Temperature-dependent magnetotransport properties for systems of few quantum wires

Physica B 227 (1996) 24-30

ELSEVIER

Temperature-dependent magnetotransport properties for systems of few quantum wires G. Ploner, J. Smoliner, G. Strasser, E. Gornik* Institut fiir Festk6rperelektronik, ~likrostukturzentrum der TU-Wien, Floragasse 7, A-1040 Wien, Austria

Abstract We have investigated temperature-dependent magnetotransport properties of quantum wires fabricated on high mobility GaAs-GaAIAs modulation doped heterostructures. Laser holography and optical lithography were used to define multiple quantum wire systems with 40 wires in parallel. These "few wire systems" turn out to have the best signal to noise ratio for systematic magnetic depopulation and magnetophonon resonance measurements. In the examined temperature range between 1.9 and 160 K it was found that the 1D subband energies increase strongly with decreasing 1D electron density and the polaron mass increases with increasing 1D subband spacing. Between 100 and 160 K, magnetophonon resonance data indicate a decline of both the subband spacing and also the polaron mass with increasing temperature. This effect is most probably due to an increase of the electron concentration with increasing temperature. Keywords: Magnetotransport properties; Quantum wires

1. Introduction M a g n e t o p h o n o n resonances in semiconductors turn out to be a useful tool to characterize various types of samples ranging from bulk material down to one-dimensional (1D) systems. In bulk material, m a g n e t o p h o n o n resonances cause weak structures in the second derivative of sample resistance (d2R/dB2), which were used to determine the effective mass and the L O phonon energy [1, 2]. First investigations on m a g n e t o p h o n o n resonances in two-dimensional electron systems were reported by Tsui et al. [3]. The structures they observed in d 2 R / d B 2 were extremely weak, but nevertheless allowed the determination of the polaron mass. Later, m a g n e t o p h o n o n resonances were also investigated by cyclotron resonance measurements [4, 5], where it was possible to determine the energy *Corresponding author.

relaxation rates. The work on magnetophonon resonances in two-dimensional systems was carried on by Brummell and coworkers [6], who studied systematically magnetophonon resonances as a function of temperature, electron concentration and magnetic field orientation. Most recently, magnetophonons were also observed in vertical transport experiments performed on GaAs-A1GaAs superlattices [7]. Theoretical considerations on magnetophonon resonances for two-dimensional electron systems were carried out by Hamaguchi et al. [8] and Mori et al. [9]. There have been several theoretical predictions of interesting magnetophonon effects in nanostructured systems such as quantum wires. Vasilopoulous et al. [10] have shown that for 1D systems, the magnetophonon resonances should be shifted to lower fields. Mori et al. [11] and Hamaguchi et al. [12] pointed out that the magnetoconductivity axx should consist of two contributions, one of which is related to the

0921-4526/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PII S092 1-4526(96)003 24-9

G. Ploner et al. / Physica B 227 (1996) 24- 30

current carried by the electron hopping motion between the localized cyclotron orbits through the electron-phonon interaction, and the other is caused by the current carried by electron motion affected by the confinement potential. The former exhibits maxima when the resonance condition is matched, whereas the latter results in minima at resonance. More recently, Ryu et al. [13, 14] proposed an alternative theoretical approach to 1D transport in the magnetophonon regime which gives in a linear response approximation analytical expressions for the oscillating magnetoconductivity and which allows the treatment of tilted magnetic fields as well. In our former experimental work [15, 16], we have experimentally investigated magnetophonon resonances in multiple quantum wire systems realized on GaAs-A1GaAs heterostructures. It was shown that magnetophonon resonances can be used to determine the 1D subband spacings and also the polaron mass, which was found to be enhanced compared to 2D systems. Moreover, also the influence of hydrostatic pressure was investigated [17]. However, all these measurements indicated a strong enhancement of subband energies at elevated temperatures, which up to now remains unexplained. In the present paper we study magnetotransport phenomena in the temperature range between 1.9 and 160 K for systems consisting of a small number of quasi-one-dimensional quantum wires in parallel. Comparing the subband energies determined by magnetic depopulation experiments at 1.9 K with magnetophonon resonance (MPR) experiments at 115 K, we find reasonable agreement of the obtained values within the accuracy of our measurements. Furthermore, when studying systematically the dependence of the subband spacing on electron density using magnetic depopulation experiments as well as MPR, we find that the 1D subband energies increase with decreasing 1D electron density and that the results obtained from magnetophonon resonance data are in good agreement with the behavior observed in low-temperature measurements. From the M P R data it is shown that the polaron mass increases with increasing subband energies. The M P R data obtained from temperature-dependent measurements up to 160 K

25

indicate that both the subband spacing and also the polaron mass decline slowly with growing temperature. Two different types of samples have been used in our experiments. The first type, in the following referred to as G73, is a modulation-doped heterostructure with a carrier density of 2.7 x 1011 cm 2 at T = 4 . 2 K and a mobility of 650000cm2/Vs. The other, referred to as G104, has a density of 1.1 x 1011 cm -2 and a mobility of 1.6 × 106 cm2/V s. Bar shaped mesas with a length of 200 lam and a width of 20 gm were fabricated on both samples and ohmic contacts were aligned using a layered GeAuNi metallization. On the mesas, an array of 40 quantum wires were fabricated using Laser holography and subsequent wet chemical etching. The holographic gratings had a period of 475 nm for all samples. From scanning electron microscope images, the geometrical width of the wires was determined to be 200 nm.

2. Results and discussion First we demonstrate that it is possible to determine 1D-subband spacings from magnetophonon resonance (MPR) experiments which are in accordance with the results obtained from magnetic depopulation measurements. For this purpose, we compare the magnetic depopulation data obtained for sample G73 at T = 1.9 K and a high-temperature measurement performed on the same sample at T = 118 K. All experiments discussed below are made in a configuration where a magnetic field is applied perpendicular to the heterostructure. Fig. l(a) shows the longitudinal resistance Rx~ of sample G73 as a function of magnetic field. The typical features of 1D systems, such as the large magneto size peak at B = 0.4 T and a number of characteristic minima due to magnetic depopulation of magnetoelectric hybrid levels are clearly observed. In Fig. l(b) the Landau index assigned to these minima is plotted over the inverse of the corresponding magnetic field values together with the best fit to the data obtained on the basis of a parabolic confinement potential [18]. The values for the subband spacing Eo and the one-dimensional (1 D) carrier density obtained from this fit are

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G. Ploner et al. /Physica B 227 (1996) 24-30

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Fig. 2. (a) Derivative of Rxx with respect to the magnetic field measured at T = 118 K for sample G73. Several magnetophonon resonances can be clearly resolved. (b) Plot of B 2 taken from the positions of the maxima in the high-temperature magnetoresistance versus the inverse squared index N. Evaluation of the slope of the resulting straight line and its intersection with the B2-axis according to Eq. (1) yields a subband spacing of 1.5 -I- 0.3 meV at high temperatures and a polaron mass of 0.077 electron masses.

1.1 _ 0.2 m e V a n d 4.4 +__0.2 x 10 6 c m - 1, respectively. Fig. 2(a) s h o w s the d e r i v a t i v e of R~x w i t h respect to m a g n e t i c field B as it h a s b e e n m e a s u r e d at

118 K, r e v e a l i n g s o m e p r o n o u n c e d o s c i l l a t o r y beh a v i o r d u e to M P R . I n Fig. 2(b), a n i n d e x N has b e e n a s s i g n e d to the p o s i t i o n s of the m a x i m a in the m a g n e t o r e s i s t a n c e a c c o r d i n g to the c o n d i t i o n

G. Ploner et al. / Physica B 227 (1996) 24-30

NhQeff = hf2LO, where hOLo = 36.6 meV is the LO phonon energy in bulk GaAs, h~2e2ff= h~,-2eyclotro n 2 + h~C~o 2 and hf2o is the subband spacing of

the 1D wires. The squared magnetic field positions B 2 of the corresponding maxima have been plotted as a function of 1/N 2 and the solid line in Fig. 2(b) is due to a fit of the data according to B2

(m*°l~ 2E~°

(m*°l']2 E 2 .

(1)

From the slope of the line we obtain a polaron mass and a subband spacing E0 = 1.5 _+ 0.3 meV, which is in good agreement with the low temperature value of Eo = 1.1 _+ 0.2 meV. As mentioned above, former experiments revealed an extremely large discrepancy between the low-temperature and high-temperature subband spacings, which could not be explained. The results of magnetotransport measurements presented in this paper, however, indicate that a possible reason for this effect may be found in the fact that in the experiments reported in Refs. [14] and [15] very large arrays of quantum wires were used. Thereby induced inhomogeneities in the sample structure may possibly have led to a distortion of the measured resonant structures which obscured the position of the resonant structures. For the experiments reported in this paper, therefore systems of few quantum wires in parallel have been used. Moreover, the enhancement of the subband energies at high temperatures seems to depend on the wafer and the fabrication parameters, but no systematic behavior could be found. If we compare the hightemperature and the low-temperature results for our new sample, however, the enhancement of the subband energies at high temperatures is just in the order of 30%. As this value is reproducible for all investigated samples, this offers the possibility to determine the 1D subband energies also for samples with low electron densities, where magnetic depopulation usually fails due to the low number of occupied subbands. In a second experiment, the influence of the density of charge carriers on the 1D subband spacing has been investigated. For this purpose, an array of wires was produced on a sample with low electron density (G104), which were completely depleted at m*ol =0.077me

27

4.2 K. Charge carriers of varying density were then generated by illuminating the array using a red light emitting diode. In those cases where the carrier densities were large enough to yield a sufficient number of occupied subbands, the electron density and subband spacing were determined from magnetic depopulation measurements. In the cases of very low carrier densities and small numbers of occupied subbands magnetophonon resonance experiments have been used. Since the number of carriers generated by illumination at low temperature is partially reduced on heating by thermally activated recombination processes, we adopted the following procedure for the high-temperature (MPR)-experiments: (a) cooling the sample to liquid helium temperature and illuminating it by the LED, (b) heating to a temperature which shows the most pronounced MPR effect (typically between T = 110 and T = 120 K) and waiting for equilibrium (i.e. until no changes in the sample resistance were observed), (c) performing a measurement of R~x as a function of magnetic field and (d) cooling down again slowly to liquid helium temperature (or below) to check for the remaining density of charge carriers. It is assumed that the carrier densities which are operational in steps (c) and (d) are identical. Fig. 3 shows the experimentally determined behavior of the subband energies for sample G104 as a function of the 1D carrier density NI D, indicating a strong increase of Eo with decreasing NID. In those cases where the electron density was large enough to give sufficient numbers of occupied subbands, we determined NID from a fit to the SdH data (cf. Ref. [18]) recorded at T = 2 K. At very low carrier densities, however, low-temperature SdH data could no longer be evaluated with sufficiently low experimental error. In these cases we also estimated N1D from the measured sample resistance by extrapolating the relation between N1D and sample resistance obtained at higher densities. In Fig. 3, the values of E0 for I D densities larger than 2 x 106 c m - l were determined by magnetic depopulation measurements at T = 1.9 K, those in the range between 1 and 2 x 1 0 6 c m - 1 from MPR experiments at T = 115 K. In this lowdensity regime the low-temperature magnetoresistance displays only few and weak structures in the

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G. Ploner et al. / Physica B 227 (1996) 2 4 - 3 0

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Fig. 3. Plot of the measured subband energies Eo as a function of ID electron density for sample G104. The inset shows a typical set of high-temperature data obtained for this sample. Note that in this case the positions of the minima in R~x have to be considered in order to obtain consistent results. The solid line is only a guide to the eye.

Rx~(B) curves at 2 K. The inset in Fig. 3 shows, however, that particularly in this low-density regime the oscillatory nature of the longitudinal resistance is very pronounced at T = l 1 5 K and allows the resolution of up to five resonant structures in the derivative of Rx~ with respect to B. It is remarkable that consistent results are obtained for sample G104 only if minima in R,~ are considered for the evaluation of our data according to Eq. (1). In contrast to that, maxima of the magnetoresistance had to be used for sample G73, where the confinement is much weaker. The latter seems to be in contrast to predictions of Hamaguchi et al. [12] and Mori et al. [11] which, in the case of weak confinement, postulate maxima in the magnetoconductivity when the condition for resonant optical phonon scattering is matched. To avoid this apparent conflict one might argue as follows: The subband spacings observed for sample G104, where a maximum value of 3.3 meV has been found, indicate that the confine-

ment for this sample is in an intermediate regime. Numerical calculations performed by the authors cited above indicate that phonon-assisted impediment of the skipping orbit motion becomes the dominant factor in the expression for the magnetoconductivity at confinement energies above approximately 5 meV. Thus, the behavior observed for sample G104 is in agreement with theoretical predictions, yielding minima in Rxx(B) corresponding to maxima in axe(B) as postulated if scattering between localized states dominates the magnetoconductivity. In the case of G73, however, the measured subband spacing (1 meV) indicates such a weak confinement that at elevated magnetic fields, where M P R are observed, the magnetic confinement dominates over the electrical confinement. Thus, the behavior of the electronic motion is 2D-like leading to a tensorial relationship between conductivity and resistivity such that p ~ oc a~x. Hence Pxx displays the same features as ax~ leading to a maximum in the resistivity whenever the conductivity has a maximum. The scatter of the values for Eo obtained from MPR measurements reveals the limitations of the method used to vary the electronic density in the wires. The way in which thermal recombination and generation processes influence the carrier density at high temperatures depends partially on parameters which are difficult to control experimentally. Small changes in the carrier density occurring, for example, while the cooling process after the high-temperature measurement may have a large effect on Eo plotted as a function of N1D; particularly in the low density regime where the dependence of Eo on N1D is very strong. This interpretation becomes even more plausible when m*o~is plotted against Eo as shown in Fig. 4. Obviously, the polaron mass increases when the carrier density is reduced, reflecting an enhancement of the electron LO-phonon coupling and leading to a clear correlation between Eo and m~o]. Note that also the value of m*o] obtained from the measurement on sample G73 (indicated by an arrow in Fig. 4) is in good agreement with the behavior observed for sample G104. Fig. 5 shows the dependence of E0 and m*o~ as a function of temperature. The slight reduction of the subband energy as the temperature is raised

29

G. Ploner et al. / Physica B 227 (1996) 24 30

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therefore also expected for nanostructured samples. In the latter case any etched surface regions provide an additional source of electronic defect states from which electrons m a y be easily detrapped at elevated temperatures. In Fig. 5(b) the 1D polaron mass is plotted as a function of temperature. In accordance with the foregoing argument the polaron mass also decreases slightly with increasing temperature. This behavior is consistent with the results shown in Fig. 4, which yielded a decreasing polaron mass at decreasing subband energies, since both phenomena reflect the dependence of subband spacing and electron-LO phonon coupling from the density of charge carriers. Note that the value at 160 K is somewhat obstructing this interpretation. It has to be mentioned, however, that at the highest temperatures examined in this experiment, increasing thermal noise is superimposed to the magnetoresistance oscillations which enhances the experimental error considerably. As shown above, samples with few quantum wires turn out to be the most suitable system for systematic magnetotransport investigations since they reveal the best signal to noise ratio and also most reproducible data. Our results indicate that all effects become larger with increasing subband

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30

G. Ploner et al. / Physica B 227 (1996) 24 30

spacing and decreasing electron density. If one compares the peak size of the magnetophonon resonances in the present work with the peaks in previous publications [15-17], one can see that the magnetophonon resonances are directly evident in the magnetoresistance provided the 1D subband spacing is large enough. The fabrication of samples with high subband spacing through deep etching, however, is difficult since the quantum wires are often completely depleted in this case. Therefore, future work will systematically investigate the effects which occur if the electron concentration is decreased by external parameters such as back gate voltages or hydrostatic pressure. In detail, we will study the transition from the weak confinement regime to the strong confinement regime induced by a reduction of the electron concentration through back gate voltages. This should result in an increase of the magnetophonon resonance amplitudes and also in clear maxima in the magnetoresistance as predicted by Mori et al. [11]. Such experiments will also clarify, whether the strong enhancement of the 1D subband energies at high temperatures for certain samples is correlated to the electron density or just to the fabrication process.

3. Summary In summary, it has been shown that magnetophonon resonances can be used to determine subband spacings with reasonable accuracy to characterize ID systems with low numbers of subbands occupied. Traditional methods, such as magnetic depopulation experiments have too large experimental errors or even fail in this case. Furthermore, we found experimental evidence for the model considerations of Mori et al. [11], who claimed that in a weak confinement potential, the hopping motion of electrons between localized states leads to a minimum in the longitudinal resistance when the condition for resonant LO phonon scattering is matched. For very weak confinement, however, a 2D-like behavior seems to be dominant, where the tensor relation between Rxx(B) and ~rx~(B) leads to maxima in the resistivity at the magnetophonon resonance positions. Finally, our data show that the electron-phonon coupling is strongly dependent on the electron density in the

quasi-lD systems leading to a distinct increase of the polaron mass with declining 1D-carrier density.

Acknowledgements This work was sponsored by Oesterreichische Nationalbank Project No. 4874 and 6sterreichische Gesellschaft for Mikroelektronik (GMe). The authors are grateful to M. Heiblum, who provided samples in the early stage of the experiments.

References [1] R.A. Stradling and R.A. Wood, Solid State Commun. 6 (1968) 701. [2] P.G. Harper, J.W. Hodby and R.A. Stradling, Rep. Prog. Phys. 37 (1973) 1. [3] D.C. Tsui, T. Englert, A.Y. Chao and A.C. Gossard, Phys. Rev. Lett. 44 (1980) 341. [4] P. Warmenbol, F.M. Peeters, X. Wu and J.T. Devreese, Phys. Rev. B 40 (1989) 6258. [5] P. Warmenbol, F.M. Peeters and J.T. Devreese, Phys. Rev. B 37 (1988) 4694. [6] M.A. Brummell, D.R. Leadley, J.R. Nicholas, J.J. Harris and C.T. Foxon, Surf. Sci. 196 (1988) 451. [7] H. Noguchi, H. Sakaki, T. Takamasu and N. Miura, Phys. Rev. B 45 (1992) 12148. [8] C. Hamaguchi, N. Moil, H. Murata and K. Taniguchi, High Magnetic Fields in Semiconductor Physics 1I, Vol. 87 (Springer, Berlin, 1989). [9] N. Mori, H. Murata, K. Taniguchi and C. Hamaguchi, Phys. Rev. B 38 (1988) 7622. [10] P. Vasilopoulous, P. Warmenbol, F.M. Peeters and J.T. Devreese, Phys. Rev. B 40 (1989) 1810. [11] N. Mori, H. Momose and C. Hamaguchi, Phys. Rev. B 45 (1992) 4536. [12] C. Hamaguchi, N. Moil, H. Momose, T. Ezaki, G. Berthold, J. Smoliner, E. Gornik, G. B6hm, G. Weimann, T. Suski and P. Wisniewski, Physica B 201 (1994) 339. [13] J.Y. Ryu and R.F. O'Connell, Phys. Rev B 48 (1993) 9126. [14] Jai Yon Ryu, G.Y. Hu and R.F. O'Connell, Phys. Rev B 49 (1994) 10437. [15] G. Berthold, J. Smoliner, M. Hauser, C. Wirner, E. Gornik, G. B6hm, G. Weimann, C. Hamaguehi, N. Mori and H. Momose, Proc. 2nd Int. Syrup. on New Phenomena in Mesoscopic Structures, published as workbook only. [16] G. Berthold, J. Smoliner, C. Wirner, E. Gornik, G. B6hm, G. Weimann, M. Hauser, C. Hamaguchi, N. Mori and H. Momose, Semicond. Sci. Technol. 8 (1993) 735. [17] G. Berthold, J. Smoliner, E. Gornik, G. B6hm, G. Weimann, T. Suski, P. Wisniewski, C. Hamaguchi, N. Mori and H. Momose, EP2DS 10, Newport (1993) Surf. Sci. 305 (1994) 637. [18] K.F. Berggren, G. Roos and H. van Houten, Phys. Rev. B 37 (1988) 10118.