electron waves add up [17], which effectively increases the probability of sojourn of electron in this region. The intensity of this effect is directly measured by the quantum correction (Go - GF) to the conductance which consequently has to be strongly correlated with L, and fully controlled by its temperature dependence &(Z’). Based on empirical arguments (see below) the following form of this correlation may be used tentatively = G” - G;.
Vol. 101, No. 4
movement of all the other electrons. The phase shift due to the interaction of electron with time dependent electric field is known as electrostatic Aharonov-Bohm effect [18, 191 and complete breaking of the phase (A# = 1) can be described by the relation 1 = (e/~){~(~))?~.
l/L,(T)
GAS
(1)
Let us analyze the possible temperature dependence of the phase coherence length L+(T). As mentioned above it can be defined as a length of the path on which the electron phase changes by about l[A$(L$) =r I]. The very nature of the phase breaking process responsible for it is the elastic interaction of the electron with the electromagnetic field inevitably present in the crystal. In the absence of the external magnetic field and magnetically active impurities it is prevailingly the fluctuating electrical field produced by thermally excited vibration of the lattice and/or by chaotic
(2)
where {~(~)} is the variation of time dependent electrostatic potential in the places through which the electron (center of the wave packet) moves during the time T+ (for details see refs 18 and 19). Of course, the evaluation of this variation is extremely difficult, however, fluctuations of the electric potential in the crystal should be approximately -kT/e to ensure the maintenance of the equilibrium in the electron gas. Thus writing k(To + T)/e instead of ($(I)) in equation (2), we obtain a reasonable estimate for r+, namely T+ = h/&T, + T).
(3)
The parameter To is added in order to express the fact that there always exist so-called zero point quantum movements [20] of the electrons and of the crystal lattice, so that the dephasing process continues to operate even at zero temperature. In highly disordered materials with permittivity EQ in the vicinity of the metal-insulator transition the Fermi velocity should be close to the velocity on the shallow impurity orbit vB = e2/4n&. Assuming that L, = vBT.$ (this assumption will be addressed later), we immediately obtain the formula L, =
e’ 4m,,k( To + T ) ’
(4)
The substitution of equation (4) into equation (1) yields the correct, actually observed type of the dependence of quantum correction to the conductance on temperature with an undetermined, at this moment, proportionality factor. Let us turn now our attention to another experimental result. Figure 2 presents an example of typical low-field magnetoresistance curves taken at four different temperatures. Close to the beginning the dependences vary only slightly and after passing a certain point they start to decrease with an approximately constant slope. The behavior of magnetoresistance just described is observed very often in 2D systems provided that the measurements are performed with sufficient accuracy in the millitesla range. As far as we know only the structures with appreciable spin-orbit interaction make an exception (see e.g. refs. 3, 8, 9). It can be seen that the slope at higher fields is larger for lower temperatures while the point at which the curves start to deviate is shifted to lower fields at lower temperatures. To describe these facts more precisely we introduce the field Bk at which the tendency to decrease just prevails over the effect
Vol. 101, No. 4
ANOMALY
IN TWO-DIMENSIONAL
ELECTRON
GAS
245
3.8
3.6
3.5 0
1
2
3
4
.j
T (W 3.4 ’ 0
I 20
I 40
GO
’
Fig. 3. Temperature dependence plotted in Bij2 vs T coordinates.
B (mT)
Fig. 2. The dependence of the longitudinal magnetoresistance on the magnetic field perpendicular to the sample measured at four different temperatures (from top to bottom 1.28 K, 2.38 K, 3.45 K, 4.2 K). The method of the determination of Bk is shown (see the text). constraining the resistance to remain constant. It is defined as an intersection of the straight line parallel to the B-axis corresponding to the initial resistance value with the extrapolation of the straight line fitting the curve at higher fields (see Fig. 2). The 1.28K trace shows a slope change at about 40mT. However, we are only interested in the field at which the resistance begins to change rapidly, so this slope change at higher field does not affect our arguments. It is a well-known fact that NMR in the weak localization regime is a consequence of the so-called magnetic Aharonov-Bohm effect [18, 191 consisting of the phase shift of electron wave due to the magnetic field. If electron wave is elastically split by some obstacle inside the crystal in such a way that it creates a closed path, the partial electronic wave and its time reversal propagating on the same loop in the opposite direction can interfere constructively again at the splitting point. Applying a magnetic field perpendicular to this loop, the phases of both constituents are shifted proportionally to the magnetic flux through the loop and at a sufficiently high field the interference is destroyed. The complete dephasing takes place on loops the spatial extent of which is characterized by the magnetic length x = (h/e@.!
(5)
It is obvious that the influence of magnetic field destroying interference patterns in the closed path cannot be observed in loops the extent of which is appreciably larger than L, for which the phase coherence of electron waves is already broken. The critical magnetic field Bk for which the phase shifts due to the magnetic and phase coherence length controlled effects
of the parameter
Bk
are of the same significance can be determined from the condition L, = X which may be rewritten with regard to equations (4) and (5) as
This formula can be directly compared with the experiment. Figure 3 depicts the temperature dependence of the parameter Bk in coordinates B:‘2 vs T. It is apparent that the experimental data are successfully fitted by a straight line with a slope of 2 x 10e2 T’” K-i in remarkably good agreement with the formula (6) while for InP (E = 12 [21]) the numerical factor on the left-hand side of equation (6) is practically the same (-1.8 x 10m2 T”* K-l). Moreover, the intercept with the axes provides the parameter To = 2.4K and the zero temperature value BA’2 = 4.8 x lop2 T”*. (Naturally, only one of these two quantities is independent.) Now taking for granted formula (1) and considering the relations (4) and (6) a strong correlation should exist between the quantum correction to the conductance and the parameter Bf2 measured at various temperatures. Indeed, plotting GD vs B:‘2, we obtain a diagram (Fig. 4) showing that this correlation does exist and is even linear with a slope of -4 x lo3 Tu2Q. This confirms the correctness of the form of equation 1) and enables us to determine the b missing constant Go (= 1.24 x 1O-4 Q-‘) in equation (1). If we now express the quantum correction GO - Gi: in 2D natural conductance units e2/2?r2h and Bf” in terms of BAi2, we obtain the correlation mentioned above in a very lucid form: 2n2h(Go
- GF)le2 = const (Bk/Bo)
I/2
.
(7)
On the basis of experimental data given it can be easily proved that the const = 1 almost exactly. This equality provides the evidence of the fact that the quantum correction to the conductance and parameter
ANOMALY
246
IN HO-DIMENSIONS
i’ / I
0.05 r 0.0 L.___J -... 1.45 1.4
I-__A_.___J 1.5 1.55
1.6
G'(IO-"0-l) Fig. 4. Illustration of the correlation between the parameter Bk and the conductivity GD plotted in GD vs B:j2 coordinates. B:12 are only two different measures of the same physical phemomenon, i.e. of the thermally agitated electron phase breaking process. Moreover, using equation (7), the proportionality factor in equation (1) can be determined and we can write (li/eBo)“2/L~(T)
= 27r2h(Go - GF)/e2.
(8)
This equation together with relation (4) describes the observed temperature dependence of conductance very satisfactorily. These rather striking experimental facts ought to be explained consistently in more detail. Let us start with the temperature dependence of conductance. It is evident at first glance that its behavior cannot be accounted for by classical theory of Landau-Baber type [22] because the predicted correction terms are of just opposite sign and that the quantum mechanical corrections due to the weak localization have to be used. Notice at this place an essential difference between the diffusion of Brownian particle which randomly changes its state during every collision [23] and the movement of an electron on the loop in WL regime. In the latter case the electron does not perform random walk but its path is quite stable and definite for the time r$. For this time interval the movement of the electron is non-dissipative and persistent, in some sense similar to the current in a superconducting loop and this in itself cannot be described by a usual square root diffusion formula [23] but rather by the already used relation L, cc r+. If we now treat the time rm as proportional to an inverse of the probability of the escape of an electron from a weakly localized orbit, we obtain quite a simple and natural inte~retation of equation (I), i.e. that the quantummechanical enhancement of conductance is directly proportional to the probability of releasing the electrons from weakly localized orbits. Admitting once this hypothetical picture of the
ELECTRON
GAS
Vol. 101, No. 4
conducting 2D system, one immediately also sees the inadequacy of scaling arguments [24] for this particular case leading to the logarithmic dependence of conductance on temperature [3, 241 (see also the insert of Fig. 1). The scaling procedure requires, namely, that the change of the dispersion of edge states of hypercube (square in 2D) accompanying its doubling can be treated as a small perturbation [25, 261. When the length of such a hypercube edge is smaller than or comparable of this new confinement with I+, the introduction should completely change the wave function within this region and cannot be considered as a small perturbation any more. On the other side, for the hypercubes (squares) appreciably larger than L, the creation of new edge states has practically nothing to do with the coherent loops of extent L, and consequently is without influence on the corresponding quantum correction. Moreoever, contradiction between the accurate solution for a random electronic 2D system similar to ours in the magnetic field and the scaling arguments and the necessity of comparison of L, with other characteristic lengths in the system for the determination of applicability of scaling theory have been pointed out recently in [27]. As concerns the rather astonishing agreement between numerical factors in equations (6) and (8) and experimental data the authors believe that this is due to the fact that the condition L, = h is only virtually related to the geometry of particular loops. Actually, using this condition, we compare the local Aharonov-Bohm electron phase shifts corresponding to two different components of the same electromagnetic 4-vector, so that the geometry is effectively excluded from the problem. In conclusion, we have studied the temperature dependence of low-field magnetoresistance anomaly and conductance in WL regime in highly disordered 2 DEG in the vicinity of metal-insulator transition. Experimentally observed linear temperature dependence of conductance contradicting the generally accepted scaling theory as well as the strong correlation between quantum correction to conductivity and lowfield magnetoresistance anomaly has been accounted for in terms of the competition between phase-breaking processes due to the thermally agitated electrostatic and magnetic Aharanov-Bohm effect. The model proposed provides simple universal formulae containing only minimum of fitting parameters which can be easily compared with experiment.
Ack~~w~edge~e~ts-me authors are grateful to K. Melichar, J. Pangriic and J. Sidakova for technical assistance. The work was partly supported by the Grant Agency of the Czech Republic (contract numbers 202/ 961002 1 and 202/95/0194).
Vol. 101, No. 4
ANOMALY
IN TWO-DIMENSIONAL
REFERENCES 1. 2.
3. 4. 5. 6. 7. 8. 9. 10.
11. 12. 13. 14.
Wegner, F., 2. Phys. B35, 1979, 207. Altshuler, B.L. and Aronov, A.G., in ElectronElectron Interactions in Disordered Systems (Edited by A.L. Efros and M. Pollak), NorthHolland, Amsterdam, 1985, p. 1. Bergmann, G., Phys. Rev. B28, 1983, 2914. Altshuler, B.L., Khmel’nitskii, D.E., Larkin, A.L. and Lee, P.A., Phys. Rev. B22, 1980, 5142. Zhao, H.L., Spivak, B.Z., Gelfand, M.P. and Feng, S., Phys. Rev. B44, 1991, 10760. Dorin, V.V., Phys. Lett. A183, 1993, 233. Chang, A.M., Baranger, H.U., Pfeiffer, L.N. and West, K.W., Phys. Rev. Lett. 73, 1994, 2111~. Chin, A. and Kurdak, C., Chang, A.M., Chang, T.Y., Phys. Rev. B46, 1992, 6846. Poole, D.A., Pepper, M. and Hughes, A., J. Phys. C: Solid State Phys. 15, 1982, L1137. Katine, J.A., Berry, M.J., Westervelt, R.M. and Gossard, A.C., Superlattices and Microstructures 16, 1994, 211. Asche, M., Friedland, K.-J., Kleinert, P. and Kostial, H., Semicond. Sci. Technol. 7, 1992, 923. Shahbazyan, T.V. and Raikh, M.E., Phys. Rev. Lett. 73, 1994, 1408. Mare& J.J., Feng, X., Koch, F., Kohl, A.. and KriStofik, J., Phys. Rev. B50, 1994, 5213. Feng, X., MareS, J.J., Raikh, M.E., Koch, F., Griitzmacher, D. and Kohl, A., Surf Sci. 263, 1992, 147.
15.
16. 17. 18. 19. 20. 21.
22. 23. 24.
25. 26. 27.
ELECTRON
GAS
247
Keithley, J.F., Yeager, J.R. and Erdman, R.J., Low Level Measurements, 3rd edn, Keithley Instr. Inc., Cleveland, OH, 1984, p. 57. Aoki, H. and Ando, T., Phys. Rev. Lett. 54, 1985, 831. L&y-Leblond, J.M. and Balibar, F., Quantics, North-Holland, Amsterdam, 1990, p. 145. Aharonov, Y. and Bohm, D., Phys. Rev. 115,1959, 485. Aharonov, Y. and Bohm, D., Phys. Rev. 123,1961, 1511. Mertzbacher, E., Quantum Mechanics, 2nd edn, Wiley, New York, 1970, p. 57. Mares, J.J., KriStofik, J., Smid, V., Zeman, J. and PekBrek, L., in Semi-Insulating III-VMaterials (Edited by A.G. Milnes and C.J. Milner), Adam Hilger, New York, 1990, p. 275. Baber, W.G., Proc. R. Sot. A158, 1937, 383. MacDonald, D.K.C., Noise and Fluctuations, Wiley, New York, 1962, p. 22. Abrahams, E., Anderson, P.W., Licciardello, D.C. and Ramakrishnan, T.V., Phys. Rev. Lett. 42,1979, 673. Mott, N.F. and Kaveh, M., J. Phys. C: Solid State Phys. 14, 1981, L659. Thouless, D.J., Phys. Rev. Lett. 39, 1977, 1167. Azbel’, M.Ya., Phys. Rev. B45, 1992, 4208.
PII: ~~38-1~8~94)~593-5
LOW-FIELD MAGNETORESISTANCE ANOMALY IN TWO-DIMENSIONAL ELECTRON GAS J.J. Mares, J. KriStofik and P. Hub& institute of Physics, Academy of Sciences of the Czech Republic, Cukrovamicka 10, Praha 6, Czech Republic (Received 1 July 1996; accepted 20 ~e~t~~ber 1996 by A.L. Eji-5s) A strong correlation between the low-field magnetoresistance anomaly and the temperature dependence of the quantum correction to the conductance in a disordered 2D system was experimentally observed in the mT range. The behavior contradicting the generally accepted logarithmic scaling was accounted for by a phase-brea~ng process due to the Aharonov-Bohm effect. Copyright 0 1996 Elsevier Science Ltd Keywords: A. quantum wells, A. semiconductors, D. electron-eIectron interactions, D. electronic transport, D. quantum localization.
Since the eighties, when the notion of weak localization (WL) [l-3] was worked out, the suppression of electron bank-pattering by an external magnetic field in structures with two-dimensional electron gas (2DEG) and its consequence - the giant negative magnetoresistan~ (NMR) of these systems [4-61 - have been well established and, generally speaking, well understood effects. However, some peculiar features of these phenomena, e.g. weak dependence of ma~etoresistan~e on or even its independence from the magnetic field in the miillitesla range, were studied to a considerably lesser extent and mainly theoretically and were interpreted by various authors in a rather controversial way. This effect, called a little bit inapprop~ately “low-field ma~etoresistance anomaly”, bears in fact quite a universal character in 2D systems. It can be observed in disordered metals [7- 1l] as well as in structures with hopping transport [5,12] and from this point of view it is, as we believe, worth looking for a universal explanation of the phenomenon, in case it exists. The present paper is devoted to the investigation of the temperature dependence of NMR and its low-field anomaly in a highly disordered multi-~-layer semiconductor system in the vicinity of the metal-insulator transition. This system was proved [13, 141 to be very suitable for such a study because the effect is well pronounced in the temperature range above 1 K and very easily measurable with high reliability. The aim of this paper is to provide an additional experimental material in this field and assess the validity of some theoretical approaches. For our experiments we have chosen, as a typical
disordered 2D system, an InP-based sulphur-doped MOVPE-grown multi-&layer structure showing at helium temperatures characteristic features of the presence of 2DEG, such as strongly anisotropical high field magnetotransport with QHE [14]. The 2D subsystem consists of ten &layers of an intended donor concentration of 4 x 101’ rns2 per sheet and 10008, apart embedded into the semiinsulating material. Measurements were performed on usual Hall bar samples [14] provided with AuGe alloyed contacts. For the resistance measurements the four probe d.c. method was used employing a Keithley 220 current source and a Keithley 619 electrometer [15]. To ensure the linearity and to avoid the self-heating and high electric field effects it was proved to be sufficient to keep the measuring current at 1 fi. Temperature in the range 1.2-4.2 K inside the sample space of the cryostat was controlled by keeping constant He vapor pressure and measured at zero magnetic field by a carbon resistor calibrated with respect to a Ge normal thermometer. Magnetic fiefd generated by a small superconducting magnet was determined from the current flowing through the series resistance. A special care was taken to eliminate transient effects, so that the experimental points were read under the steady state conditions. This ensured the reproducibility and identity of magnetoresistance curves taken in both sweeping directions. High temperature transport measurements enabled us to determine the actual electron mobility and donor concentration in our samples which were found to be respectively -0.5 m* Vs.-’ and -4.5 x 10” m-’ per one
243
ANOMALY
244 1.6 _--
--
IN TWO-DIMENSIONAL __-.--
ELECTRON
Fig. 1. Temperature dependence of the sheet conductance per layer GO. The insert shows the same dependence plotted in coordinates corresponding to the logarithmic scaling law [24]. B-layer, which means that the samples belonged to the metallic side of the metal-insulator transition of those systems [ 131. Sheet conductance per layer G” (see Fig. 1) at low temperatures and zero magnetic field is, however, linearly increasing with temperature. Suppression of transport by lowering the temperature which is the very symptom of localization is, in our case of metal, the sign of the so-called weak localization. The central role in weak localization phenomena is played by the phase coherence length L, [16]. This length characterizes the extent of the region in which the partial electron waves preserve their coherence, i.e. the ability to interfere. Within this region the amplitudes and not the intensities of electron waves add up [17], which effectively increases the probability of sojourn of electron in this region. The intensity of this effect is directly measured by the quantum correction (Go - GF) to the conductance which consequently has to be strongly correlated with L, and fully controlled by its temperature dependence &(Z’). Based on empirical arguments (see below) the following form of this correlation may be used tentatively = G” - G;.
Vol. 101, No. 4
movement of all the other electrons. The phase shift due to the interaction of electron with time dependent electric field is known as electrostatic Aharonov-Bohm effect [18, 191 and complete breaking of the phase (A# = 1) can be described by the relation 1 = (e/~){~(~))?~.
l/L,(T)
GAS
(1)
Let us analyze the possible temperature dependence of the phase coherence length L+(T). As mentioned above it can be defined as a length of the path on which the electron phase changes by about l[A$(L$) =r I]. The very nature of the phase breaking process responsible for it is the elastic interaction of the electron with the electromagnetic field inevitably present in the crystal. In the absence of the external magnetic field and magnetically active impurities it is prevailingly the fluctuating electrical field produced by thermally excited vibration of the lattice and/or by chaotic
(2)
where {~(~)} is the variation of time dependent electrostatic potential in the places through which the electron (center of the wave packet) moves during the time T+ (for details see refs 18 and 19). Of course, the evaluation of this variation is extremely difficult, however, fluctuations of the electric potential in the crystal should be approximately -kT/e to ensure the maintenance of the equilibrium in the electron gas. Thus writing k(To + T)/e instead of ($(I)) in equation (2), we obtain a reasonable estimate for r+, namely T+ = h/&T, + T).
(3)
The parameter To is added in order to express the fact that there always exist so-called zero point quantum movements [20] of the electrons and of the crystal lattice, so that the dephasing process continues to operate even at zero temperature. In highly disordered materials with permittivity EQ in the vicinity of the metal-insulator transition the Fermi velocity should be close to the velocity on the shallow impurity orbit vB = e2/4n&. Assuming that L, = vBT.$ (this assumption will be addressed later), we immediately obtain the formula L, =
e’ 4m,,k( To + T ) ’
(4)
The substitution of equation (4) into equation (1) yields the correct, actually observed type of the dependence of quantum correction to the conductance on temperature with an undetermined, at this moment, proportionality factor. Let us turn now our attention to another experimental result. Figure 2 presents an example of typical low-field magnetoresistance curves taken at four different temperatures. Close to the beginning the dependences vary only slightly and after passing a certain point they start to decrease with an approximately constant slope. The behavior of magnetoresistance just described is observed very often in 2D systems provided that the measurements are performed with sufficient accuracy in the millitesla range. As far as we know only the structures with appreciable spin-orbit interaction make an exception (see e.g. refs. 3, 8, 9). It can be seen that the slope at higher fields is larger for lower temperatures while the point at which the curves start to deviate is shifted to lower fields at lower temperatures. To describe these facts more precisely we introduce the field Bk at which the tendency to decrease just prevails over the effect
Vol. 101, No. 4
ANOMALY
IN TWO-DIMENSIONAL
ELECTRON
GAS
245
3.8
3.6
3.5 0
1
2
3
4
.j
T (W 3.4 ’ 0
I 20
I 40
GO
’
Fig. 3. Temperature dependence plotted in Bij2 vs T coordinates.
B (mT)
Fig. 2. The dependence of the longitudinal magnetoresistance on the magnetic field perpendicular to the sample measured at four different temperatures (from top to bottom 1.28 K, 2.38 K, 3.45 K, 4.2 K). The method of the determination of Bk is shown (see the text). constraining the resistance to remain constant. It is defined as an intersection of the straight line parallel to the B-axis corresponding to the initial resistance value with the extrapolation of the straight line fitting the curve at higher fields (see Fig. 2). The 1.28K trace shows a slope change at about 40mT. However, we are only interested in the field at which the resistance begins to change rapidly, so this slope change at higher field does not affect our arguments. It is a well-known fact that NMR in the weak localization regime is a consequence of the so-called magnetic Aharonov-Bohm effect [18, 191 consisting of the phase shift of electron wave due to the magnetic field. If electron wave is elastically split by some obstacle inside the crystal in such a way that it creates a closed path, the partial electronic wave and its time reversal propagating on the same loop in the opposite direction can interfere constructively again at the splitting point. Applying a magnetic field perpendicular to this loop, the phases of both constituents are shifted proportionally to the magnetic flux through the loop and at a sufficiently high field the interference is destroyed. The complete dephasing takes place on loops the spatial extent of which is characterized by the magnetic length x = (h/e@.!
(5)
It is obvious that the influence of magnetic field destroying interference patterns in the closed path cannot be observed in loops the extent of which is appreciably larger than L, for which the phase coherence of electron waves is already broken. The critical magnetic field Bk for which the phase shifts due to the magnetic and phase coherence length controlled effects
of the parameter
Bk
are of the same significance can be determined from the condition L, = X which may be rewritten with regard to equations (4) and (5) as
This formula can be directly compared with the experiment. Figure 3 depicts the temperature dependence of the parameter Bk in coordinates B:‘2 vs T. It is apparent that the experimental data are successfully fitted by a straight line with a slope of 2 x 10e2 T’” K-i in remarkably good agreement with the formula (6) while for InP (E = 12 [21]) the numerical factor on the left-hand side of equation (6) is practically the same (-1.8 x 10m2 T”* K-l). Moreover, the intercept with the axes provides the parameter To = 2.4K and the zero temperature value BA’2 = 4.8 x lop2 T”*. (Naturally, only one of these two quantities is independent.) Now taking for granted formula (1) and considering the relations (4) and (6) a strong correlation should exist between the quantum correction to the conductance and the parameter Bf2 measured at various temperatures. Indeed, plotting GD vs B:‘2, we obtain a diagram (Fig. 4) showing that this correlation does exist and is even linear with a slope of -4 x lo3 Tu2Q. This confirms the correctness of the form of equation 1) and enables us to determine the b missing constant Go (= 1.24 x 1O-4 Q-‘) in equation (1). If we now express the quantum correction GO - Gi: in 2D natural conductance units e2/2?r2h and Bf” in terms of BAi2, we obtain the correlation mentioned above in a very lucid form: 2n2h(Go
- GF)le2 = const (Bk/Bo)
I/2
.
(7)
On the basis of experimental data given it can be easily proved that the const = 1 almost exactly. This equality provides the evidence of the fact that the quantum correction to the conductance and parameter
ANOMALY
246
IN HO-DIMENSIONS
i’ / I
0.05 r 0.0 L.___J -... 1.45 1.4
I-__A_.___J 1.5 1.55
1.6
G'(IO-"0-l) Fig. 4. Illustration of the correlation between the parameter Bk and the conductivity GD plotted in GD vs B:j2 coordinates. B:12 are only two different measures of the same physical phemomenon, i.e. of the thermally agitated electron phase breaking process. Moreover, using equation (7), the proportionality factor in equation (1) can be determined and we can write (li/eBo)“2/L~(T)
= 27r2h(Go - GF)/e2.
(8)
This equation together with relation (4) describes the observed temperature dependence of conductance very satisfactorily. These rather striking experimental facts ought to be explained consistently in more detail. Let us start with the temperature dependence of conductance. It is evident at first glance that its behavior cannot be accounted for by classical theory of Landau-Baber type [22] because the predicted correction terms are of just opposite sign and that the quantum mechanical corrections due to the weak localization have to be used. Notice at this place an essential difference between the diffusion of Brownian particle which randomly changes its state during every collision [23] and the movement of an electron on the loop in WL regime. In the latter case the electron does not perform random walk but its path is quite stable and definite for the time r$. For this time interval the movement of the electron is non-dissipative and persistent, in some sense similar to the current in a superconducting loop and this in itself cannot be described by a usual square root diffusion formula [23] but rather by the already used relation L, cc r+. If we now treat the time rm as proportional to an inverse of the probability of the escape of an electron from a weakly localized orbit, we obtain quite a simple and natural inte~retation of equation (I), i.e. that the quantummechanical enhancement of conductance is directly proportional to the probability of releasing the electrons from weakly localized orbits. Admitting once this hypothetical picture of the
ELECTRON
GAS
Vol. 101, No. 4
conducting 2D system, one immediately also sees the inadequacy of scaling arguments [24] for this particular case leading to the logarithmic dependence of conductance on temperature [3, 241 (see also the insert of Fig. 1). The scaling procedure requires, namely, that the change of the dispersion of edge states of hypercube (square in 2D) accompanying its doubling can be treated as a small perturbation [25, 261. When the length of such a hypercube edge is smaller than or comparable of this new confinement with I+, the introduction should completely change the wave function within this region and cannot be considered as a small perturbation any more. On the other side, for the hypercubes (squares) appreciably larger than L, the creation of new edge states has practically nothing to do with the coherent loops of extent L, and consequently is without influence on the corresponding quantum correction. Moreoever, contradiction between the accurate solution for a random electronic 2D system similar to ours in the magnetic field and the scaling arguments and the necessity of comparison of L, with other characteristic lengths in the system for the determination of applicability of scaling theory have been pointed out recently in [27]. As concerns the rather astonishing agreement between numerical factors in equations (6) and (8) and experimental data the authors believe that this is due to the fact that the condition L, = h is only virtually related to the geometry of particular loops. Actually, using this condition, we compare the local Aharonov-Bohm electron phase shifts corresponding to two different components of the same electromagnetic 4-vector, so that the geometry is effectively excluded from the problem. In conclusion, we have studied the temperature dependence of low-field magnetoresistance anomaly and conductance in WL regime in highly disordered 2 DEG in the vicinity of metal-insulator transition. Experimentally observed linear temperature dependence of conductance contradicting the generally accepted scaling theory as well as the strong correlation between quantum correction to conductivity and lowfield magnetoresistance anomaly has been accounted for in terms of the competition between phase-breaking processes due to the thermally agitated electrostatic and magnetic Aharanov-Bohm effect. The model proposed provides simple universal formulae containing only minimum of fitting parameters which can be easily compared with experiment.
Ack~~w~edge~e~ts-me authors are grateful to K. Melichar, J. Pangriic and J. Sidakova for technical assistance. The work was partly supported by the Grant Agency of the Czech Republic (contract numbers 202/ 961002 1 and 202/95/0194).
Vol. 101, No. 4
ANOMALY
IN TWO-DIMENSIONAL
REFERENCES 1. 2.
3. 4. 5. 6. 7. 8. 9. 10.
11. 12. 13. 14.
Wegner, F., 2. Phys. B35, 1979, 207. Altshuler, B.L. and Aronov, A.G., in ElectronElectron Interactions in Disordered Systems (Edited by A.L. Efros and M. Pollak), NorthHolland, Amsterdam, 1985, p. 1. Bergmann, G., Phys. Rev. B28, 1983, 2914. Altshuler, B.L., Khmel’nitskii, D.E., Larkin, A.L. and Lee, P.A., Phys. Rev. B22, 1980, 5142. Zhao, H.L., Spivak, B.Z., Gelfand, M.P. and Feng, S., Phys. Rev. B44, 1991, 10760. Dorin, V.V., Phys. Lett. A183, 1993, 233. Chang, A.M., Baranger, H.U., Pfeiffer, L.N. and West, K.W., Phys. Rev. Lett. 73, 1994, 2111~. Chin, A. and Kurdak, C., Chang, A.M., Chang, T.Y., Phys. Rev. B46, 1992, 6846. Poole, D.A., Pepper, M. and Hughes, A., J. Phys. C: Solid State Phys. 15, 1982, L1137. Katine, J.A., Berry, M.J., Westervelt, R.M. and Gossard, A.C., Superlattices and Microstructures 16, 1994, 211. Asche, M., Friedland, K.-J., Kleinert, P. and Kostial, H., Semicond. Sci. Technol. 7, 1992, 923. Shahbazyan, T.V. and Raikh, M.E., Phys. Rev. Lett. 73, 1994, 1408. Mare& J.J., Feng, X., Koch, F., Kohl, A.. and KriStofik, J., Phys. Rev. B50, 1994, 5213. Feng, X., MareS, J.J., Raikh, M.E., Koch, F., Griitzmacher, D. and Kohl, A., Surf Sci. 263, 1992, 147.
15.
16. 17. 18. 19. 20. 21.
22. 23. 24.
25. 26. 27.
ELECTRON
GAS
247
Keithley, J.F., Yeager, J.R. and Erdman, R.J., Low Level Measurements, 3rd edn, Keithley Instr. Inc., Cleveland, OH, 1984, p. 57. Aoki, H. and Ando, T., Phys. Rev. Lett. 54, 1985, 831. L&y-Leblond, J.M. and Balibar, F., Quantics, North-Holland, Amsterdam, 1990, p. 145. Aharonov, Y. and Bohm, D., Phys. Rev. 115,1959, 485. Aharonov, Y. and Bohm, D., Phys. Rev. 123,1961, 1511. Mertzbacher, E., Quantum Mechanics, 2nd edn, Wiley, New York, 1970, p. 57. Mares, J.J., KriStofik, J., Smid, V., Zeman, J. and PekBrek, L., in Semi-Insulating III-VMaterials (Edited by A.G. Milnes and C.J. Milner), Adam Hilger, New York, 1990, p. 275. Baber, W.G., Proc. R. Sot. A158, 1937, 383. MacDonald, D.K.C., Noise and Fluctuations, Wiley, New York, 1962, p. 22. Abrahams, E., Anderson, P.W., Licciardello, D.C. and Ramakrishnan, T.V., Phys. Rev. Lett. 42,1979, 673. Mott, N.F. and Kaveh, M., J. Phys. C: Solid State Phys. 14, 1981, L659. Thouless, D.J., Phys. Rev. Lett. 39, 1977, 1167. Azbel’, M.Ya., Phys. Rev. B45, 1992, 4208.