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GaN heterostructures
Physics Letters A 366 (2007) 267–270 www.elsevier.com/locate/pla
Observation of the transition from diffusive regime to ballistic regime of the 2DEG transport property in Alx Ga1−x N/GaN heterostructures K. Han a , B. Shen a,∗ , N. Tang a , Y.Q. Tang a , X.W. He a , Z.X. Qin a , Z.J. Yang a , G.Y. Zhang a , T. Lin b , B. Zhu b , W.Z. Zhou b , J.H. Chu b a State Key Laboratory of Artificial Microstructure and Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, China b National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, China
Received 17 January 2007; accepted 8 February 2007 Available online 13 February 2007 Communicated by V.M. Agranovich
Abstract Electron–electron interaction effect of the two-dimensional electron gas (2DEG) in Alx Ga1−x N/GaN heterostructures has been investigated by means of magnetotransport measurements at low temperatures. From the temperature dependence of the longitudinal conductivity of the heterostructures, a clear transition region has been observed. Based on the theoretical analysis, we conclude that this region corresponds to the transition from the diffusive regime to the ballistic regime of the 2DEG transport property. The interaction constant is determined to be −0.423, which is consistent with the theoretical prediction. However, the critical temperature for the transition, which is 8 K in Alx Ga1−x N/GaN heterostructures, is much higher than the theoretical prediction. © 2007 Elsevier B.V. All rights reserved. Keywords: Alx Ga1−x N/GaN heterostructure; 2DEG; Magnetoresistivity; Electron–electron interaction
1. Introduction Due to the large conduction band offset and the strong polarization-induced electrical field at the heterointerface [1,2], an Alx Ga1−x N/GaN heterostructure has a deep quantum well and a high density of the two-dimensional electron (2DEG). The 2DEG sheet density at the heterointerface is as high as 1013 cm−2 , which is much higher than that in Alx Ga1−x As/GaAs heterostructures. The superposition of the wave functions becomes stronger, and many quantum effects arise. Magnetotransport measurement provides a good way to study the quantum effects related to the 2DEG in the heterostructures [3–7]. Based on the semi-classical Drude model, the longitudinal resistivity has no magnetic field dependence while the Hall resistivity increases linearly with increasing magnetic field in the * Corresponding author.
E-mail address: [email protected] (B. Shen). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.02.032
magnetotransport measurements of semiconductor heterostructures. However, many quantum effects have been observed in semiconductor heterostructures by magnetotransport measurements, and the magnetic field dependence of the magnetoresistivity departures from the Drude model. One of these quantum effects in semiconductor heterostructures is the negative magnetoresistivity (NMR) in the magnetic fields. Two main mechanisms have been developed to explain the NMR. One is the weak localization (WL) effect [8,9], which is resulted from the interference of electron waves propagating in opposite directions along closed paths. The other is the electron–electron interaction (EEI) [4,5,10]. The WL effect can be suppressed by magnetic field, when it is higher than the critical field Btr , 2 which is given by Btr = h/(2el ¯ e ), where le is the elastic scattering length. Meanwhile, the EEI effect can be present in stronger magnetic field than the WL effect. The EEI effect manifests itself quite differently in the temperature dependence in two regimes, which concerns the quasiparticle interaction time h¯ /kB T and the momentum relaxation time τ . The two regimes are the diffusive regime (kB T τ/h¯ 1) and the ballistic regime
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K. Han et al. / Physics Letters A 366 (2007) 267–270
(kB T τ/h¯ 1). In the diffusive regime, the EEI correction of the longitudinal conductivity is logarithmic temperature dependence [11], while in the ballistic regime, the EEI correction of the longitudinal conductivity is linear temperature dependence. In this study, we have investigated the EEI correction of the 2DEG in Alx Ga1−x N/GaN heterostructures by means of magnetotransport measurements at low temperatures. A clear transition from the diffusive regime to the ballistic regime of the 2DEG transport property is found. The experimental results are compared with the theory predictions.
Fig. 1 shows the magnetoresistivity of an Al0.11 Ga0.89 N/ GaN heterostructure as a function of magnetic field normal
to the interface at 1.5 K, 6 K, and 10 K, respectively. The NMR at the magnetic fields lower than 5 T is observed clearly [4]. The strong Shubnikov–de Haas (SdH) oscillations at high magnetic fields are observed [3]. The oscillations become weaker with increasing temperature. From the analysis of the SdH oscillations, the sheet density and the mobility of the 2DEG in the heterostructures are determined to be 5.4 × 1012 cm−2 and 13 500 cm2 /Vs, respectively, at 1.5 K. They are almost constant in the measured temperature range. The temperature dependence of the longitudinal conductivity σxx of the heterostructure is plotted in Fig. 2. When the magnetic field is lower than Btr , the WL effect and the EEI effect are co-existing, and they all contribute to the conductivity. However, the two effects show different magnetic field dependence. In order to study the EEI correction, the magnetic field is applied to exclude the weak-localization correction. The critical magnetic field Btr to suppress the weak-localization effect is just several Gauss in our heterostructures. Fig. 2 shows the temperature dependence of the conductivity at 0.01 T, which is much stronger than the critical field. At 0.01 T, the WL correction vanishes, and thus the temperature dependence is mainly attributed to the EEI correction. The curve shows a clear transition to the linear behavior. When the temperature is higher than 8 K, a linear region appears. Meanwhile an obvious divergence from the linear dependence appears when the temperature is below 8 K. Recently, Zala et al. developed a systematic theory about the EEI correction of the conductivity that bridged the gap between the two theories in the diffusive and ballistic regimes known before (the ZNA theory) [13]. In this theory, the elastic (coherent) electron scattering from the modulated density of other electrons (Friedel oscillation) is caused by an impurity with short range potential. Zala et al. proved that the diffusive regime and the ballistic regime were due to the same physical processcoherent scattering by the Friedel oscillation, and presented the cross-over function between the diffusive and the ballistic limits. The EEI correction to the conductivity in the ZNA theory is given as follow:
Fig. 1. Magnetoresistivity of an Al0.11 Ga0.89 N/GaN heterostructure as a function of magnetic field normal to the heterointerface at 1.5 K, 6 K, and 10 K, respectively.
Fig. 2. Temperature dependence of the longitudinal conductivity of an Al0.11 Ga0.89 N/GaN heterostructure measured at 0.01 T.
2. Experiment Al0.11 Ga0.89 N/GaN heterostructures used in the study were grown by means of metal organic chemical vapor deposition (MOCVD) on the (0001) surface of sapphire substrates. A nucleation GaN buffer layer was grown at 488 ◦ C, followed by a 2.0-µm-thick unintentionally doped GaN (iGaN) layer deposited at 1071 ◦ C. Then, an unintentionally doped Al0.11 Ga0.89 N/GaN (i-AlGaN) layer with a thickness of 30 nm was grown at 1080 ◦ C. High-resolution Xray diffraction reciprocal space mapping indicates that the Al0.11 Ga0.89 N layer pseudomorphically has been grown on GaN [12]. For the magnetotransport measurements, Hall-bar structures with two current contacts and six potential probes were made by the standard photolithography technique. Low-damage reactive ion beam etching was used to form the Hall-bar structures. Ohmic contacts were made by evaporating Ti/Al/Ni/Au metal multilayer structure in an electron–beam evaporate system, followed by the rapid annealing at 900 ◦ C for 10 seconds in N2 ambient. We have performed the magnetotransport measurements at variable temperatures from 1.5 K to 40 K with the magnetic field changing from 0 to 13 T. 3. Experimental results and discussions
K. Han et al. / Physics Letters A 366 (2007) 267–270
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EF e 2 kB T τ e2 3 ln δσc = . (1) 1 − f (kB T τ/h¯ ) − π h¯ h¯ 8 kB T 2π 2 h¯ Eq. (1) is the charge channel correction, where kB is the Boltzmann constant, T the temperature, τ the momentum relaxation time and EF the Fermi energy. Meanwhile, the EEI correction in the triplet channel δσT is: 3F0σ e2 kB T τ 3 σ δσT = 1 − t kB T τ/h¯ ; F0 [1 + F0σ ] π h¯ h¯ 8 2 σ ln(1 + F0 ) e EF ln , −3 1− (2) F0σ kB T 2π 2 h¯ where F0σ is the interaction constant in the triplet channel. The detailed expressions of f (x) and t (x; F0σ ) can be found in Ref. [13], and fortunately they can be neglected for all the practical applications. In these expressions, the linear temperature dependence term is due to the renormalization of τ (T ) by Friedel oscillation. This contribution dominates in the ballistic limit kB T τ/h¯ 1. In the diffusive limit kB T τ/h¯ 1, the conductivity correction is determined by the logarithmic term. A transition between the two regimes occurs in the intermediate region. To our knowledge, no experiments about the transition between the ballistic and the diffusive regimes have been reported in Alx Ga1−x N/GaN heterostructures up to now. One of the reasons is that the temperature at which the transition expected to occur is given by kB T τ/h¯ ≈ 0.1 [14,15]. Experimentally, the transition temperature is too low to reach. In our sample, the value of the parameter kB T τ/h¯ varies from 0.35 to 9.26 in the measured temperature range, which is from 1.5 K to 40 K. According to the theory, the experiment should be carried out in the intermediate and the ballistic regimes when the value of the parameter kB T τ/h¯ is between 0.1 and 10. Our experimental conditions are just in this region. There should be no transition occurring in our experiments. However we find a transition in the temperature dependence of the longitudinal conductivity of the heterostructure, as shown in Fig. 2. At low temperatures, we believe that the divergence from the linear dependence is due to the usual logarithmic correction in the diffusive regime, and the linear relation identifies the transition from the diffusive regime to the ballistic regime. But the region where the crossover occurs is different from the theoretical prediction kB T τ/h¯ ≈ 0.1. In our heterostructures, the transition occurs at about 8 K in temperature, where the parameter kB T τ/h¯ is about 1.8. The value of the parameter is much larger than the theoretical prediction. According to the ZNA theory, in the ballistic regime, the conductivity correction of the EEI has a linear dependence on temperature as follows: 3F0σ e2 kB T τ . δσ ee (T ) = (3) 1+ π h¯ h¯ 1 + F0σ The constant F0σ is the only alterable parameter in the theory, which determines the results, but the exact value of F0σ cannot be calculated theoretically. Thus, we use the interaction constant F0σ as a fitting parameter in the linear regime according to Eq. (3). As a result, we determine F0σ to be −0.423.
Fig. 3. Calculation of the temperature dependence of the longitudinal conductivity of an Al0.11 Ga0.89 N/GaN heterostructure under the condition F0σ = −0.423 based on the ZNA theory.
In the ZNA theory, the relationship between the conductivity correction δσ ee and the parameter kBh¯T τ is non-monotonous in a narrow region −0.45 < F0σ < −0.25. We plot the theoretic curve σ ee (T ) when F0σ is −0.423 in Fig. 3. Compared with the experimental results, the theoretical and the experimental curves of σ ee (T ) have the same nonmonotonous behavior, but the temperatures of the transition are different from each other. The theoretical temperature is much lower than the experimental result, which is lower than 0.1 K. The temperature scale of the theoretical prediction is also less than the experimental result for the intermediate region of the transition. This difference shows that it is much easier for us to observe the transition between two regimes in Alx Ga1−x N/GaN heterostructures used in this study. The influence of magnetic field on the conductivity correction in the ballistic regime has also been studied. A lot of works have been done on the conductivity correction due to the EEI in the diffusive regime both theoretically and experimentally. They pointed out that this correction had no magnetic field dependence. In contrast to the diffusive regime, the behavior of the conductivity correction is not well understood in the ballistic regime. The theory of the interaction correction between temperature and conductivity correction leaves MR in a perpendicular magnetic field out of consideration [13]. It is unclear whether the correction to the conductivity at zero magnetic field can hold in finite magnetic field. We find that the EEI correction in the ballistic regime can hold the linear relation with temperature in finite perpendicular magnetic field in our heterostructures. But the slope of the linear temperature dependence of the conductivity alters, even reverses its sign as raising the magnetic field, as shown in Figs. 4(a) and 4(b). The changing shows that the EEI correction has magnetic field dependence in the ballistic regime. We cannot propose proper explanation on this phenomenon, and more detailed research is required. 4. Conclusions We have studied the magnetotransport properties of the 2DEG with high mobility in Al0.11 Ga0.89 N/GaN heterostructures. The clear transition region has been observed from the
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K. Han et al. / Physics Letters A 366 (2007) 267–270
Alx Ga1−x N/GaN heterostructures, is much higher than the theoretical prediction. Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 60325413), National Basic Research Program of China (No. 2006CB604908), the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (No. 705002), the Research Fund for the Doctoral Program of Higher Education in China (20060001018), and Beijing Natural Science Foundation (No. 4062017). References
Fig. 4. (a) The longitudinal conductivity of an Al0.11 Ga0.89 N/GaN heterostructure in the ballistic regime as a function of temperature at various magnetic field of 0, 0.5 T, 1 T, and 2 T, respectively. (b) The slope of the temperature dependence of the longitudinal conductivity in an Al0.11 Ga0.89 N/GaN heterostructure in the ballistic regime as a function of magnetic field.
temperature dependence of the longitudinal conductivity of the heterostructures. Based on the theoretical analysis, we conclude that this region corresponds to the transition from the diffusive regime to the ballistic regime of the 2DEG transport property. The interaction constant is determined to be −0.423, which is consistent with the theoretical prediction. However, the critical temperature for the transition, which is 8 K in
[1] M.A. Khan, Q. Chen, M.S. Shur, B.T. Mermott, J.A. Higgins, J. Burm, W. Schaff, L.F. Eastman, Electron. Lett. 32 (1996) 357. [2] Y.-F. Wu, B.P. Keller, S. Keller, D. Kapolnek, P. Kozodoy, S.P. Denbaars, U.K. Mishra, Appl. Phys. Lett. 69 (1996) 1438. [3] Z.W. Zheng, B. Shen, R. Zhang, Y.S. Gui, C.P. Jiang, Z.X. Ma, G.Z. Sheng, S.L. Guo, Y.D. Zheng, T. Someya, Y. Arakawa, Phys. Rev. B 62 (2000) R7739. [4] M.A. Paalanen, D.C. Tsui, J.C.M. Hwang, Phys. Rev. Lett. 51 (1983) 2226. [5] K.K. Choi, D.C. Tsui, S.C. Palmateer, Phys. Rev. B 33 (1986) 8216. [6] G.M. Minkov, O.E. Rut, A.V. Germamenko, A.A. Sherstobitov, V.I. Shashkin, O.I. Khrykin, V.M. Daniltsev, Phys. Rev. B 64 (2001) 235327. [7] V.T. Renard, I.V. Gornyi, O.A. Tkachenko, V.A. Tkachenko, Z.D. Kvon, E.B. Olshanetsky, A.I. Toropov, J.-C. Portal, Phys. Rev. B 72 (2005) 075313. [8] A.F. Brana, C. Diaz-Paniagua, F. Batallan, J. Appl. Phys. 88 (2000) 932. [9] A.V. Buyanov, J.P. Bergman, J.A. Sandberg, B.E. Sernelius, P.O. Holtz, B. Monemar, Phys. Rev. B 58 (1998) 1442. [10] B.L. Altshuler, A.G. Aronov, in: A.L. Efros, M. Pollak (Eds.), Electron– Electron Interactions in Disordered Systems, North-Holland, Amsterdam, 1985. [11] A. Gold, V.T. Dolgopolov, Phys. Rev. B 33 (1986) 1076. [12] N. Tang, B. Shen, M.J. Wang, K. Han, Z.J. Yang, K. Xu, G.Y. Zhang, T. Lin, B. Zhu, W.Z. Zhou, J.H. Chu, Appl. Phys. Lett. 88 (2006) 172112. [13] G. Zala, B.N. Narozhny, I.L. Aleiner, Phys. Rev. B 64 (2001) 214204. [14] G. Zala, B.N. Narozhny, I.L. Aleiner, Phys. Rev. B 64 (2001) 201201(R). [15] I.V. Gornyi, A.D. Mirlin, Phys. Rev. Lett. 90 (2003) 076801.
Observation of the transition from diffusive regime to ballistic regime of the 2DEG transport property in Alx Ga1−x N/GaN heterostructures K. Han a , B. Shen a,∗ , N. Tang a , Y.Q. Tang a , X.W. He a , Z.X. Qin a , Z.J. Yang a , G.Y. Zhang a , T. Lin b , B. Zhu b , W.Z. Zhou b , J.H. Chu b a State Key Laboratory of Artificial Microstructure and Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, China b National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, China
Received 17 January 2007; accepted 8 February 2007 Available online 13 February 2007 Communicated by V.M. Agranovich
Abstract Electron–electron interaction effect of the two-dimensional electron gas (2DEG) in Alx Ga1−x N/GaN heterostructures has been investigated by means of magnetotransport measurements at low temperatures. From the temperature dependence of the longitudinal conductivity of the heterostructures, a clear transition region has been observed. Based on the theoretical analysis, we conclude that this region corresponds to the transition from the diffusive regime to the ballistic regime of the 2DEG transport property. The interaction constant is determined to be −0.423, which is consistent with the theoretical prediction. However, the critical temperature for the transition, which is 8 K in Alx Ga1−x N/GaN heterostructures, is much higher than the theoretical prediction. © 2007 Elsevier B.V. All rights reserved. Keywords: Alx Ga1−x N/GaN heterostructure; 2DEG; Magnetoresistivity; Electron–electron interaction
1. Introduction Due to the large conduction band offset and the strong polarization-induced electrical field at the heterointerface [1,2], an Alx Ga1−x N/GaN heterostructure has a deep quantum well and a high density of the two-dimensional electron (2DEG). The 2DEG sheet density at the heterointerface is as high as 1013 cm−2 , which is much higher than that in Alx Ga1−x As/GaAs heterostructures. The superposition of the wave functions becomes stronger, and many quantum effects arise. Magnetotransport measurement provides a good way to study the quantum effects related to the 2DEG in the heterostructures [3–7]. Based on the semi-classical Drude model, the longitudinal resistivity has no magnetic field dependence while the Hall resistivity increases linearly with increasing magnetic field in the * Corresponding author.
E-mail address: [email protected] (B. Shen). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.02.032
magnetotransport measurements of semiconductor heterostructures. However, many quantum effects have been observed in semiconductor heterostructures by magnetotransport measurements, and the magnetic field dependence of the magnetoresistivity departures from the Drude model. One of these quantum effects in semiconductor heterostructures is the negative magnetoresistivity (NMR) in the magnetic fields. Two main mechanisms have been developed to explain the NMR. One is the weak localization (WL) effect [8,9], which is resulted from the interference of electron waves propagating in opposite directions along closed paths. The other is the electron–electron interaction (EEI) [4,5,10]. The WL effect can be suppressed by magnetic field, when it is higher than the critical field Btr , 2 which is given by Btr = h/(2el ¯ e ), where le is the elastic scattering length. Meanwhile, the EEI effect can be present in stronger magnetic field than the WL effect. The EEI effect manifests itself quite differently in the temperature dependence in two regimes, which concerns the quasiparticle interaction time h¯ /kB T and the momentum relaxation time τ . The two regimes are the diffusive regime (kB T τ/h¯ 1) and the ballistic regime
268
K. Han et al. / Physics Letters A 366 (2007) 267–270
(kB T τ/h¯ 1). In the diffusive regime, the EEI correction of the longitudinal conductivity is logarithmic temperature dependence [11], while in the ballistic regime, the EEI correction of the longitudinal conductivity is linear temperature dependence. In this study, we have investigated the EEI correction of the 2DEG in Alx Ga1−x N/GaN heterostructures by means of magnetotransport measurements at low temperatures. A clear transition from the diffusive regime to the ballistic regime of the 2DEG transport property is found. The experimental results are compared with the theory predictions.
Fig. 1 shows the magnetoresistivity of an Al0.11 Ga0.89 N/ GaN heterostructure as a function of magnetic field normal
to the interface at 1.5 K, 6 K, and 10 K, respectively. The NMR at the magnetic fields lower than 5 T is observed clearly [4]. The strong Shubnikov–de Haas (SdH) oscillations at high magnetic fields are observed [3]. The oscillations become weaker with increasing temperature. From the analysis of the SdH oscillations, the sheet density and the mobility of the 2DEG in the heterostructures are determined to be 5.4 × 1012 cm−2 and 13 500 cm2 /Vs, respectively, at 1.5 K. They are almost constant in the measured temperature range. The temperature dependence of the longitudinal conductivity σxx of the heterostructure is plotted in Fig. 2. When the magnetic field is lower than Btr , the WL effect and the EEI effect are co-existing, and they all contribute to the conductivity. However, the two effects show different magnetic field dependence. In order to study the EEI correction, the magnetic field is applied to exclude the weak-localization correction. The critical magnetic field Btr to suppress the weak-localization effect is just several Gauss in our heterostructures. Fig. 2 shows the temperature dependence of the conductivity at 0.01 T, which is much stronger than the critical field. At 0.01 T, the WL correction vanishes, and thus the temperature dependence is mainly attributed to the EEI correction. The curve shows a clear transition to the linear behavior. When the temperature is higher than 8 K, a linear region appears. Meanwhile an obvious divergence from the linear dependence appears when the temperature is below 8 K. Recently, Zala et al. developed a systematic theory about the EEI correction of the conductivity that bridged the gap between the two theories in the diffusive and ballistic regimes known before (the ZNA theory) [13]. In this theory, the elastic (coherent) electron scattering from the modulated density of other electrons (Friedel oscillation) is caused by an impurity with short range potential. Zala et al. proved that the diffusive regime and the ballistic regime were due to the same physical processcoherent scattering by the Friedel oscillation, and presented the cross-over function between the diffusive and the ballistic limits. The EEI correction to the conductivity in the ZNA theory is given as follow:
Fig. 1. Magnetoresistivity of an Al0.11 Ga0.89 N/GaN heterostructure as a function of magnetic field normal to the heterointerface at 1.5 K, 6 K, and 10 K, respectively.
Fig. 2. Temperature dependence of the longitudinal conductivity of an Al0.11 Ga0.89 N/GaN heterostructure measured at 0.01 T.
2. Experiment Al0.11 Ga0.89 N/GaN heterostructures used in the study were grown by means of metal organic chemical vapor deposition (MOCVD) on the (0001) surface of sapphire substrates. A nucleation GaN buffer layer was grown at 488 ◦ C, followed by a 2.0-µm-thick unintentionally doped GaN (iGaN) layer deposited at 1071 ◦ C. Then, an unintentionally doped Al0.11 Ga0.89 N/GaN (i-AlGaN) layer with a thickness of 30 nm was grown at 1080 ◦ C. High-resolution Xray diffraction reciprocal space mapping indicates that the Al0.11 Ga0.89 N layer pseudomorphically has been grown on GaN [12]. For the magnetotransport measurements, Hall-bar structures with two current contacts and six potential probes were made by the standard photolithography technique. Low-damage reactive ion beam etching was used to form the Hall-bar structures. Ohmic contacts were made by evaporating Ti/Al/Ni/Au metal multilayer structure in an electron–beam evaporate system, followed by the rapid annealing at 900 ◦ C for 10 seconds in N2 ambient. We have performed the magnetotransport measurements at variable temperatures from 1.5 K to 40 K with the magnetic field changing from 0 to 13 T. 3. Experimental results and discussions
K. Han et al. / Physics Letters A 366 (2007) 267–270
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EF e 2 kB T τ e2 3 ln δσc = . (1) 1 − f (kB T τ/h¯ ) − π h¯ h¯ 8 kB T 2π 2 h¯ Eq. (1) is the charge channel correction, where kB is the Boltzmann constant, T the temperature, τ the momentum relaxation time and EF the Fermi energy. Meanwhile, the EEI correction in the triplet channel δσT is: 3F0σ e2 kB T τ 3 σ δσT = 1 − t kB T τ/h¯ ; F0 [1 + F0σ ] π h¯ h¯ 8 2 σ ln(1 + F0 ) e EF ln , −3 1− (2) F0σ kB T 2π 2 h¯ where F0σ is the interaction constant in the triplet channel. The detailed expressions of f (x) and t (x; F0σ ) can be found in Ref. [13], and fortunately they can be neglected for all the practical applications. In these expressions, the linear temperature dependence term is due to the renormalization of τ (T ) by Friedel oscillation. This contribution dominates in the ballistic limit kB T τ/h¯ 1. In the diffusive limit kB T τ/h¯ 1, the conductivity correction is determined by the logarithmic term. A transition between the two regimes occurs in the intermediate region. To our knowledge, no experiments about the transition between the ballistic and the diffusive regimes have been reported in Alx Ga1−x N/GaN heterostructures up to now. One of the reasons is that the temperature at which the transition expected to occur is given by kB T τ/h¯ ≈ 0.1 [14,15]. Experimentally, the transition temperature is too low to reach. In our sample, the value of the parameter kB T τ/h¯ varies from 0.35 to 9.26 in the measured temperature range, which is from 1.5 K to 40 K. According to the theory, the experiment should be carried out in the intermediate and the ballistic regimes when the value of the parameter kB T τ/h¯ is between 0.1 and 10. Our experimental conditions are just in this region. There should be no transition occurring in our experiments. However we find a transition in the temperature dependence of the longitudinal conductivity of the heterostructure, as shown in Fig. 2. At low temperatures, we believe that the divergence from the linear dependence is due to the usual logarithmic correction in the diffusive regime, and the linear relation identifies the transition from the diffusive regime to the ballistic regime. But the region where the crossover occurs is different from the theoretical prediction kB T τ/h¯ ≈ 0.1. In our heterostructures, the transition occurs at about 8 K in temperature, where the parameter kB T τ/h¯ is about 1.8. The value of the parameter is much larger than the theoretical prediction. According to the ZNA theory, in the ballistic regime, the conductivity correction of the EEI has a linear dependence on temperature as follows: 3F0σ e2 kB T τ . δσ ee (T ) = (3) 1+ π h¯ h¯ 1 + F0σ The constant F0σ is the only alterable parameter in the theory, which determines the results, but the exact value of F0σ cannot be calculated theoretically. Thus, we use the interaction constant F0σ as a fitting parameter in the linear regime according to Eq. (3). As a result, we determine F0σ to be −0.423.
Fig. 3. Calculation of the temperature dependence of the longitudinal conductivity of an Al0.11 Ga0.89 N/GaN heterostructure under the condition F0σ = −0.423 based on the ZNA theory.
In the ZNA theory, the relationship between the conductivity correction δσ ee and the parameter kBh¯T τ is non-monotonous in a narrow region −0.45 < F0σ < −0.25. We plot the theoretic curve σ ee (T ) when F0σ is −0.423 in Fig. 3. Compared with the experimental results, the theoretical and the experimental curves of σ ee (T ) have the same nonmonotonous behavior, but the temperatures of the transition are different from each other. The theoretical temperature is much lower than the experimental result, which is lower than 0.1 K. The temperature scale of the theoretical prediction is also less than the experimental result for the intermediate region of the transition. This difference shows that it is much easier for us to observe the transition between two regimes in Alx Ga1−x N/GaN heterostructures used in this study. The influence of magnetic field on the conductivity correction in the ballistic regime has also been studied. A lot of works have been done on the conductivity correction due to the EEI in the diffusive regime both theoretically and experimentally. They pointed out that this correction had no magnetic field dependence. In contrast to the diffusive regime, the behavior of the conductivity correction is not well understood in the ballistic regime. The theory of the interaction correction between temperature and conductivity correction leaves MR in a perpendicular magnetic field out of consideration [13]. It is unclear whether the correction to the conductivity at zero magnetic field can hold in finite magnetic field. We find that the EEI correction in the ballistic regime can hold the linear relation with temperature in finite perpendicular magnetic field in our heterostructures. But the slope of the linear temperature dependence of the conductivity alters, even reverses its sign as raising the magnetic field, as shown in Figs. 4(a) and 4(b). The changing shows that the EEI correction has magnetic field dependence in the ballistic regime. We cannot propose proper explanation on this phenomenon, and more detailed research is required. 4. Conclusions We have studied the magnetotransport properties of the 2DEG with high mobility in Al0.11 Ga0.89 N/GaN heterostructures. The clear transition region has been observed from the
270
K. Han et al. / Physics Letters A 366 (2007) 267–270
Alx Ga1−x N/GaN heterostructures, is much higher than the theoretical prediction. Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 60325413), National Basic Research Program of China (No. 2006CB604908), the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (No. 705002), the Research Fund for the Doctoral Program of Higher Education in China (20060001018), and Beijing Natural Science Foundation (No. 4062017). References
Fig. 4. (a) The longitudinal conductivity of an Al0.11 Ga0.89 N/GaN heterostructure in the ballistic regime as a function of temperature at various magnetic field of 0, 0.5 T, 1 T, and 2 T, respectively. (b) The slope of the temperature dependence of the longitudinal conductivity in an Al0.11 Ga0.89 N/GaN heterostructure in the ballistic regime as a function of magnetic field.
temperature dependence of the longitudinal conductivity of the heterostructures. Based on the theoretical analysis, we conclude that this region corresponds to the transition from the diffusive regime to the ballistic regime of the 2DEG transport property. The interaction constant is determined to be −0.423, which is consistent with the theoretical prediction. However, the critical temperature for the transition, which is 8 K in
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