# Weak antilocalization in high mobility GaInAs/InP two-dimensional electron gases with strong spin-orbit coupling

###### Abstract

We have studied the spin-orbit interaction in a high mobility two-dimensional electron gas in a GaInAs/InP heterostructure as a function of an applied gate voltage as well as a function of temperature. Highly sensitive magnetotransport measurements of weak antilocalization as well as measurements of Shubnikov–de Haas oscillations were performed in a wide range of electron sheet concentrations. In our samples the electron transport takes place in the strong spin precession regime in the whole range of applied gate voltages, which is characterized by the spin precession length being shorter than the elastic mean free path. The magnitude of the Rashba spin-orbit coupling parameter was determined by fitting the experimental curves by a simulated quantum conductance correction according to a model proposed recently by Golub [Phys. Rev. B 71, 235310 (2005)]. A comparison of the Rashba coupling parameter extracted using this model with the values estimated from the analysis of the beating pattern in the Shubnikov–de Haas oscillations showed a good agreement.

###### pacs:

71.70.Ej, 72.25.Rb, 73.63.Hs## I Introduction

Two-dimensional electron gases (2DEGs) with an InAs or high In-content GaInAs channel layer are very promising candidates for spintronic applications, because they show a strong Rashba spin-orbit interaction along with a high electron mobility.Nitta et al. (1997); Engels et al. (1997); Th. Schäpers et al. (1998) These properties are essential for the realization of various spintronic devices, e.g., spin field effect transistors, Datta and Das (1990); Schliemann et al. (2003) spin-filters, Bulgakov et al. (1999); Koga et al. (2002a) or spin-splitters.Ohe et al. (2005) The strength of the Rashba coupling can be estimated from measurements of Shubnikov–de Haas oscillations by analyzing the characteristic beating pattern. Nitta et al. (1997); Engels et al. (1997); Th. Schäpers et al. (1998) However, the latter can also be evoked by inhomogeneities of the sheet carrier concentration Brosig et al. (1999); Thillosen et al. (2006) or due to a slightly occupied second lowest subband;Leadley et al. (1992) thus, the observance of a beating pattern is not an unambiguous indication of the presence of spin-orbit coupling. On the other hand, the weak localization effect is very sensitive not only to an applied magnetic field but also to spin-orbit coupling. The latter results in a non-monotonous dependence of the quantum correction to the conductivity on magnetic field. In case of strong spin-orbit coupling the quantum correction to the conductivity can even change its sign. The observation of such an effect, also referred to as weak antilocalization (WAL),Hikami et al. (1980); Bergmann (1982) is an unambiguous indication of the presence of spin-orbit interaction. Thus, weak antilocalization measurements open the way for the experimental determination of the contributions to the spin splitting: the linear and cubic Dresselhaus terms,Dresselhaus (1955) associated with the lack of crystal inversion symmetry, and the Rashba term,Bychkov and Rashba (1984) resulting from the structural inversion asymmetry. The latter can be controlled by applying an external electric field and is thus in particular interesting for spintronic devices.Nitta et al. (1997); Engels et al. (1997)

Until recently, only theoretical models describing the weak spin precession regime, also called ”diffusion” regime, were available,Iordanskii et al. (1994); Pikus and Pikus (1995); Knap et al. (1996) where the elastic mean free path is much shorter than the spin precession length .Koga et al. (2002b); Ch. Schierholz et al. (2002) However, is often found in high mobility 2DEGs comprising a strong spin-orbit interaction. Under such conditions an experimental observation of the weak antilocalization requires very sensitive magnetotransport measurements, since the width of the WAL peak is often less than 1 mT. From theoretical point of view, extensions of the ”diffusion” models to the case of strong spin precession were only limited to an even narrower range of magnetic fields and thus do not describe the whole WAL curve.Miller et al. (2003) Very recently, this problem was solved by the model developed by GolubGolub (2005) and later extended by Glazov and Golub,Glazov and Golub (2006) which is valid for both the weak and strong spin precession regimes in 2DEGs.

Taking this into account, we have utilized this model to extract the spin-orbit coupling in high-mobility GaInAs/InP samples. By fitting the WAL curves at different gate voltages we obtained the dependence of the Rashba spin-orbit coupling parameter on the sheet carrier concentration. The results of these measurements were compared with the values extracted from the analysis of the beating pattern of the Shubnikov–de Haas oscillations. We found, that despite of the fact that the fitting procedure is time-consuming due to the numerical complexity, the weak antilocalization measurements might be more advantageous for the determination of the Rashba coupling parameter, since they are also applicable if no beating pattern of the Shubnikov–de Haas oscillations can be observed, e.g. in case of low mobility.

## Ii Experimental

The GaInAs/GaInAs/InP heterostructure used in our investigation was grown by metal organic vapor phase epitaxy. A 2DEG was formed within the strained GaInAs channel layer. A sketch of the layer sequence is given in Fig. 2 (inset). Conventional 200-m-wide Hall bar structures with voltage probes separated by 160 m were defined by optical lithography and reactive ion etching. Subsequently, the AuGe ohmic contacts were deposited and annealed by rapid temperature processing. Metallic top gates separated from the semiconductor surface by an insulating HSQ layer (hydrogen silsesquioxane) covered the complete Hall bar structure and allowed us to control the sheet carrier concentration in a wide range, even down to a complete depletion of the 2DEG.

The magnetotransport measurements were performed in a He-cryostat with a superconducting magnet at temperatures down to 0.4 K utilizing a lock-in technique. Since the WAL effect is strongly temperature dependent, special attention was payed to the electrical power dissipated in the 2DEG. Depending on the changes of the resistance of the sample at different gate voltages the ac bias current was varied to avoid a heating of the 2DEG. In order to perform weak antilocalization measurements in well-controlled magnetic fields being less than 15 mT, an additionally mounted small superconducting coil was used.

## Iii Discussion

In order to characterize the 2DEG, Shubnikov–de Haas oscillations were measured in a wide range of gate voltages at 0.5 K. As shown in Fig. 1 a), the magnetoresistance vs. inverse magnetic field curves reveal an increase of the oscillation period as well as a change of the beating pattern if the gate voltage is decreased from 0 to V. By performing a fast Fourier transform analysis of the Shubnikov–de Haas oscillations we determined the corresponding electron sheet densities. No indication of a second subband occupation was found. As can be seen in Fig. 1 b), in the gate voltage range from 0 to -5.9 V we were able to change the carrier concentration from cm to cm. On a long-time scale, i.e. within days, a slight variation of the sheet carrier concentration determined at the same applied gate voltage was observed, possibly due to the presence of the electron states with a long relaxation time at the interface between the semiconductor and the gate insulator. However, the longitudinal resistance as well as the electron mobility remained unique functions of the carrier concentration [c.f. Fig. 1b)]. As a consequence, we took rather than as a reference for the following analysis of the weak antilocalization effect.

For our heterostructure we assumed that the spin-orbit coupling is dominated by the Rashba effect.Th. Schäpers et al. (1998) Indeed, by performing self-consistent band structure calculations and applying the theory presented, e.g. in Ref. [Winkler, 2003], the strength of the Dresselhaus and Rashba spin-orbit coupling at several gate voltages was estimated. At zero gate voltage, the ratio between the coupling parameter of the linear Dresselhaus term ( eVm) and the Rashba term ( eVm) is 0.24. With increasing this ratio becomes even less. The cubic Dresselhaus term, which is dependent strongly on , at zero gate voltage is comparable with the linear one and decreases rapidly with reduced . Thus, in our sample the spin-orbit coupling is dominated by the Rashba effect. By varying the built-in electrical field and, consequently, the bending of the conductance and valence band profile the magnitude of the Rashba coupling parameter can be controlled.

In 2DEGs, the Rashba spin-orbit coupling parameter can be determined from the position of the nodes of the beating pattern in the Shubnikov–de Haas oscillations:Das et al. (1989); Th. Schäpers et al. (1998)

(1) |

where is the Fermi wave vector, is the effective electron mass, and is the magnetic field where the th node is observed. The values of the spin-orbit coupling parameter determined by this method are presented in Fig. 2. As can be seen, increases with decreasing because of the larger asymmetry of the quantum well profile at more negative gate voltages. As the sheet carrier concentration becomes smaller than 2 cm, the second node (see Fig. 1) cannot be resolved anymore and, consequently, one faces the limitations of the beating pattern analysis technique for the extraction of the Rashba parameter. With the known values of the spin precession length can be calculated: . The values of at different electron concentrations are shown in Fig. 3 a). For comparison, the corresponding values of the elastic mean free path are plotted in Fig. 3 b).

We now turn to the measurements of the weak antilocalization
effect. As can be seen in Fig. 4, a clear weak
antilocalization peak was resolved at for gate voltages
ranging from -3.0 to -5.9 V. Here, the magnitude of the
quantum conductance correction was obtained by subtracting the
conductance at zero magnetic field from the experimental
determined magnetoconductance. ^{\theendnote}^{\theendnote}endnote: \theendnoteThis is acceptable because
the change of the magnetoconductance due to the Lorentz force,
which is a classical effect, is negligible in the range of
magnetic field applied here. Increasing the gate voltage resulted
in a strong increase of the sample resistance and simultaneously
in a broadening of the WAL peak. Under such conditions, the
experimental observation of the WAL effect becomes easier,
compared to resolving the beating pattern of the Shubnikov–de
Haas oscillations, in particular at the highest gate voltages. At
V the WAL effect could not be resolved
unambiguously.

In order to extract the Rashba coupling parameter from the weak antilocalization measurements the experimental curves were fitted by numerically calculated ones. The choice of the proper model is governed by the transport regime in the sample. In our case, a comparison of the elastic mean free path and the spin relaxation length extracted from the findings of the Shubnikov–de Haas oscillations revealed that is larger than . This corresponds to the regime of strong spin precession, so that the model in Refs. [Golub, 2005; Glazov and Golub, 2006] has to be applied. It has been shown that the calculated weak antilocalization curves do not differ from each other significantly if the ratio between the coupling parameters and is less than approximately 0.6.Glazov and Golub (2006) Since in our case the ratio is less than 0.24, we can readily neglect the Dresselhaus terms in the further analysis. At a given electron concentration, the number of free fitting parameters in the simulations could be reduced from four to two, since the elastic mean free path and the elastic scattering time were determined directly from the Shubnikov–de Haas measurements and were kept constant during the fitting procedure. The initial value of one of the free parameters, the phase coherence length , was estimated according to Refs. [Al’tshuler et al., 1982; Giuliani and Quinn, 1983; Fukuyama and Abrahams, 1983; Choi et al., 1987] and adjusted during the fitting procedure. As can be seen in Fig. 4, a good fit to the experimental curves has been achieved for all gate voltages. The corresponding values of are shown in Fig. 2.

Knowing the fitting parameters in particular the dependence of on , the corresponding dependence of the spin precession length was determined [see Fig. 3 a)]. High electron concentrations, i.e. small negative gate voltages, result in which is an order of magnitude shorter than . Applying higher negative gate voltages, i.e. lowering , leads to the rapid shortening of the and, due to the enhancement of the spin-orbit interaction, to the shortening of . Despite the decrease of with , is always larger than the corresponding value of . Thus the strong spin precession regime of the electron transport is preserved in the whole range of electron concentrations studied here.

A comparison of the values of obtained by the analysis of the beating pattern with the ones determined from the WAL measurements reveals a good agreement, although the latter are slightly higher. The origin of this discrepancy is not clear, yet. Possibly, the inaccuracy in the determination of the minimum position of the WAL curve caused by the additional contribution of other magnetotransport effects results in an overestimation of the Rashba coupling parameter. As long as the spin-orbit interaction is the only cause of the beatings in the Shubnikov–de Haas oscillations, these two methods are complementary in a certain range of carrier densities. However, at high carrier densities the observation of WAL is more difficult, since the longitudinal resistance of the sample and consequently the signal-to-noise ratio drops rapidly, whereas the characteristic beating pattern in the Shubnikov–de Haas oscillations can be resolved easily. In contrast, at low electron concentrations the WAL peak can be easily measured, whereas the position of the nodes cannot be resolved anymore.

Temperature dependent measurements, presented in Fig. 5 a) for a gate voltage of V, show a fast suppression of the WAL peak with increasing temperature. Also here, a good agreement between the experiment and the model of Glazov and GolubGlazov and Golub (2006) can be readily seen. The only parameter which was varied during the fitting procedure was the phase coherence time [see Fig. 5 b)].

At low temperatures the phase breaking rate is essentially determined by electron-electron scattering, which can be divided into two contributions. The first one, being connected to a large-energy-transfer scattering mechanism, leads to a -dependence of and is dominant at .Fukuyama and Abrahams (1983); Giuliani and Quinn (1983); Zheng and Sarma (1996) The second contribution originating from a small-energy-transfer mechanism depends linearly on and is most significant at .Al’tshuler et al. (1982); Fukuyama and Abrahams (1983) As can be seen in Fig. 5 b), the experimental points of the phase breaking rate could be fitted well to a combination of and dependencies. A comparison with the slopes corresponding to and [cf. Fig. 5b)] confirms that at higher temperatures ( K) large-energy-transfer scattering with dominates. Whereas, at lower temperatures a deviation from a slope proportional to towards a linear temperature dependence is observed, in particular at V. In fact, at increasing negative gate voltages a shift of the crossover temperature towards larger values is expected, owing to the decrease of with decreasing electron concentration. In detail, at gate voltages of V, V, and V the cross-over temperature was determined to be 5.4 K, 3.9 K, and 2.4 K, respectively. In accordance with the theoretical prediction Al’tshuler et al. (1982); Giuliani and Quinn (1983); Fukuyama and Abrahams (1983); Zheng and Sarma (1996) an increased scattering rate is found for larger negative gate voltages, i.e. lower and lower . However, a direct comparison of the values of extracted from the fit to the Golub modelGolub (2005) with the theoretically determined valuesChoi et al. (1987) reveals, that the latter is larger by a factor of about two. Probably, this discrepancy is connected to uncertainties in the determination of at large negative gate voltages, which is caused by the weak dependency of the quantum conduction correction on . As we observed in our calculations, has a strong effect on the magnitude of the quantum correction at magnetic fields smaller then 0.1 mT, i.e. close to the resolution limit of the experiment. In addition, inaccuracies in the determination of and from the Shubnikov–de Haas oscillations become more pronounced at large negative gate voltages. In our simulations we found that these inaccuracies mainly affect the precision in the determination of .

## Iv Conclusions

In conclusion, we have studied weak antilocalization in a GaInAs/InP 2DEG as a function of electron concentration. Experimental curves were fitted by a universal model which describes both the weak and the strong spin precession regime of the electron transport. Satisfactory fits were achieved in a wide range of gate voltages as well as at different temperatures. We have shown, that the dependence of spin-orbit coupling on the gate voltage can be successfully studied in high mobility 2DEGs by analyzing weak antilocalization measurements. This is a reliable method which is complementary to the beating pattern analysis of the Shubnikov–de Haas oscillations.

###### Acknowledgements.

The authors are deeply grateful to L. E. Golub and M. M. Glazov (A. F. Ioffe Physico-Technical Institute, St. Petersburg, Russia) for fruitful discussions.## References

- Nitta et al. (1997) J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys. Rev. Lett. 78, 1335 (1997).
- Engels et al. (1997) G. Engels, J. Lange, Th. Schäpers, and H. Lüth, Phys. Rev. B 55, R1958 (1997).
- Th. Schäpers et al. (1998) Th. Schäpers, G. Engels, J. Lange, Th. Klocke, M. Hollfelder, and H. Lüth, J. Appl. Phys. 83, 4324 (1998).
- Datta and Das (1990) S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).
- Schliemann et al. (2003) J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90, 146801/1 (2003).
- Bulgakov et al. (1999) E. N. Bulgakov, K. N. Pichugin, A. F. Sadreev, P. Středa, and P. Šeba, Phys. Rev. Lett. 83, 376 (1999).
- Koga et al. (2002a) T. Koga, J. Nitta, H. Takayanagi, and S. Datta, Phys. Rev. Lett. 88, 126601 (2002a).
- Ohe et al. (2005) J. I. Ohe, M. Yamamoto, T. Ohtsuki, and J. Nitta, Phys. Rev. B 72, 041308(R) (2005).
- Brosig et al. (1999) S. Brosig, K. Ensslin, R. J. Warburton, C. Nguyen, B. Brar, M. Thomas, and H. Kroemer, Phys. Rev. B 60, R13989 (1999).
- Thillosen et al. (2006) N. Thillosen, S. Cabañas, N. Kaluza, V. A. Guzenko, H. Hardtdegen, and Th. Schäpers, Phys. Rev. B 73, 241311(R) (2006).
- Leadley et al. (1992) D. R. Leadley, R. Fletcher, R. J. Nicholas, F. Tao, C. T. Foxon, and J. J. Harris, Phys. Rev. B 46, 12439 (1992).
- Hikami et al. (1980) S. Hikami, A. I. Larkin, and Y. Nagaoka, Progr. Theor. Phys. 63, 707 (1980).
- Bergmann (1982) G. Bergmann, Solid State Comm. 42, 815 (1982).
- Dresselhaus (1955) G. Dresselhaus, Phys. Rev. 100, 580 (1955).
- Bychkov and Rashba (1984) Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984).
- Iordanskii et al. (1994) S. V. Iordanskii, Yu. B. Lyanda-Geller, and G. E. Pikus, JETP Lett. 60, 199 (1994).
- Pikus and Pikus (1995) F. G. Pikus and G. E. Pikus, Phys. Rev. B 51, 16928 (1995).
- Knap et al. (1996) W. Knap, C. Skierbiszewski, A. Zduniak, E. Litwin-Staszewska, D. Bertho, F. Kobbi, J. L. Robert, G. E. Pikus, F. G. Pikus, S. V. Iordanskii, V. Mosser, K. Zekentes and Yu. B. Lyanda-Geller, Phys. Rev. B 53, 3912 (1996).
- Koga et al. (2002b) T. Koga, J. Nitta, T. Akazaki, and H. Takayanagi, Phys. Rev. Lett. 89, 046801 (2002b).
- Ch. Schierholz et al. (2002) Ch. Schierholz, R. Kürsten, G. Meier, T. Matsuyama, and U. Merkt, phys. stat. sol. (b) 223, 436 (2002).
- Miller et al. (2003) J. B. Miller, D. M. Zumbühl, C. M. Marcus, Yu. B. Lyanda-Geller, D. Goldhaber-Gordon, K. Campman, and A. C. Gossard, Phys. Rev. Lett. 90, 076807 (2003).
- Golub (2005) L. E. Golub, Phys. Rev. B 71, 235310 (2005).
- Glazov and Golub (2006) M. M. Glazov and L. E. Golub, Semicond. 40, 1209 (2006).
- Das et al. (1989) B. Das, D. C. Miller, S. Datta, R. Reifenberger, W. P. Hong, P. K. Bhattacharya, J. Singh, and M. Jaffe, Phys. Rev. B 39, 1411 (1989).
- Winkler (2003) R. Winkler, Spin-orbit coupling effects in two-dimensional elctron and hole systems (Springer-Verlag, Berlin, Heidelberg, New York, 2003).
- Al’tshuler et al. (1982) B. L. Al’tshuler, A. G. Aronov, and D. E. Khmelnitsky, J. Phys. C 15, 7367 (1982).
- Giuliani and Quinn (1983) G. F. Giuliani and J. J. Quinn, Phys. Rev. B 26, 4421 (1982).
- Fukuyama and Abrahams (1983) H. Fukuyama and E. Abrahams, Phys. Rev. B 27, 5976 (1983).
- Choi et al. (1987) K. K. Choi, D. C. Tsui, and K. Alavi, Phys. Rev. B 36, 7751 (1987).
- Zheng and Sarma (1996) L. Zheng and S. Das Sarma, Phys. Rev. B 53, 9964 (1996).