Non-linear current through a barrier in 1D wires with finite-range arXiv:cond-mat/9704143v1 [cond-mat.mes-hall] 16 Apr 1997 interactions Margit Steiner∗ and Wolfgang H¨usler∗∗ a I. Institut f¨r Theoretische Physik der Universit¨t Hamburg, u a Jungiusstr. 9, D-20355 Hamburg, Germany The transport properties of a tunnel barrier in a one-dimensional wire are investigated at finite voltages and temperatures. We generalize the Luttinger model to account for finite ranges of the interaction. This leads to deviations from the power law behaviour first derived by Kane and Fisher1 . At high energies the influence of the interaction disappears and the Coulomb blockade is suppressed. The crossover in the voltage or in the temperature dependence can provide a direct measure for the range of the interaction. Keywords : electron-electron interactions, electronic transport, tunnelling. Since the discovery of the vanishing transmittivity of a tunnel barrier in a one-dimensional (1D) wire due to the repulsive electron–electron interaction1 new interest has emerged in the transport properties of 1D electron systems2–10 . Indeed, the influence of the electron correlations shows up strikingly in non-linear current voltage relations which are investigated experimentally in narrow, semiconducting wires11 . Local interactions, v(x−x′ ) = v0 δ(x−x′ ) are described within the Luttinger model for which the power-law1–5 ωc I(V ) = eRT eV ωc 2/g−1 (1) has been predicted for the current-voltage relation through a tunnel barrier with tunnel resistance RT , at zero temperature T = 0 . A similar behaviour has been found for the linear conductance ∼ T 2/g−2 as a function of temperature1,2,5 . The exponent depends on 0 Present addresses : ∗ Imperial College, Math. Phys. Department, 180, Queen’s Gate, London SW7 2BZ, UK ∗∗ Theoretical Physics Institute, 116 Church Str. SE Minneapolis MN 55455, U.S.A. 1 the strength v0 of the interaction, g ≡ (1 + v0 /πvF )−1/2 ( vF : Fermi velocity). Repulsion corresponds to g < 1 . The Luttinger model limits the energies to values well below the upper cut-off ωc which can be identified with the Fermi energy and can be small in semiconducting devices at low electron densities. At larger voltages, Eq. (1) formally describes currents that exceed even the value V /RT for non-interacting electrons. This indicates that some new energy or length scale must become important at higher energies. Here we shall identify the finite range of the e − e–interaction to cause an asymptotic approach of the current towards V /RT at large voltages. In the case of very strong interaction the cross-over can occur in an oscillatory manner while for more realistic interaction strengths the current stays below V /RT for all voltages. The low energy, and hence long wave length properties, are well described within the Luttinger model where the range of the interaction 1/α is assumed to be shorter than even the inter-electron spacing a ≡ πvF /2ωc for spinless electrons. Finite voltages, however, introduce a wave length 1/∆k = vF /eV on which eventually a finite value for 1/α can be experienced. Large voltages change the momenta at the Fermi points by |∆k| that exceeds the scale α on which the Fourier components of the interaction vanish, v (∆k > α) ≈ 0 so ˆ that V > αvF /e suppress the effect of interactions. Accordingly, at high temperatures T > αvF , I(V ) becomes independent of T so that the differential conductance approaches the constant 1/RT , like in the non-interacting case (we do not consider the effect of lattice vibrations here12 ). Finite temperatures may even reduce the current. Both the voltage and the temperature dependencies of the current show the important common feature of a crossover which in principle allows to extract the range 1/α of the interaction. The importance of finite range interactions has been found already in zero dimensional systems, quantum dots, where ‘crystallisation’ of the charge density distribution can occur leading to qualitative changes in the low energy excitation spectra13–15 as compared to what is expected for a contact interaction16 . 2 For convenience we concentrate on the interaction17 α ′ v(x − x′ ) = v0 e−α|x−x | 2 . (2) In the presence of metallic gates close to the 1D channel a more realistic form would vary ∼ |x − x′ |−3 at large distances, however, we expect qualitatively the same results for the latter finite range interaction as for (2), cf. below. The non-Fermi liquid behaviour of the charge excitations in a 1D wire is expressed most conveniently by the Hamiltonian9,18–20 vF 2 dx [(Π(x))2 + (∂x θ(x))2 ] + 1 2π Hw = dx dx′ (∂x θ(x))v(x − x′ )(∂x′ θ(x′ )) (3) where the Fermi-fields are expressed1,18 through Bose fields, Π(x) ≡ ∂x φ(x) and θ(x) , with [φ(x) , θ(x′ )] = −(i/2)sgn(x − x′ ) . The spatial derivative, ∂x θ , measures the fluctuations of the charge density, and the time derivative, ∂t θ , is proportional to the current. Here, we account for the dispersion relation of the charged modes in the wire, as it can be obtained from (3) by spatial Fourier transform ω(k) = vF |k| 1 + v (k) ˆ πvF . (4) The Fourier transform of the interaction potential (2) v (k) = v0 ˆ α2 k 2 + α2 (5) is constant v (k) = v0 in the limit α → ∞ of the Luttinger liquid used in previous calculations ˆ where it merely renormalizes the sound velocity → vF /g . The tunnel barrier can be described1 by √ Hb = Ub 1 − cos(2 πθ(x = 0)) . (6) Furthermore, we assume an electrostatic potential (V /2)sgn(x) dropping discontinuously at the location x = 0 of the tunnel barrier, 3 eV HV = √ θ(x = 0) , π as in1–3,7,8,10 . In the limit of weak tunnelling it has been demonstrated9 that the selfconsistently adjusted chemical potential indeed varies most pronouncedly close to x = 0 . The dc–current e I = − √ ∂t q(t) π (7) can be expressed in terms of the field θ at x = 0 , q(t) ≡ θ(t, x = 0) where both, the quantum average and the dynamics refer to the full Hamiltonian Hw + Hb + HV . Since Hw is purely quadratic in θ(x) all of the contributions away from the impurity x = 0 can be integrated out to obtain the reduced dynamics for q(t) . We are interested in the probability for transitions of q from a value θi to θf during the time t which can be expressed as a double integral θf θi Dq θi θf Dq ′ exp (iS[q]) exp (−iS[q ′ ]) F[q, q ′ ] (8) over paths q and q ′ with endpoints q(0) = q ′ (t) = θi and q(t) = q ′ (0) = θf . The action S[q] contains all contributions to the Hamiltonian at x = 0 while the influence of the bulk modes, x = 0 , is exactly accounted for in the functional21 F[q, q ′ ] = exp − t 0 dt′ t 0 dt′′ (q(t′ ) − q ′ (t′ )) × ˙ ˙ (w(t′ − t′′ )q(t′′ ) − w ∗ (t′ − t′′ )q ′ (t′′ )) ˙ ˙ (9) where w(t) = ∞ 0 dω βω J(ω) (1 − cos ωt) coth + i sin ωt 2 ω 2 4 , (10) and 1/β is the temperature. It will be important for the following that the non-linear dispersion (4) of the bulk modes, shown in the inset of Figure 1, cause a non-Ohmic dissipative influence. The function J(ω) includes all of the details of the environmental modes in their efficiency to damp the frequency ω of the motion of θ(x = 0, t) . Within the Feynman–Vernon technique21 the quantum state for q is assumed to be initially known (e.g. θi = q(0) = 0 ) before the exact time evolution is switched on. With √ time t , q acquires probability Pm (t) to assume the value θf = m π (cf. (8)) where m elementary charges have been transferred through the barrier. The long time behaviour of the probability density distribution defines the stationary dc–current I = −e m lim ∂t Pm (t) , m t→∞ according to (7), assuming ergodicity for the whole system. √ For large Ub the potential (6) has deep minima at θ = m π so that integer m contribute mainly to the saddle points of the action S[q] in (8). In this limit the charge is transferred in integer units. Step like instantons dominate the path integral (8)22,23 for the low current properties, each instanton contributing with a factor ±i∆/2 proportional to the tunnelling amplitude2 . The influence functional F [q, q ′ ] (9) introduces a temperature dependent coupling w(ti − tj ) between instantons centred at times ti and tj so that the sum over all possible instanton configurations in general cannot be performed analytically. For a barrier of low transmittivity the most important configurations are instanton – anti-instanton pairs that contribute in order ∆2 . This leads to an expression for the current3 I(V ) = e ∆2 (1 − e−βeV ) 4 dt eieV t e−w(t) , (11) when the detailed balance property ∂t P−1 = eβeV ∂t P+1 is used24 . 2 The value of ∆ can be related to Ub , cf.10 . Through the one instanton action, ∆ depends in principle also on α . 5 For the case of Ohmic dissipation, J(ω) ∝ ω , corresponding to a contact interaction, considerable progress has been made. To order ∆2 the current has been calculated in3 and to any order for weak interaction 1−g ≪ 1 in25 . Recently, the extension to arbitrary interaction strength has been achieved using conformal field theory techniques7 and by systematically exploiting the duality symmetry between low and high transmittivities10 . 20 16 !(k)= vF J ( ! ) = vF 8 4 12 0 0 4 8 k= 8 g = 0.3 g = 0.6 g = 0.9 g=1 4 0 0 2 4 6 8 != vF 10 Figure 1: Effective density J(ω) of charged modes in the wire that damp the motion of θ(x = 0, t) at the frequency ω , according to Eq. (17) for different g . J(ω) is the crucial ingredient for the non-linear current. The inset shows the dispersion relation ω(k) according to Eq. (4). Natural units for wave vectors and frequencies are α and αvF , respectively. How the electron–electron interaction influences the transport properties is determined by J(ω) (cf. (10,11)). Its relationship to the bulk mode dispersion ω(k) can most easily be deduced from the partition function of the wire (3) Z = Tr e−βHw = D[θ(x, τ )] e−Sw [θ] = ˆ ˆ D[θ(k, τ )] e−Sw [θ] where ˆ Sw [θ(k, τ )] = ˆ with θ(k, τ ) = 1 2vF dk 2π β 0 2 ˆ ˆ dτ θ(−k, τ ) −∂τ + ω 2 (k) θ(k, τ ) dx θ(x, τ )eikx . 6 (12) The modes θ(x = 0, τ ) act as a harmonic thermal environment on the mode of interest, q(τ ) ≡ θ(x = 0, τ ) , Z= D[θ(x = 0, τ )] e−Sw [θ] ≡ D[q(τ )] D[q] ̺[q] . (13) The functional ̺[q] ∝ exp − β 0 dτ β 0 ˆ dτ ′ q(τ )K(τ − τ ′ )q(τ ′ ) (14) for the reduced density contains the retarding effects, described by the Kernel26 K(ωn ) = β 0 ˆ dτ K(τ )e−iωn τ = dk vF 2 + ω 2 (k) 2π ωn −1 (15) with ωn = 2πn/β . Analytic continuation, J(ω) = −ℑm lim K(−iω + δ) (16) δ→0 relates J(ω) directly to K(ωn ) 24 . The asymptotic behaviours J(ω → 0) ≈ 2|ω|/g and J(ω → ∞) ∼ 2ω can readily 2 2 be deduced from (15) in view of ω 2(|k| ≪ α) ≈ vF k 2 /g 2 and ω 2 (|k| ≫ α) ≈ vF k 2 . Here, g ≡ (1 + v0 /πvF )−1/2 has been defined in analogy to the Luttinger model with v0 = v(k = 0) ˆ (cf. (5)). For ω(k) as in (4) the integration (15) with (16) can be carried out analytically, yielding √ J(˜ ) ω (N+ (˜ ) + ω N− (˜ ))M(˜ ) ω ˜ ω ω = 2 2˜ ω 2 )(|N (˜ )|2 + |N (˜ )|2 ) αvF (1 + ω ˜ + ω − ω (17) where M(˜ ) = ω 4˜ 2 + (˜ 2 − 1/g 2)2 ω ω ω ω N+ (˜ ) = −˜ 2 + 1/g 2 + M(˜ ) ω 1/2 ω ω N− (˜ ) = −˜ 2 + 1/g 2 − M(˜ ) ω 1/2 . The time (αvF )−1 for electrons of velocity vF needed to traverse the interaction range establishes the natural frequency scale of the problem, ω = ω/αvF . ˜ 7 Figure 1 illustrates the result (17). At small ω ≪ αvF , J(ω) ∼ 2ω/g and the currentvoltage relation (1) is recovered at long wave lengths and low energies. With large ω ≫ αvF , J(ω) approaches the linear behaviour, J(ω) → 2ω , that corresponds to the non-interacting case, g = 1 , for the reasons motivated initially. Note, however that J(ω) does not simply interpolate between either of the Ohmic asymptotics > but crosses the value 2ω so that the damping J(ω ∼ αvF ) < 2ω is smaller than it would be in the absence of interactions. g = 0.4 g = 0.7 g=1 8 I=( vF =2 0 2 e) 10 6 4 2 0 0 2 4 6 8 10 eV= vF Figure 2: Current through the tunnel barrier versus applied voltage for different g , ∆′ ≡ ∆/ωc . The power-law at low voltages agrees with the Luttinger liquid behaviour. The crossover to the linear relation, I(V ) = V /RT , manifests the finite range of the e − e–interaction. The current-voltage relation, obtained according to (11) for zero temperature, is depicted in Figure 2. The crossover behaviour of J(ω) shows up in a transition from the power law at low voltages, I(V ) ∼ V 2/g−1 to the linear tunnel resistance behaviour, I(V ) → V /RT . It takes place on the voltage scale αvF (2/g − 1)/e . The high voltage limit does not show any offset that would correspond to a Coulomb blockade since the charging energy Ec /e = lim (V /RT − I(V )) ∝ V →∞ 8 ∞ 0 dω J(ω) −2 =0 ω (18) vanishes. The proportionality in (18) follows from the short time behaviour of w(t) to the order ∼ t2 (cf. 10) and the right hand side expression vanishes since, for any dispersion (4), dω J(ω)/2ω equals the number of modes in the wire, cf. (15,16). Two conditions are usually considered as being sufficient to establish a Coulomb blockade27,28 : the suppression of quantum fluctuations of the charges by low transmittivities and the presence of a nearby dissipative environment of high impedance for which the bulk modes serve3 . Although both conditions are fulfilled in the present system no charging effects appear. The vanishing lateral extension of a 1D wire does not suffice to accumulate charging energy. Near a single barrier, a finite cross section is required for the capacitance C to be finite so that I(V ) ∼ (V − Ec /e)/RT at high voltages, with Ec = e2 /2C 29 . This is consistent with the result for a selfconsistent determination of the charge distribution along the wire9 . Eq. (18) holds for any interaction potential of finite range. Another example is the screened Coulomb interaction v(x − x′ ) = ′ e2 e−α|x−x | / κ (x − x′ )2 + d2 ( κ : dielectric constant, d : width of the wire) which is again determined by two parameters, α and αv0 /2 → e2 /κd . At low voltages I(V ) obeys a power law and at V > 2αvF ( 1 + 2e2 K0 (αd)/κπvF − 1)/e ( K0 : modified Bessel function) a crossover to the effectively non-interacting behaviour occurs. Only true long-range interaction α → 0 changes the power-law behaviour at low voltages6,19,30 and the divergence of v (k → 0) ˆ suppresses the crossover. In principle, we can also infer the current-voltage characteristics for the case of a weak barrier and attractive, finite–range interactions by taking advantage of the exact duality relation31 between the weak and the strong barrier limit as it has been proven32 for arbitrary J(ω) . In our case, the dual J(ω) interpolates between J(ω ≪ αvF ) = 2gω and J(ω ≫ αvF ) = 2ω , since g maps to 1/g , so that the interaction strength again vanishes at high energies. Correspondingly, to second order in the barrier height, the current exhibits a 2 crossover from the Luttinger liquid behaviour I(V ) ∼ (1 − c(Ub )V 2g−2 )V at small voltages 2 to the Ohmic behaviour, I(V ) ∼ (1 − c(Ub ))V , at large voltages. 9 1.0 3 0.3 0.5 0.1 0 (@I=@V )=( vF 2e =2) 1 g = 0.7 0.0 0.0 a 1.0 2.0 5 3 1.0 0.1 0.5 1 0.3 0.0 0.0 g = 0.4 b 0.5 T=( vF (2=g 2)) Figure 3: Temperature dependence of the differential conductance for different voltages eV /αvF = 0.1, . . . , 5 as indicated. In a and b g = 0.7 and g = 0.4 , respectively. At low temperatures and low voltages the power law behaviours1 are recovered. Also the temperature dependence of the differential conductance ∂I/∂V , Figure 3, reveals a crossover around T ≈ αvF (2/g − 2) as can be deduced from a high temperature expansion up to the cubic term in (10) and (11). At small T the linear conductance varies ∼ T 2/g−2 , in agreement with1 , while the non-linear conductance can even decrease with temperature. At high temperatures ∂I/∂V approaches the constant value 1/RT , irrespective of the voltage, 10 as has been demonstrated already for weak interaction5 . To summarise, we have shown that finite ranges of the e − e–interaction change the non-linear current through a tunnelling barrier in a 1D wire qualitatively, compared to the simple power-law behaviour. The latter is usually considered to be the main characteristic for one-dimensionality, but is only valid for short range interactions. 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