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ISSN 0021 3640, JETP Letters, 2013, Vol. 97, No. 3, pp. 149­154. © Pleiades Publishing, Inc., 2013. Original Russian Text © S.I. Dorozhkin, A.A. Kapustin, S.S. Murzin, 2013, published in Pis'ma v Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki, 2013, Vol. 97, No. 3, pp. 170­ 175.

Observation of Crossover from Weak Localization to Antilocalization in the Temperature Dependence of the Resistance of a Two Dimensional System with Spin­Orbit Interaction
S. I. Dorozhkin*, A. A. Kapustin, and S. S. Murzin
Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, Moscow region, 142432 Russia * e mail: dorozh@issp.ac.ru
Received November 9, 2012; in final form, January 9, 2013

A nonmonotonic temperature dependence of the resistance with a maximum in the temperature range of 2­ 4 K whose position depends on the hole density has been observed in hole channels of silicon field effect tran sistors. The spin­orbit hole relaxation time and the temperature dependences of the phase relaxation time of the electron wave have been obtained from the measurements of the alternating sign anomalous magnetore sistance. The nonmonotonic temperature dependence of the resistance can be described by the formulas of weak localization theory with these parameters. The maximum appears owing to a temperature induced change in the relation between the measured times. As a result, the localization behavior of the conductivity at high temperatures is changed to the antilocalization behavior at low temperatures. The inclusion of quan tum corrections to the conductivity caused by the electron­electron interaction improves quantitative agree ment between the experiment and calculation. Thus, it has been demonstrated that, in contrast to the widely accepted concept, there is a region of the parameters where the electron­electron interaction does not change the antilocalization (metallic) type of the temperature dependence of the resistance. DOI: 10.1134/S002136401303003X

The spin­orbit interaction triggering the spin relaxation mechanism significantly modifies quantum corrections to the conductivity caused by localization effects [1] and the electron­electron interaction effects [2, 3] in disordered systems. In the weak local ization theory, where the physical mechanism is the interference of electron waves at their scattering on impurities, the sign of the correction to the conductiv ity depends on the relation between the phase relax ation time of the electron wave and the spin relax ation time owing to the spin­orbit interaction so (in this work, we discuss only two dimensional systems of charge carriers). At so , spin relaxation is insignif icant and the temperature dependence of the resistiv ity (T) is determined by the localization of carriers and has the insulating form. In the opposite limit so , quantum correction to the conductivity of nonin teracting electrons is positive (the so called antilocal ization effect). Since the spin relaxation time is inde pendent of the temperature and the phase relaxation time is determined by inelastic processes, in the absence of other mechanisms of the temperature dependence of the resistance in the system with the spin­orbit interaction, an increase in (T) with a decrease in the temperature could lead (see, e.g., review [4]) to a nonmonotonic dependence (T) with the maximum at (T) ~ so. We certainly observed this behavior of the resistance. The use of the times (T)

and so determined from anomalous magnetoresis tance curves made it possible to satisfactorily describe the observed temperature dependence by the formulas of the weak localization theory. At first glance, the obtained result contradicts the theory of quantum cor rections taking into account the electron­electron interaction. This theory predicts the universal insulat ing behavior for systems with strong spin­orbit inter action (see Fig. 2 in [2] and Fig. 41 in review [3]). This behavior was confirmed in many investigations of thin metal films (see, e.g., [5]) and in hole channels of sili con field effect transitions on the Si(111) surface [6]. The effect of the spin­orbit interaction on the elec tron­electron interaction is characterized by the parameter Tso/ . The indicated contradiction is likely explained by the fact that the temperature Tm at which the resistance is maximal corresponds to the transient regime Tmso/ > 1, whereas the prediction of the universal insulating behavior was made in [2] /T. under the condition so The nonmonotonic temperature dependences of the resistance with the maximum were observed in two dimensional systems with a high mobility of charge carriers in the transition region in the electron density from the insulating to the metallic state both in systems with weak spin­orbit interaction [7] and in systems where noticeable spin­orbit interaction could be expected [8, 9]. The nonmonotonic temperature

149


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â

DOROZHKIN et al.

â

â

Fig. 1. Temperature dependences of change in the resistiv ity (per square) in (open symbols) zero magnetic field for three hole surface densities ps. The resistivities at T = 4.2 K were 2477, 3302, and 4902 per square for ps = 1.9 â 1012, 1.5 â 1012, and 1.1 â 1012 cm­2, respectively. The closed symbols depict the temperature dependences of the resistivity in the field H = 0.2 and 0.4 T for ps = 1.5 â 1012 and 1.1 â 1012 cm­2, respectively.

dependence of the resistance with the maximum in systems with weak spin­orbit interaction can appear [10, 11] owing to the renormalization (dependences on the temperature and disorder) of the Fermi liquid parameter describing the electron­electron interac tion. However, such a renormalization is usually important in systems with a low carrier density, partic ularly in the case of the presence of valley degeneracy, and is hardly significant in our samples. The non monotonic temperature dependence of the resistance under the conditions most corresponding to our experiment was observed on hole channels in multi layer Ge/SiGe heterostructures [12]. However, the origin of the maximum in the temperature depen dence of the resistance was beyond the scope of works [8, 9, 12]. Remarkable results were obtained in [13], where the metallic temperature dependence of the conductivity was observed in hole channels of GaAs/InGaAs/GaAs heterostructures, whereas cross over was likely masked by the electron­phonon scat tering.

Fig. 2. (a) (Solid lines) Experimental magnetic field dependences of the resistivity of the sample xx for ps = 1.5 â 1012 cm­2 at various temperatures. The line for T = 4.2 K is shifted down by 26.6 in order to avoid its inter section with other lines. The dashed lines are the calcula tion of the magnetoresistance by Eq. (1) with the spin relaxation times so and phase relaxation times shown in panel (b) by circles and squares, respectively. The solid straight line in panel (b) corresponds to the power law temperature dependence = 27.9T­p ps with p = 1.29 and the dashed straight line in panel (b) corresponds to the value so = 2.62 ps. These straight lines are the least squares fits of the respective experimental points.

The results of this work were obtained for accumu lation hole channels of silicon field effect transistors fabricated on the Si(110) surface. The hole mobility depended on their density and was about 103 cm2/(V s) in the density range under study. Most of the experi ments were performed in a cryostat with the evacua tion of 4He vapor in the temperature range of 1.3­7 K. Some measurements were carried out in a cryostat with the evacuation of 3He vapor in the temperature range of 0.5­7 K. Anomalous magnetoresistance, Hall resistance, and Shubnikov­de Haas oscillations were measured in magnetic fields created by a super conducting solenoid. We studied two samples made of one silicon wafer. The results were the same for both samples. The hole density in the samples was deter mined from the period of Shubnikov­de Haas oscilla tions, as well as from the magnetic field dependence of the Hall resistance.
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â

â â

â

â

â â

Fig. 4. Evolution of the anomalous magnetoresistance with variation of the hole density. The approximations of the experimental dependences by Eq. (1) are shown by dashed lines (T = 4.2 K). Fig. 3. (Points) Experimental temperature dependences of the conductivity recalculated from the data shown in Fig. 1 and the corresponding theoretical curves, which were obtained with the times so and determined from the measured anomalous magnetoresistance (the correspond ing values for ps = 1.5 â 1012 cm­2 are given in the caption of Fig. 2, and so = 4.5 ps and = 20.3T­1.46 ps for ps = 1.1 â 1012 cm­2). The dotted lines calculated by Eq. (2) are the temperature dependences of the conductivity in zero magnetic field determined by the weak localization effects. The solid lines were calculated as additive contributions from weak localization (Eq. (2)) and from the electron­ electron interaction (Eqs. (3) and (4) with the interaction parameters given in the main text). The dashed lines are the calculated corrections to the conductivity in magnetic fields of 0.2 and 0.4 T for ps = 1.5 â 1012 and 1.1 â 1012 cm­2, respectively.

shows the typical results of such measurements for the hole density ps = 1.5 â 1012 cm­2 at various tempera tures. As can be seen in Fig. 2, a weak (H 0.1 T) magnetic field suppresses the antilocalization correc tion. As a result, the temperature dependence becomes insulating. The observed nonmonotonic magnetic field dependence of the resistance is charac teristic of the weak localization effects in the system with a quite fast spin­orbit relaxation. To quantita tively describe the anomalous magnetoresistance curves, we used the Hikami­Larkin­Nagaoka for mula [1] ( H ) ­ ( 0 ) = e 2
2 2

H + H so 1 + 2 H (1)

Figure 1 shows the temperature dependence of the resistivity of a sample in zero and classically weak magnetic fields for three different hole densities ps. All curves measured in zero field have a maximum, which shifts toward lower temperatures with a decrease in ps. In magnetic field, the temperature dependence of the resistivity remains monotonic and insulating through out the entire temperature range. Theories of quantum corrections should obviously be used to explain such effects. To reveal the role of weak localization effects, we measured the anomalous magnetoresistance observed in classically weak magnetic fields. Figure 2
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1 1 H + 2 H so 1 1 H + + ­ + 2 2 H 2 2 H ­ ln H + H so 1 H + 2 H so 1 H ­ ln + ln . H 2 H 2H

Here, is the quantum correction to the conductiv ity of a two dimensional system; is the digamma c , where D is the hole diffu function; and Hx = 4 eD x sion coefficient and x is one of the used subscripts. For


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small changes in the magnetoresistance, xx(H) ­ (H = 0) ­2(H = 0)[(H) ­ (0)]. We used Eq. (1) to describe the anomalous magne toresistance in our samples because of the results reported in [14], where it was shown that this formula remains valid for systems with the spin­orbit splitting of the spectrum proportional to the third power of the wave vector of two dimensional carriers whose spin relaxation occurs through the D'yakonov­Perel' mechanism [15], which is most efficient in semicon ductors. Theoretical calculations performed for two dimensional hole systems within the effective mass approximation predict the cubic wave vector depen dence of the spin­orbit splitting associated with the asymmetry of the potential well in which two dimen sional holes are located (see, e.g., review [16] and ref erences therein). However, our choice is not the only possible. The spin­orbit relaxation in hole systems can also occur in the absence of the spin­orbit split ting of the two dimensional spectrum owing to the elastic scattering between the states of light and heavy holes mixed because of the quantum confinement effect. This was demonstrated in [17] and was used to describe the experiment on the anomalous magne toresistance in hole channels of GaAs/AlGaAs het erostructures in [18]. A reason for our choice is the observation of beats of Shubnikov­de Haas oscilla tions in our samples (for more details of this effect, see [19]), which certainly indicates that spin degeneracy is lifted in this system owing to the spin­orbit interac tion. Accurate independent determination of the spin and phase relaxation times from the experimental curves is possible only for a nonmonotonic magnetic field dependence of the magnetoresistance. This cir cumstance limits the hole density values for which such a procedure can be performed (see Fig. 4). The so and values determine the position and height of the maximum of the magnetoresistance in the mag netic field, respectively. We approximated the experi mental curves by the calculated lines trying to repro duce the position and height of the maximum and obtained a satisfactory agreement for reasonable so and values. In particular, in agreement with the commonly accepted point of view, the spin­orbit relaxation time is independent of the temperature, whereas the phase relaxation time increases with a decrease in the temperature as T­p (see Fig. 2b). Deviations of the calculated magnetoresistance from the experimental data in strong magnetic fields observed in Fig. 2a can be attributed to various rea sons. First, Eq. (1) is applicable only in the so called c , where diffusion approximation when H < Htr = 4 eD is the carrier momentum relaxation time. Under the experimental conditions, Htr decreases from 0.77 to 0.41 T with an increase in the hole density from ns =

0.98 â 1012 cm­2 to ns = 1.68 â 1012 cm­2, respectively. Deviations from this formula are manifested even at H < Htr (see, e.g., [20] and references therein). Sec ond, there is a classical temperature dependent mech anism of the magnetoresistance for a system consisting of two groups of carriers (in our case, these are holes corresponding to two branches of the spectrum) that was successfully used to explain the temperature dependence of the magnetoresistance in two dimen sional hole channels of GaAs/AlGaAs heterostruc tures [21]. For weak magnetic fields, this mechanism makes a positive contribution to the magnetoresis tance proportional to the magnetic field squared. Thus, the difference between the experimental and theoretical curves in stronger fields can be attributed to this mechanism. Determining the time so and interpolating the experimental dependence (T) by the power law temperature dependence, we can try to describe the experimental temperature dependences of the resis tance. However, it is more convenient to consider the conductivity , which is inversely proportional to the resistivity. Figure 3 shows the experimental data and theoretical results obtained with the weak localization theory [14] for the conductivity (we note that the sign in the corresponding formula in [14] should be changed to the opposite): ( H = 0 ) = e 2
2 2

1 ­ ln + ln + so 2

1 2 + ln + . 2 so

(2)

This equation provides the minimum in correction to the conductivity at = so( 5 + 1)/2 1.62so. The dotted lines calculated by Eq. (2) with the parameters so and dependences (T) obtained from the measure ments of the anomalous magnetoresistance for the corresponding hole densities reproduce the qualitative behavior and scale of change in the conductivity with the temperature. It is noteworthy that the relations Tso/ 1 and 1.3 at ps = 1.5 â 1012 and 1.1 â 1012 cm­2, respectively, are valid for temperatures Tm at which the maxima of the resistance are observed in the experi ment. For this reason, it can be assumed that the spin­ orbit interaction under our conditions hardly affects the electron­electron interaction. We numerically calculated corrections from the electron­electron interaction using formulas from [22], where the inter action was described in all regimes, including the pre viously considered diffusive (T/ 1) [3] and ballis 1). These calculations show that the tic (T/ observed nonmonotonic temperature dependence cannot be reproduced if it is assumed, according to [2], that the triplet term in the diffusion regime is sup pressed by the spin­orbit interaction. However, the triplet term can improve agreement with the experi
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ment, as will be shown below. The significance of the triplet term at a small parameter Tso/ 0.05 was mentioned in recent experimental work [23] specially devoted to this problem. For these reasons, we used equations from [22] including both the singlet, C = e T 1 ­ 3 f ( T / ) ­ e 8 2 and triplet, T =
2 ~ 3 e T 3 F0 ~ 1 ­ t ( T / ; F0 ) ~ 8 1 + F0 2 2 2 2

ln

EF , T

(3)

(4)

3e ­2 2

E 1 1 ­ ln ( 1 + F 0 ) ln F , T F0

density is accompanied by an increase in the positive contribution to the magnetoresistance primarily owing to a decrease in the spin relaxation time. It is worth noting that the theory of quantum cor rections [22] owing to the electron­electron interac tion predicts the nonmonotonic temperature depen dence of the resistance near the parameter value rs ~ 3.5. However, this dependence has a minimum. The appearance of the minimum is due to opposite signs of the contributions to the conductivity from the ballistic (proportional to the temperature) and diffusive (pro portional to the logarithm of the temperature) terms. It is important that the position of the minimum in the temperature depends both on the transport time and on the interaction parameters. For example, the extre ~ mum at F = F 0 = ­0.4 is reached at T/ 0.03 (see
0

terms. Here, EF is the Fermi energy of the two dimen sional system. The functions f and t in Eqs. (3) and (4), respectively, are cumbersome and specified by Eqs. (3.36) and (3.44) in [22], respectively. The Fermi ~ liquid parameters F and F 0 are determined by the
0

dimensionless interaction parameter that is the Cou lomb to kinetic energy ratio rs = 2 e2/( VF), where VF is the Fermi velocity of two dimensional charge carriers and is the dielectric constant of the medium surrounding the two dimensional system (rs was cal culated with the dielectric constant of silicon = 11.5). In the hole density range used in the experi ment, rs is between 2.7 and 3.8. The Fermi liquid ~ parameters F and F 0 are given by the expressions
0

[22] 1 rs ~ F0 = ­ , 2 rs + 2 F0 = ­
2 1 rs arctan r s / 2 ­ 1 , r2 ­ 2 s

(5) rs > 2 .
2

(6)

~ Using the parameters F 0 = ­0.41 and F 0 = ­0.36 cal culated by these formulas with the hole density ps = 1.1 â 1012 cm­2, we obtain good agreement with the experiment (the lower solid line in Fig. 3) without the fitting parameters (the only fit is the vertical displace ment of the curves up to coincidence of absolute values of the conductivity at the minimum). To describe the dependence at ps = 1.5 â 1012 cm­2 (rs = 3.0), we had ~ to use the parameters F = ­0.35 and F 0 = ­0.29 0

Fig. 8 in [22]), i.e., seemingly in the pure diffusive regime. The maxima in the temperature dependence of the conductivity calculated by Eqs. (3) and (4) are at T 1.6 and 23 K for ps = 1.1 â 1012 and 1.5 â 1012 cm­2, respectively. This explains the opposite signs and sig nificantly different magnitudes of the calculated shifts of the minimum of the conductivity owing to correc tions from the interaction for different hole densities in Fig. 3. To conclude, we have shown that the mechanism of the observed crossover in the temperature dependence of the resistivity is the transition from weak localiza tion to antilocalization caused by a change in the rela tion between the spin­orbit and phase relaxation times. Our analysis indicates that there is a region of parameters where antilocalization is manifested in the temperature dependence of the resistivity for two dimensional systems with spin­orbit interaction, in contrast to the commonly accepted opinion that the behavior of such systems is universally insulating, as was predicted by Altshuler and Aronov [2], consider ing the quantum corrections associated with the elec tron­electron interaction. We are grateful to L.E. Golub, G.M. Min'kov, I.L. Aleiner, A.V. Germanenko, P.M. Ostrovskii, V.F. Gantmakher, and V.T. Dolgopolov for stimulating discussions. This work was supported in part by the Division of Physical Sciences, Russian Academy of Sciences (program "Spin Phenomena in Solid Nano structures and Spintronics"). REFERENCES
1. S. Hikami, A. I. Larkin, and Y. Nagaoka, Prog. Theor. Phys. 63, 707 (1980). 2. B. L. Altshuler and A. G. Aronov, Solid State Commun. 46, 429 (1983). 3. B. L. Altshuler and A. G. Aronov, in Electron­Electron Interactions in Disordered Systems, Ed. by A. L. Efros and M. Pollak (North Holland, Amsterdam, 1985). 4. V. F. Gantmakher and V. T. Dolgopolov, Phys. Usp. 51, 3 (2008).

corresponding to rs = 2.0. We did not try to describe the nonmonotonic temperature dependence at ps = 1.9 â 1012 cm­2, because it is impossible to obtain reli able values so and at this density in view of the absence of a pronounced maximum of the anomalous magnetoresistance (see Fig. 4). An increase in the hole
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DOROZHKIN et al. 15. M. I. D'yakonov and V. I. Perel', Sov. Phys. JETP 23, 1053 (1971); Sov. Phys. Solid State 13, 3023 (1971). 16. R. Winkler, Springer Tracts Mod. Phys. 191, 1 (2003). 17. N. S. Averkiev, L. E. Golub, and G. E. Pikus, J. Exp. Theor. Phys. 86, 780 (1998). 18. S. Pedersen, C. B. Sorensen, A. Kristensen, et al., Phys. Rev. B 60, 4880 (1999). 19. S. I. Dorozhkin and E. B. Olshanetsky, JETP Lett. 46, 502 (1987). 20. A. Zduniak, M. I. Dyakonov, and W. Knap, Phys. Rev. B 56, 1996 (1997). 21. S. S. Murzin, S. I. Dorozhkin, G. Landwehr, and A. C. Gossard, JETP Lett. 67, 113 (1998). 22. G. Zala, B. N. Narozhny, and I. L. Aleiner, Phys. Rev. B 64, 214204 (2001). 23. G. M. Minkov, A. V. Germanenko, O. E. Rut, et al., Phys. Rev. B 85, 125303 (2012).

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Translated by R. Tyapaev

JETP LETTERS

Vol. 97

No. 3

2013