Impurity pinning in transport through 1D Mott–Hubbard and spin gap insulators

PERGAMON

Solid State Communications 114 (2000) 9–13 www.elsevier.com/locate/ssc

Impurity pinning in transport through 1D Mott–Hubbard and spin gap insulators V.V. Ponomarenko 1, N. Nagaosa* Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan Received 1 December 1999; accepted 17 December 1999 by T. Tsuzuki

Abstract A low energy crossover [V.V. Ponomarenko, N. Nagaosa, Phys. Rev. Lett. 81 (1998) 2304] induced by Fermi liquid reservoirs in transport through a 1D Mott–Hubbard insulator of finite length L is examined in the presence of impurity pinning. Under the assumption that the Hubbard gap 2M is large enough: M . TL ; vc =L (vc: charge velocity in the wire) and the impurity backscattering rate G 1 p TL ; the conductance vs. voltage/temperature displays a zero-energy resonance. Transport through a spin gapped 1D system is also described availing of duality between the backscattered current of this system and the direct current of the Mott–Hubbard insulator. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: A. Nanostructures; D. Electron–electron interactions; D. Electronic transport

Recent developments in the nano-fabrication technique have allowed to produce relatively clean one channel wires of 1–10 mm length. In these wires, 1D Tomonaga– Luttinger Liquid (TLL) behavior [2] was observed in transport measurements [3–5]. Then it has been suggested that the correlated insulating behavior may be tuned with this setup [6,7]. This behavior of the 1D electron systems is expected [8] at half-filling where any Umklapp scattering has to open a Hubbard gap 2M in the charge mode spectrum of the infinite wire if the forward scattering is repulsive. Earlier consideration [1] predicted crossover from Fermi liquid transport carried by the reservoir electrons at low energy to that of a Mott–Hubbard insulator in the one channel wire of long enough length L : M . TL ; vc =L (vc: charge velocity in the wire). Charge in this insulator is carried by soliton excitations of the condensate emerging in the charge mode. This insulator, which is probably the easiest for the experimental observation, features two marginal quantities [9,10]: charge of the soliton is unchanged e (e ˆ " ˆ 1 below) and the exponent 1/2 characterizing the * Corresponding author. Tel.: 181-3-5841-6844; fax: 181-35841-6844. 1 On leave of absence from A.F. Ioffe Physical Technical Institute, 194021, St. Petersburg, Russian Federation. Present address: Department of Physics and Astronomy, SUNY at Stony Brook, NY 11794, USA.

electron–soliton transition brings about similarity with the free electron tunneling through a resonant level. In this paper we investigate the effect of impurity backscattering of low rate G p M on the above crossover. For low energy following Schmid [11] the problem is mapped through a Duality Transform onto the model of a point scatterer with pseudospin imbeded in TLL. It is solved exactly by fermionization. The result shows that the impurity enlarges the width of a zero energy resonance in the conductance up to G 1 1 G 2 where the exponentially p small G 2 (G 2 < TL M e 22M=TL at half filling) is the rate of tunneling of the condensate phase. The linear bias conductance at zero temperature G ˆ G 2 =‰p…G 1 1 G 2 †Š reveals an exponential enhancement of the amplitude of the p impurity potential G 1 =G 2 by the condensate tunneling. This suppression of the resonant conductance does not affect the saturation of the current at J ˆ G 2 above the crossover as far as G 1 1 G 2 p TL : Finally, the result is addressed to the case when the gap opens in the spin mode of the wire. It occurs if the effective interaction between electrons inside the wire is attractive [8]. Although it is not expected for the semidconductor nanostructure, it has been observed in quasi1D organics [12]. The current backscattering in this system coincides with the direct one of the Mott–Hubbard insulator multiplied by a factor G 01 =G 02 where G 0i has the same meaning for this system as G i above. So, the backscattering current is zero in the absence of impurities, and the current is maximum

0038-1098/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S0038-109 8(99)00573-6

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V.V. Ponomarenko, N. Nagaosa / Solid State Communications 114 (2000) 9–13

V=p: Impurity appearance gives rise to backscattering together with simultaneous tunneling between the spin mode vacua of the wire. The conductance is suppressed below a crossover energy and recovers above it. It results in a constant shift DJ ˆ 2G 01 of the current at high voltage. Transport through a one-channel wire confined between two leads could be modeled by a 1D system of electrons whose pairwise interaction is local and switched off outside the finite length of the wire. Applying bosonization and spin-charge separation we can describe the charge and p spin density fluctuations rb …x; † ˆ …2x fb …x; t††=… 2p†; b ˆ c; s; respectively, with (charge and spin) bosonic fields f c,s. Without impurities their Lagrangian symmetrical under the spin rotation reads " !2 !2 ) ( Z X vb …x† 1 2t fb …t; x† 2x fb …t; x† p  p  L ˆ dx 2 2gb …x† vb2 4p 4p bˆc;s !# p E2 U 2mb 2 F b w…x† cos x 1 2fb …t; x† (1) pvF vb where w…x† ˆ u…x†u…L 2 x† specifies a one channel wire of the length L adiabatically attached to the leads x . L; x , 0; and vF(EF) denotes the Fermi velocity (energy) in the channel. The parameter mc ; m varies the chemical potential inside the wire from its zero value at the half-filling and ms ˆ 0: The constants of the forward scattering differ inside the wire gb …x† ˆ gb for x [ ‰0; LŠ from those in the leads gb …x† ˆ 1; and an Umklapp scattering (backscattering) of the strength Uc(Us) is introduced inside the wire. The velocities vc,s(x) change from vF outside the wire to some constants vc,s inside it. We can eliminate them rescaling the spacial coordinate xold in theR charge and spin Lagrangians of Eq. (1) into xnew ; x0old dy=vc;s …y†: As a result, the new coordinate will have an inverse energy dimension and the length of the wire becomes different for the charge mode L ! 1=TL and spin mode L ! 1=T 0L : Applying renormalization-group results of the uniform sine-Gordon model at energies larger than TL or T 0L we come to the renormalized values of the parameters in Eq. (1). For the repulsive interaction when initially gs . 1 . gc ; the constant Us of backscattering flows to zero and gs to 1, brining the spin mode into the regime of the free TLL. The constant Uc of Umklapp process increases, reaching vF =vc at the energy cut-off corresponding to the mass of the soliton M if the chemical potential m is less than M. Meanwhile, gc flows to its free fermion value gc ˆ 1=2 [1]. These values of the parameters specify the Mott–Hubbard insulator. For the attractive interaction initially gs , 1 , gc ; and both constants Uc,s flow vice versa resulting in the spin gap insulator and TLL of some gc . 1 in the charge mode. Both cases of the interaction remain to some extent symmetrically conjugated under the spin-charge exchange even after accounting extent symmetrically conjugated under the spin-charge exchange even after accounting for a week backscattering on a point impurity potential inside the wire

0 , x0 , L : Limp ˆ 2

2Vimp cos pa



   fc …t; xc † fs …t; xs † p 1 w0 cos p …2† 2 2

where xc;s ˆ x0 =vc;s ; w0 ; w 1 mxc includes a phase of the scatterer w . The amplitude of the potential Vimp specifies 2 †; and a . transmittance coefficient [13] as 1=…1 1 Vimp 1=EF is momentum cut-off assumed to be determined by the Fermi energy. Below we will first elaborate the case of the Mott–Hubbard insulator adjusting results to the transport through the spin gap insulator at the end. Duality transform. An effective model for energies lower than some cut-off D 0 specified below may be read off following Schmid [11] from the expression for the partition function Z associated to the Lagrangian (1) plus (2). Without impurities the spin and charge modes are decoupled. After integrating out f c in the reservoirs the charge mode contribution into Z describes rare tunneling between neighbor degenerate vacua of the massive charge mode p in the wire characterized by the quantized values of 2fc …t; x† 1 2mx ˆ 2pm; m is integer. Variation of m by a ˆ ^1 relates to passage of a (anti)soliton through the wire ((anti)-instanton in imaginary time t ). The tunneling amplitude has been found [1] by mappingp into a free fermionic model to be P e2s0 =TL ; s0 ˆ M 2 2 m2 ; M sin 4; …m p M† with the pre-factor P and the energy cut-off D 0 proportional to TL on approaching the perturbative regime TL , M: The calculation p [9,10]  by instanton techniques has corrected P ˆ C × D 0 …sin3 4 MTL †1=4 with the constant C of the order of 1. The parameter D 0 is a high-energy cut-off to the long-time asymptotics of the p kink–kink interaction: F…t† ˆ ln{ t2 1 1=D 02 } created by p the reservoirs. It varies with m from D 0 . MTL at m ˆ 0 to 0 D . …M=m†TL if m . TL : A crucial modification to this consideration produced by the impurity under the assumption EF Vimp p M ensues from the shift of the m m-vacuum. Since pit  is equal to …21† …2Vimp =pa† cos w cos …fs …t; xs †= 2† the neighbor vacua become nondegenerate. This can be accounted for by ascribing opposite values of the pseudospin variable s ˆ ^1 to the neighbor vacua which are the eigenvalues of the third component of the Pauli matrix s 3. The energy splitting pbecomes an operator s3 …2Vimp =pa† cos w cos …fs …t; x0 †= 2† acting on the pseudospins, and every (anti-)instanton tunneling rotates a s 3-value into its opposite with the Pauli matrix s 1. The partition function then can be written as Z/

∞ X Z X Nˆ0 aj ˆ^

Dfs

e2S0 ‰fs Š N!

  Z Y  N × Trs T dti P e 2s0 =TL s1 …ti †  X iˆ1 ai aj × exp F…ti 2 tj † 2 i;j   2 cos w Vimp Z fs …t; x0 † p 1 dt3 …t† cos pa 2

…3†

V.V. Ponomarenko, N. Nagaosa / Solid State Communications 114 (2000) 9–13

Rb

R

Here S0 ‰fs Š ˆ 0 dt dx{…2t fs …t; x†† 1 …2x fs …t; x†† }= …8p† is the free TLL Euclidean action. All tP -integrals run from 0 to inverse temperature b ˆ 1=T and j aj ˆ 0: To have all s 1,3-matrices time-ordered under the sign T, we attributed each of them to a corresponding time t assuming that their time evolution is trivial. Noticing that the interaction F coincides with the pair correlator of some bosonic field u c whose evolution is described with the free TLL action S0 ‰uc Š; pwe  re-write Eq. (3) ascribing a factor exp…7uc …tj ; 0†= 2† to the t j (anti-)instanton, respectively: " ( ( Z Z / Trs T Dfs Duc exp 2 S0 ‰fs Š 2 S0 ‰uc Š 2

! uc …tj ; 0† p 2 !#))# f s …t; x s † p 2

2

" Z 1 2 dt × P e2s0 =TL s1 …t† cos

cos w Vimp 1 s3 …t† cos pa

11

0

(equal to the energy ones) D and EF, respectively. Substitution of these fields into Eq. (5) produces a free-electron Hamiltonian where the interaction reduces to tunneling between the c c,s fermions and the Majorana one j 2 ( ; j , below). Application of a voltage V between the left and right reservoirs forces us to use the real-time representation. Since peach instanton tunneling transfers charge Df c =… 2p† ˆ 1e between the reservoirs the p p voltage may be accounted for by a shift uc = 2 ! uc = 2 1 Vt of the cos-argument in the real-time form of the action from Eq. (4). Assuming that it remains small enough, V , TL , M; we will neglect its effect on the other parameters. The realtime Lagrangian associated with the fermionized Hamiltonian (5) reads: X Z LF ˆ ij2 t j…t† 1 i dx c a1 …2t 1 2x †c a aˆc;s

p 2 G 2 ‰c c1 …0; t†j…t† e 2iVt 1 h:c:Š p 2 G 1 ‰c s1 …0; t†j…t† 1 h:c:Š

…4†

It is easy to recognize a standard Hamiltonian form Z ˆ cst × Tr{e2bH } in Eq. (4) with p H ˆ H0 ‰fs …x†Š 1 H0 ‰uc …x†Š 2 2P e2s0 =TL s1 cos …uc …0†= 2† ! 2Vimp fs …xs † p 2 cos…w†s3 cos (5) pa 2 Here fs …x† and uc …x† are Schro¨dinger’s bosonic operators related to the variables f s(t ,x) and u c(t ,x) of the functional integration in Eq. (4). The operator H0 ‰fs …x†Š …H0 …uc …x†; vc ††; a function of the field fs …x† …uc …x†† and its conjugate is a free TLL Hamiltonian …g ˆ 1† corresponding to the free TLL action S0 ‰fs Š …S0 ‰uc Š† in Eq. (4), respectively. The Dual model specified by Eq. (5) is equivalent to the initial one, Eq. (1), at low energy. It relates to a Point Scatterer with internal degree of freedom in TLL. Fortunately, this in general rather complicated model may be solved easily through fermionization in our particular case. This simplification stems from the marginal behavior of the Mott–Hubbard insulator: the charge of the transport carriers does not change on passing from the low energies to the higher ones despite the nature of the carriers. Fermionization. From the commutation relations and hermiticity, the Pauli matrices can be written as sa ˆ i P …21†a11 2 b;g ea;b;g j b j g with Majorana fermions j 1,2,3 and anti-symmetrical tensor e : e 123 ˆ 1: Since the interaction in Eq. (5) is point-like localized and its evolution involves only the appropriate time-dependent correlators, we can fermionize it making use of: s r p p D0 EF c c …0† ˆ j 1 e i…uc …0†= 2† ; c s …0† ˆ j e i…fs …xs †= 2† 2p 2p 3 Here c c,s(0) is the x ˆ 0 value of the charge (spin) fermionic field, respectively. These fields have linear dispersions taken after the related bosonic fields with momentum cut-offs

…6†

where the rate of impurity scattering is G 1 ˆ …2EF =p†  …cos…w†Vimp † 2 and the rate of instanton tunneling is G 2 ˆ p 2pC 2 TL M sin3 4 e22s0 =TL : The current p flowing through the channel is J ˆ 22LF =2…Vt† ˆ 2i G 2 ‰c c1 …0; t†j…t† e 2iVt 2 h:c:Š: Its calculation with the nonequilibrium Lagrangian (6) needs avail of the Keldysh technique. Transforming voltage into the non-zero chemical potential of the fermions equal to 2V, we can write in standard notations [14] p Z dv J ˆ 2 G2 Re Gc.c j …v† 2p      Z dv v1V ˆ G2 GjA …v† 1 G j. …v† Im 2 1 2 f T 2p …7† where f is the Fermi distribution function. To find the Green functions G jA ; G j. of the Majorana field, its free Green functions: g jR…A† ˆ 2=…v ^ i0†; g jk;l ˆ ^2pid…v† are substituted into the appropriate Dyson equations [14]: GR…A† ˆ gR…A† …1 1 S R…A† GR…A† †; Gg ˆ …1 1 GR S R †gg …1 1 S A GA † 1 GR S g GA

…8†

~ The self-energies for the where g stands for k; l; c; c: Majorana field are: S jR…A† ˆ 7i…G 1 1 G 2 †; S j. ˆ 2i…G 2 …2 2 S ^ f ……v ^ V†=T†† 1 2…1 2 f …v=T††: Their substitution into Eq. (8) and into Eq. (7), subsequently, results in the current: Jˆ

2G 2 …G 1 1 G 2 † Z f ……v 2 V†=T† 2 f ……v 1 V†=T† dv …9† p v 2 1 4…G 1 1 G 2 † 2

which is the current passing through a resonant level of the half-width 2…G 1 1 G 2 † and suppressed by the factor

12

V.V. Ponomarenko, N. Nagaosa / Solid State Communications 114 (2000) 9–13

G 2 =…G 1 1 G 2 †: The typical features of this current can be illustrated with its zero temperature behavior vs. voltage and the linear bias conductance G vs. temperature, respectively: 2G 2 Jˆ arctan p



 V ; 2…G 1 1 G 2 †

  G 1 G 1 G2 1 1 G ˆ 22 c 0 2 pT p T

…10†

where c 0 …x† is the derivative of the di-gamma function, c 0 …1=2† ˆ p 2 =2; and the high temperature asymptotics of G is G 2 =…2T†: The zero-temperature conductance G ! …G 2 =p…G 1 1 G 2 †† manifested in Eq. p(9) is the function  of G 1 1 G 2 / …cos w Vimp e s0 =TL †2 EF = TL M sin3 4: Comparison with the initial transmittance of the impurity  p scatterer shows that G 1 =G 2 may be conceived as a renormalization of the initial amplitude Vimp by the instanton exponent and a power factor of the TLL for g ˆ 1=2: Spin gap insulator. In this case the chemical potential of the spin mode always lies in the center of a gap 2M. Meanwhile the charge mode is inhomogeneous TLL with the constant gc ; g . 1 …gc ˆ 1† inside (outside) the wire. Under Duality Transform applied to the massive spin mode the partition function Z 0 takes the form of Eq. (4) with the following modifications. The field fs …uc † is changed by fc …us †; respectively, and xs by xc. The factor cos w disappears, as w is accumulated by f c. The amplitude of the instanton 0 tunneling between the spin mode vacua becomes P 0 e 2M=T L ; where a new pre-factor P 0 and cut-off D 00 are specified by the same expressions which were written for P and D 0 ; respectively, after substitution of T 0L instead of TL into them. The action for the f c field is inhomogeneous: S‰fc Š ˆ Rb R 2 2 To 0 dt dx{…2t fc …t; x†† 1 …2x fc …t; x†† }=…8gc …x†p†: proceed in line with the above scheme of calculation, we have to first integrate out the quick modes of the f c field whose energies are larger than TL. Unless vc/vs becomes essentially small at g q 1; these energies lie much higher than the expected crossover. Therefore, we can integrate neglecting the weak tunneling between the spin mode 2 vacua. In the lowest order in Vimp variation of the cut-off 2 from EF to TL results in multiplication of Vimp by a factor: 22g …EF =TL † F…xc TL † where F…xc TL † accounts for the interference produced by the inhomogeneity of the forward scattering. This function may be extracted from the longtime asymptotics of keifc …x;t† e2ifc …x;0† l calculated [15] with Q S‰fc Š as: F 2 …z† ˆ const…z2 1 g2 †gr ∞ ‰…m ^ z†2 1 ^;mˆ1 2 gr2m^1 0 2 2 2 2 gr gŠ < const × ‰z 1 g †……1 2 z† 1 g †Š ; where g ˆ TL =EF and r ˆ …1 2 g†=…1 1 g† is a negative reflection coefficient. At the energies below the new cut-off TL, the inhomogeneous action S‰fc Š may be changed by the homogeneous one S0 ‰fc Š: Then exercising the above fermionization we come to a real-time Lagrangian

analogous with Eq. (6): X Z dx c a1 …2t 1 2x †c a LF ˆ ij2 t j…t† 1 i aˆc;s

q 2 G 01 ‰c c1 …0; t†j…t† e 2iVt 1 h:c:Š

…11†

q 2 G 02 ‰c s1 …0; t†j…t† 1 h:c:Š Here pthe of the instanton tunneling G 02 ˆ   rate 2 22M=T 0L 0 2pC T L M e has a manifest similarity with G 2 and the rate of impurity scattering G 01 ˆ …2EF =p†  2 …EF =TL † 12g F…xc TL †Vimp acquires an additional small factor …EF =TL †12g F…xc TL †; as gc . 1: Creation and annihilation of the fermion of the charge mode in the fermionized model describes a change of chirality of one quasiparticle inside p the wire. Therefore, the term proportional to G 01 in Eq. (11) leads to the backscattering current: Jbsc ˆ 2…2LF =2…Vt†† ˆ p 2i G 01 ‰c c1 …0; t†j…t† e 2iVt 2 h:c:Š equal to minus deviation of the direct current from its maximum V/p value. The new Lagrangian (11) and the current Jbsc transform into the old Lagrangian (6) and the current J, respectively, on changing G 01 ; G 02 by G 2,G 1. Hence the average value of Jbsc may be extracted from Eq. (9) in the same way. In particular, the linear bias conductance: G ˆ …1 2 …G 01 =pT†c 0 …1=2 1 {G 01 1 G 02 }=…pT†††=p is strongly suppressed at T ˆ 0 as G ˆ G 02 =p…G 01 1 G 02 †: Similar to the Mott–Hubbard case it shows that the initial amplitude of the potential Vimp renormalizes to r q q

G 01 =G 02 / Vimp e M=T

0

L

EF …EF =TL †12gc F…xc †= T 0L M

at low energy. Above the crossover temperature 2…G 01 1 G 02 † the conductance recovers as …1=p†…1 2 G 01 =2T†: The backscattering current related to Eq. (10) results in the constant shift of the direct current J ˆ V=p 2 G 01 above the voltage crossover 2…G 01 1 G 02 †: Finally, we have shown that a zero-energy resance in transport through a 1D Mott–Hubbard insulator of finite length predicted [1] for the clean wire stands weak impurity pinning while the rate of the impurity backscattering G 1 is small G 1 p TL : We have also described how opening a gap in the spin mode suppresses the charge transport. Acknowledgements This work was supported by the Center of Excellence. References [1] V.V. Ponomarenko, N. Nagaosa, Phys. Rev. Lett. 81 (1998) 2304. [2] F.D.M. Haldane, Phys. Rev. Lett. 47 (1981) 1840. [3] S. Tarucha, T. Honda, T. Saku, Solid State Commun. 94 (1995) 413.

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A. Schmid, Phys. Rev. Lett. 51 (1983) 1506. D. Jerom, H. Schults, Adv. Phys. 31 (1982) 293. U. Weiss, Solid State Commun. 100 (1996) 281. G.D. Mahan, Many-Particle Physics, Plenum Press, New York, 1990. [15] V.V. Ponomarenko, N. Nagaosa, Phys. Rev. Lett. 79 (1997) 1714.