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arXiv:cond-mat/9907072

Umklapp scattering in transport through a 1D wire of finite length

Vadim V. Ponomarenko

Published Jul 5, 1999• 0 citations
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Abstract

Suppression of electron current $ ΔI$ through a 1D channel of length $L$ connecting two Fermi liquid reservoirs is studied taking into account the Umklapp interaction induced by a periodic potential. This interaction opens band gaps at the integer fillings and Hubbard gaps $2m$ at some rational fillings in the infinite wire: $L \to \infty$. In the perturbative regime where $m \ll v_c/L$ ($v_c:$ charge velocity), and for small deviations $δn$ of the electron density from its commensurate values $- ΔI/V$ can diverge with some exponent as voltage or temperature $V,T$ decreases above $E_c=max(v_c/L,v_c δn)$, while it goes to zero below $E_c$. This results in a non-monotonous behavior of the conductance. In the case when the Umklapp interaction creates a large Mott-Hubbard gap $2m \gg T_L $ inside the wire, the transport is suppressed near half-filling everywhere inside the gap except for an exponentially small region of $V,T < T_L exp(-2m/T_L)$.

DOI
10.48550/arxiv.cond-mat/9907072

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Europhysics Letters (EPL)•Jan 1, 2014•2 citations
Suppression of electron current ΔI ΔIΔI through a 1D channel of length LLL connecting two Fermi liquid reservoirs is studied taking into account the Umklapp interaction induced by a periodic potential. This interaction opens band gaps at the integer fillings and Hubbard gaps 2m2m2m at some rational fillings in the infinite wire: L→∞L \to \inftyL→∞. In the perturbative regime where m≪vc/Lm \ll v_c/Lm≪vc​/L (vc:v_c:vc​: charge velocity), and for small deviations δnδnδn of the electron density from its commensurate values −ΔI/V- ΔI/V−ΔI/V can diverge with some exponent as voltage or temperature V,TV,TV,T decreases above Ec=max(vc/L,vcδn)E_c=max(v_c/L,v_c δn)Ec​=max(vc​/L,vc​δn), while it goes to zero below EcE_cEc​. This results in a non-monotonous behavior of the conductance. In the case when the Umklapp interaction creates a large Mott-Hubbard gap 2m≫TL2m \gg T_L 2m≫TL​ inside the wire, the transport is suppressed near half-filling everywhere inside the gap except for an exponentially small region of V,T<TLexp(−2m/TL)V,T < T_L exp(-2m/T_L)V,T<TL​exp(−2m/TL​).
Transport through a 1D Mott-Hubbard insulator of a finite length LLL is studied beyond the perturbative approach. At a special value of the low-energy constant of the interaction we have mapped the problem onto exactly solvable models and found current vs voltage at a high temperature T>max(m,TL)T>\mathrm{max}({m,T}_{L})T>max(m,TL​) and at a low energy T,V<TLT,V<{T}_{L}T,V<TL​ ( TL=vc/L{T}_{L}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}v}_{c}/LTL​=vc​/L; vc{v}_{c}vc​: charge velocity). The result shows that for the strong interaction creating a large Mott-Hubbard gap $2m\ensuremath{\gg}{T}_{L}$ inside the wire, the transport is suppressed near half filling everywhere inside the gap except for a small region where $V,T<{T}_{L}\mathrm{exp}(\ensuremath{-}2m/{T}_{L})$.
We consider transport through a one-dimensional conductor subjected to an external periodic potential and connected to noninteracting leads (a ``Mott quantum wire''). In the case of a strong periodic potential, the conductance is shown to jump from zero, for the chemical potential lying within the Mott-Hubbard gap, to the noninteracting value of 2e2/h{2e}^{2}/h2e2/h, as soon as the chemical potential crosses the gap edge. This behavior is strikingly different from that of optical conductivity, which varies continuously with the carrier concentration. In the case of a weak potential, the perturbative correction to the conductance due to umklapp scattering is absent away from half filling.
We investigate the transport of interacting electrons through single-mode quantum wires whose parameters are periodically modulated on the scale of the electronic Fermi wavelength. The Umklapp scattering and backscattering of electrons can be described in terms of nonuniform quantum sine-Gordon-like models that also incorporate the effects of electronic reservoirs (electrodes) adiabatically coupled to the wire. We concentrate on weak Umklapp scattering and analyze the precursors of the Mott transition. At half-filling the temperature dependence of the extra resistance $\ensuremath{\Delta}R=R\ensuremath{-}\ensuremath{\pi}\ensuremath{\Elzxh}{/e}^{2}$ of a modulated quantum wire of length LLL changes from the interaction-dependent ``bulk'' power-law $\ensuremath{\Delta}R\ensuremath{\propto}{T}^{{4K}_{\ensuremath{\rho}}\ensuremath{-}3}$ at high temperatures $(T\ensuremath{\gg}{v}_{\ensuremath{\rho}}/L)$ to the universal $\ensuremath{\Delta}R\ensuremath{\propto}{T}^{2}$ behavior at low temperatures $(T\ensuremath{\ll}{v}_{\ensuremath{\rho}}/L).$ Away from half-filling the ``bulk'' results are qualitatively incorrect even at high temperatures ${v}_{\ensuremath{\rho}}/L\ensuremath{\ll}T\ensuremath{\ll}{T}^{*}$ despite the fact that the electron coherence in the wire is absent in this regime.
Transport between two Fermi liquid reservoirs through a one channel wire of length LLL is examined when the 1D electron system has an energy gap $2M\ensuremath{\gg}{v}_{c}/L$ ( vc{v}_{c}vc​ is the charge velocity in the wire). In the spinless case transformation of reservoir electrons into solitons of fractional charge entails a crossover from a Fermi liquid regime of integer charge transfer below the crossover energy ${T}_{x}\ensuremath{\ll}{T}_{L}$ to an insulating regime of the fractional charge transfer in the current vs voltage, conductance vs temperature, and the shot noise. Similar behavior is predicted for a spin Mott insulator of filling factor $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}n/({2m}^{\ensuremath{'}})$ ( ${n,m}^{\ensuremath{'}}$ are positive integers).
The influence of umklapp scattering on the resistivity of an interacting-one-dimensional-fermion system (Luttinger liquid) is studied. By using a renormalization-group calculation and a memory-function approximation for the conductivity, it is possible to obtain its frequency and temperature dependence at arbitrary filling. At high temperature the conductivity behaves as a power law of the temperature with an exponent depending on the interactions. Away from half filling there is a crossover between this behavior and an exponential increase of the conductivity. At half filling, the low-temperature conductivity behaves as ${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\Delta}}/\mathit{T}}$, where \ensuremath{\Delta} is the gap in the charge spectrum. It is argued that to get such behavior other scattering processes or phase-breaking processes are needed since in the presence of only electron-electron scattering the conductivity should, strictly speaking, be infinite at every finite temperature, even at half filling. Finally some results on the exponents of correlation functions and on the weight of the Drude peak obtained previously for the Hubbard model are shown to be generic features of any Luttinger liquid.
We study low-energy spin and charge transport through a 1D Mott-Hubbard insulator of finite length L attached to Fermi liquid reservoirs, which, in the presence of spin accumulation, are characterized by different electrochemical potentials for electrons of opposite spin polarizations. At temperatures less than ( v c : charge velocity in the wire) and under the assumption that the Hubbard gap 2 M is large enough, M > T L , we calculate the average currents (charge and spin) and their zero-frequency correlators. The average spin (charge) current depends only on the difference (sum) of the spin-dependent voltages and even a weak electron backscattering of low rate leads to the spin current suppression at smaller than . The spin current recovers its free mode behavior at spin voltage or temperature larger than Γ. Suppression of the spin-charge correlator suggesting the appearance of spin-charge separation needs both to be larger than Γ. In the absence of the average charge current at its shot noise is proportional to the average spin backscattered current defined by V s and can be used to measure the spin accumulation in the reservoirs. The relation of these results to Kondo dot transport in the Toulouse limit is also clarified.