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Conductivity of a magnetic-field-induced Luttinger liquid
Physica B 312–313 (2002) 586–588
Conductivity of a magnetic-field-induced Luttinger liquid Shan-Wen Tsaia,*, Dmitrii L. Maslova,1, Leonid I. Glazmanb,2 a
Institute for Fundamental Theory and Department of Physics, University of Florida, P.O. Box 118440, Gainesville, FL 32611-8440, USA b Theoretical Physics Institute, University of Minnesota, Minneapolis, MN, USA
Abstract We study the effect of dilute impurities on a system of interacting electrons subject to a quantizing magnetic field. For the case of point impurities, the calculation of the scattering cross section can be mapped onto a 1D problem of tunneling conductance through a barrier for interacting electrons. We find that the electron–electron interaction produces temperature-dependent corrections to the cross-section, and thus to the tensor of conductivities. Upon summation of the most diverging corrections in all orders of the perturbation theory, a scaling behavior of the conductivities emerges. This behavior is similar to that of 1D Luttinger liquid. The scaling exponents depend on the magnetic field and are calculated explicitly in the weak-coupling limit. The limitations on the scaling behavior due to the formation of a charge-density wave are also discussed. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Non-Fermi liquid; Impurities; High magnetic field
The effective dimensionality of charge carriers in a bulk metal may be reduced from 3D to 1D by applying a strong magnetic field. Such a reduction leads to formation of a strongly correlated state [1]. If the interaction among electrons is long-range, certain characteristics of this state, e.g., tunneling density of states, exhibit a high-energy behavior similar to that of 1D Luttinger liquid [2]. Here we investigate the effect of dilute impurities on such a ‘‘magnetic-field-induced Luttinger liquid’’. We first find the conductivities for free electrons in the ultra-quantum limit (UQL), when only the lowest Landau level is occupied, and then calculate the renormalization corrections due to the electron–electron interaction. The second part is done by an exact mapping onto a pure 1D (no magnetic field) problem of tunneling for interacting electrons, solved by Yue et al. [3]. We find that in a certain temperature
range the dissipative conductivities scale as power-laws of the temperature: se pT 2ae ; where e ¼ 71 is for the electric current parallel (perpendicular) to B: The exponent a is determined by the effective strength of the electron–electron interactions. Our analysis is valid for temperatures Tb1=t; where t is the mean free time for scattering off impurities. In this regime, a typical electron–electron interaction time is shorter than t; and the effect of interaction on the impurity scattering may be considered for a single impurity at a time. First, we discuss scattering of free electrons from a single, axially symmetric impurity. As the angular momentum L is conserved in the scattering process, the scattering cross-section S is completely determined by a set of backscattering amplitudes, frm g; for each component m of L [4]: S ¼ 2pc2B
*Corresponding author. Tel.: +352-392-8755; fax: +352392-0524. E-mail address: [email protected]fl.edu (S.-W. Tsai). 1 Work supported by the NHMFL In-House Research Program, NSF Grant No. DMR-9703388 and Research Corporation (RI0082). 2 Work supported by NSF Grant No. DMR-9731756.
N X
jrm j2 ;
ð1Þ
m¼0
where cB is the magnetic length. Amplitudes rm are simply the reflection coefficients obtained from the . solution of a set of 1D Schrodinger equations. There is one such an equation for each value of m and the corresponding 1D potential is Vm ðzÞ ¼ /mjV ðrÞjmS: A
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 1 5 0 3 - 4
S.-W. Tsai et al. / Physica B 312–313 (2002) 586–588
short-range impurity potential (a5cB ; where a is the range of the potential) can be modeled by an effective delta-like potential. This is an especially simple case as all but the Vm¼0 component vanish. Now we turn to the calculation of the interaction correction to S: The free-electron scattering state Cð0Þ for a point impurity is specified by the reflection and transmission amplitudes r0 and t0 for a 1D d-function potential. The first-order correction to Cð0Þ due to the exchange and Hartree potentials is given by Z Z dCðrÞ ¼ dr0 dr00 Gðr; r0 ; EÞ½Vex ðr0 ; r00 Þ þ dðr0 r00 ÞVH ðr00 Þ Cð0Þ ðr00 Þ:
ð2Þ
The scattering problem is now formally equivalent to the tunneling problem in the UQL [2]. The latter, on its own turn, can be reduced to a pure 1D case, with no B field, solved by Yue et al. [3]. Eq. (2) gives a logarithmically divergent correction to the transmission amplitude dt ¼ at0 jr0 j2 ln jEj=W ; a ¼ ðe2 =pvF ÞðF ðkcB Þ F ð
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 þ 4kF2 cB ÞÞX0;
where E is the energy reckoned from the Fermi energy EF ; vF ðkF Þ is the Fermi velocity (momentum), W BEF is the effective bandwidth, k is the screening R N wave vector, 2 and F ðxÞ ¼ ex =2 E1 ðx2 =2Þ with E1 ðzÞ ¼ z dx e y =y: The physical reason for a singularity at E ¼ 0 is a nearly 1D form of the Friedel oscillation around an impurity in the strong magnetic field. Consequently, both the exchange and Hartree potentials decay as z 1 along the field, hence the log-singularity in dt: The higher-order corrections to the transmission amplitude can be summed up via the renormalization group procedure, derived in Ref. [3]. The flow of t is described by the equation dt=dx ¼ atð1 jtj2 Þ;
ð3Þ
where the ‘‘RG time’’ x lnðW =EÞ: Assuming that a itself does not flow, this equation results in a power-law scaling of t with E: The renormalized cross-section is now written in terms of the renormalized reflection amplitudes. For the current parallel to B; the conductivity is given by sjj ¼ ne2 t=mp1=SðjEj ¼ TÞ; so that sjj ¼ s0jjU þ ðs0jj s0jjU Þ½T=W 2aðBÞ ;
ð4Þ
where s0jj is the conductivity for free electrons and s0jjU is the conductivity for free electrons in the unitary limit, i.e., for an infinitely strong impurity. As T-0; the conductivity approaches the unitary limit but our calculation is expected to be valid only for Tb1=t: For the current perpendicular to B; the conductivity is s> ¼ e2 nF D> pSps 1 jj : When deriving Eq. (4), we assumed that the coupling constant is not renormalized. This assumption breaks down at low enough energies, as the ground state of
587
(repulsively) interacting electrons in the UQL is unstable with respect to the formation of a charge-density wave (CDW) [1]. For Eq. (4) to have a region of validity, there should exist an interval of intermediate energies, in which the renormalization of a due to CDW fluctuations is not yet important but the power-law renormalization of transmission amplitude t is already significant. Such an interval does exist for a long-range Coulomb interaction (jkcB j51). The RG equation for the flow of the coupling constant was derived in Ref. [1]. It follows from this equation that the renormalization of the coupling constant becomes significant at energies of order of the CDW gap D: If the lowest Landau level is not depleted too strongly, i.e., kF cB t1; one can show that xCDW ln W =DBvF =e2 : On the other hand, the power-law for t forms at such energy EPL that xPL ln W =EPL Ba 1 BvF =e2 jln kcB j: We see that in terms of the ‘‘RG time’’ x the power-law renormalization of t occurs earlier than the renormalization of the coupling constant: xCDW =xPL Bjln kcB j\1: Thus, there exists an energy interval ðD5E5W Þ in which the conductivity has a scaling form given by Eq. (4). This conclusion is supported by a numerical solution of the RG equation for the coupling constant [1] for the case of a long-range Coulomb interaction [5]. This solution demonstrates that the ratio xCDW =xPL can be as large as 7 if kcB is sufficiently small (¼ 0:1) and kF cB is close to 1: As the lowest Landau level is depleted further (kF cB decreases), the ratio xCDW =xPL decreases. However, the decrease is rather slow: e.g., for kcB ¼ 0:1; xCDW =xPL becomes equal to one only at kF cB ¼ 0:05: A detailed study of the crossover region between the power-law scaling and the CDW behavior is currently in progress [5]. Experiments on heavily doped semiconductors (InSb and InAs) in the UQL [6] do show rather strong Tdependences of sjj;> : The signs of observed dsjj;> =dT agree with Eq. (4). A more detailed analysis is required, however, to connect the theory presented here to these experimental results because the concentration of charged impurities is very high. In these cases localization has been argued to play an important role (see e.g., [6]). Low carrier density semi-metals such as bismuth or graphite would be better candidates for observing the effect studied here. We are grateful to V.M. Yakovenko for very instructive discussions.
References [1] S.A. Brazovskii, Zh. Eksp. Teor. Fiz. 61 (1971) 2401 (English transl.) Sov. Phys. JETP 34 (1972) 1286; V.M. Yakovenko, Phys. Rev. B 47 (1993) 8851.
588
S.-W. Tsai et al. / Physica B 312–313 (2002) 586–588
[2] C. Biagini, D.L. Maslov, M.Yu. Reizer, L.I. Glazman, Europhys. Lett. 55 (2001) 383. [3] D. Yue, L.I. Glazman, K.A. Matveev, Phys. Rev. B 49 (1994) 1966. [4] D.G. Polyakov, Zh. Eksp. Teor. Fiz. 83 (1982) 61 (English transl.) JETP 56 (1982) 33.
[5] S.-W. Tsai, D.L. Maslov, L.I. Glazman, unpublished. [6] S.S. Murzin, Usp. Fiz. Nauk. 170 (2000) 387 (English transl.) Sov. Phys. Usp. 43 (2000) 349 and references therein.
Conductivity of a magnetic-field-induced Luttinger liquid Shan-Wen Tsaia,*, Dmitrii L. Maslova,1, Leonid I. Glazmanb,2 a
Institute for Fundamental Theory and Department of Physics, University of Florida, P.O. Box 118440, Gainesville, FL 32611-8440, USA b Theoretical Physics Institute, University of Minnesota, Minneapolis, MN, USA
Abstract We study the effect of dilute impurities on a system of interacting electrons subject to a quantizing magnetic field. For the case of point impurities, the calculation of the scattering cross section can be mapped onto a 1D problem of tunneling conductance through a barrier for interacting electrons. We find that the electron–electron interaction produces temperature-dependent corrections to the cross-section, and thus to the tensor of conductivities. Upon summation of the most diverging corrections in all orders of the perturbation theory, a scaling behavior of the conductivities emerges. This behavior is similar to that of 1D Luttinger liquid. The scaling exponents depend on the magnetic field and are calculated explicitly in the weak-coupling limit. The limitations on the scaling behavior due to the formation of a charge-density wave are also discussed. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Non-Fermi liquid; Impurities; High magnetic field
The effective dimensionality of charge carriers in a bulk metal may be reduced from 3D to 1D by applying a strong magnetic field. Such a reduction leads to formation of a strongly correlated state [1]. If the interaction among electrons is long-range, certain characteristics of this state, e.g., tunneling density of states, exhibit a high-energy behavior similar to that of 1D Luttinger liquid [2]. Here we investigate the effect of dilute impurities on such a ‘‘magnetic-field-induced Luttinger liquid’’. We first find the conductivities for free electrons in the ultra-quantum limit (UQL), when only the lowest Landau level is occupied, and then calculate the renormalization corrections due to the electron–electron interaction. The second part is done by an exact mapping onto a pure 1D (no magnetic field) problem of tunneling for interacting electrons, solved by Yue et al. [3]. We find that in a certain temperature
range the dissipative conductivities scale as power-laws of the temperature: se pT 2ae ; where e ¼ 71 is for the electric current parallel (perpendicular) to B: The exponent a is determined by the effective strength of the electron–electron interactions. Our analysis is valid for temperatures Tb1=t; where t is the mean free time for scattering off impurities. In this regime, a typical electron–electron interaction time is shorter than t; and the effect of interaction on the impurity scattering may be considered for a single impurity at a time. First, we discuss scattering of free electrons from a single, axially symmetric impurity. As the angular momentum L is conserved in the scattering process, the scattering cross-section S is completely determined by a set of backscattering amplitudes, frm g; for each component m of L [4]: S ¼ 2pc2B
*Corresponding author. Tel.: +352-392-8755; fax: +352392-0524. E-mail address: [email protected]fl.edu (S.-W. Tsai). 1 Work supported by the NHMFL In-House Research Program, NSF Grant No. DMR-9703388 and Research Corporation (RI0082). 2 Work supported by NSF Grant No. DMR-9731756.
N X
jrm j2 ;
ð1Þ
m¼0
where cB is the magnetic length. Amplitudes rm are simply the reflection coefficients obtained from the . solution of a set of 1D Schrodinger equations. There is one such an equation for each value of m and the corresponding 1D potential is Vm ðzÞ ¼ /mjV ðrÞjmS: A
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 1 5 0 3 - 4
S.-W. Tsai et al. / Physica B 312–313 (2002) 586–588
short-range impurity potential (a5cB ; where a is the range of the potential) can be modeled by an effective delta-like potential. This is an especially simple case as all but the Vm¼0 component vanish. Now we turn to the calculation of the interaction correction to S: The free-electron scattering state Cð0Þ for a point impurity is specified by the reflection and transmission amplitudes r0 and t0 for a 1D d-function potential. The first-order correction to Cð0Þ due to the exchange and Hartree potentials is given by Z Z dCðrÞ ¼ dr0 dr00 Gðr; r0 ; EÞ½Vex ðr0 ; r00 Þ þ dðr0 r00 ÞVH ðr00 Þ Cð0Þ ðr00 Þ:
ð2Þ
The scattering problem is now formally equivalent to the tunneling problem in the UQL [2]. The latter, on its own turn, can be reduced to a pure 1D case, with no B field, solved by Yue et al. [3]. Eq. (2) gives a logarithmically divergent correction to the transmission amplitude dt ¼ at0 jr0 j2 ln jEj=W ; a ¼ ðe2 =pvF ÞðF ðkcB Þ F ð
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 þ 4kF2 cB ÞÞX0;
where E is the energy reckoned from the Fermi energy EF ; vF ðkF Þ is the Fermi velocity (momentum), W BEF is the effective bandwidth, k is the screening R N wave vector, 2 and F ðxÞ ¼ ex =2 E1 ðx2 =2Þ with E1 ðzÞ ¼ z dx e y =y: The physical reason for a singularity at E ¼ 0 is a nearly 1D form of the Friedel oscillation around an impurity in the strong magnetic field. Consequently, both the exchange and Hartree potentials decay as z 1 along the field, hence the log-singularity in dt: The higher-order corrections to the transmission amplitude can be summed up via the renormalization group procedure, derived in Ref. [3]. The flow of t is described by the equation dt=dx ¼ atð1 jtj2 Þ;
ð3Þ
where the ‘‘RG time’’ x lnðW =EÞ: Assuming that a itself does not flow, this equation results in a power-law scaling of t with E: The renormalized cross-section is now written in terms of the renormalized reflection amplitudes. For the current parallel to B; the conductivity is given by sjj ¼ ne2 t=mp1=SðjEj ¼ TÞ; so that sjj ¼ s0jjU þ ðs0jj s0jjU Þ½T=W 2aðBÞ ;
ð4Þ
where s0jj is the conductivity for free electrons and s0jjU is the conductivity for free electrons in the unitary limit, i.e., for an infinitely strong impurity. As T-0; the conductivity approaches the unitary limit but our calculation is expected to be valid only for Tb1=t: For the current perpendicular to B; the conductivity is s> ¼ e2 nF D> pSps 1 jj : When deriving Eq. (4), we assumed that the coupling constant is not renormalized. This assumption breaks down at low enough energies, as the ground state of
587
(repulsively) interacting electrons in the UQL is unstable with respect to the formation of a charge-density wave (CDW) [1]. For Eq. (4) to have a region of validity, there should exist an interval of intermediate energies, in which the renormalization of a due to CDW fluctuations is not yet important but the power-law renormalization of transmission amplitude t is already significant. Such an interval does exist for a long-range Coulomb interaction (jkcB j51). The RG equation for the flow of the coupling constant was derived in Ref. [1]. It follows from this equation that the renormalization of the coupling constant becomes significant at energies of order of the CDW gap D: If the lowest Landau level is not depleted too strongly, i.e., kF cB t1; one can show that xCDW ln W =DBvF =e2 : On the other hand, the power-law for t forms at such energy EPL that xPL ln W =EPL Ba 1 BvF =e2 jln kcB j: We see that in terms of the ‘‘RG time’’ x the power-law renormalization of t occurs earlier than the renormalization of the coupling constant: xCDW =xPL Bjln kcB j\1: Thus, there exists an energy interval ðD5E5W Þ in which the conductivity has a scaling form given by Eq. (4). This conclusion is supported by a numerical solution of the RG equation for the coupling constant [1] for the case of a long-range Coulomb interaction [5]. This solution demonstrates that the ratio xCDW =xPL can be as large as 7 if kcB is sufficiently small (¼ 0:1) and kF cB is close to 1: As the lowest Landau level is depleted further (kF cB decreases), the ratio xCDW =xPL decreases. However, the decrease is rather slow: e.g., for kcB ¼ 0:1; xCDW =xPL becomes equal to one only at kF cB ¼ 0:05: A detailed study of the crossover region between the power-law scaling and the CDW behavior is currently in progress [5]. Experiments on heavily doped semiconductors (InSb and InAs) in the UQL [6] do show rather strong Tdependences of sjj;> : The signs of observed dsjj;> =dT agree with Eq. (4). A more detailed analysis is required, however, to connect the theory presented here to these experimental results because the concentration of charged impurities is very high. In these cases localization has been argued to play an important role (see e.g., [6]). Low carrier density semi-metals such as bismuth or graphite would be better candidates for observing the effect studied here. We are grateful to V.M. Yakovenko for very instructive discussions.
References [1] S.A. Brazovskii, Zh. Eksp. Teor. Fiz. 61 (1971) 2401 (English transl.) Sov. Phys. JETP 34 (1972) 1286; V.M. Yakovenko, Phys. Rev. B 47 (1993) 8851.
588
S.-W. Tsai et al. / Physica B 312–313 (2002) 586–588
[2] C. Biagini, D.L. Maslov, M.Yu. Reizer, L.I. Glazman, Europhys. Lett. 55 (2001) 383. [3] D. Yue, L.I. Glazman, K.A. Matveev, Phys. Rev. B 49 (1994) 1966. [4] D.G. Polyakov, Zh. Eksp. Teor. Fiz. 83 (1982) 61 (English transl.) JETP 56 (1982) 33.
[5] S.-W. Tsai, D.L. Maslov, L.I. Glazman, unpublished. [6] S.S. Murzin, Usp. Fiz. Nauk. 170 (2000) 387 (English transl.) Sov. Phys. Usp. 43 (2000) 349 and references therein.