Non-Fermi liquid behavior in transport across carbon nanotube quantum dots
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a r X i v :c o n d -m a t /0606214v 1 [c o n d -m a t .s t r -e l ] 8 J u n 2006Non-Fermi liquid behavior in transport across carbon nanotube quantum dotsLeonhard Mayrhofer and Milena GrifoniTheoretische Physik,Universit¨a t Regensburg,93040Germany(Dated:February 6,2008)A low energy-theory for non-linear transport in finite-size single-wall carbon nanotubes,based on a microscopic model for the interacting p z electrons and successive bosonization,is presented.Due to the multiple degeneracy of the energy spectrum diagonal as well as off-diagonal (coherences)elements of the reduced density matrix contribute to the nonlinear transport.A four-electron periodicity with a characteristic ratio between adjacent peaks,as well as nonlinear transport features,in quantitative agreement with recent experiments,are predicted.PACS numbers:PACS numbers:73.63.Fg,71.10.Pm,73.23.HkSince their recent discovery single-wall carbon nan-otubes (SWNTs),cf. e.g.[1],have attracted a lot of experimental and theoretical attention.In particular,as suggested in the seminal works [2,3],due to the pecu-liar one-dimensional character of their electronic bands,metallic SWNTs are expected to exhibit Luttinger liquid behavior at low energies,reflected in power-law depen-dence of various quantities and spin-charge ter experimental observations have provided a confir-mation of the theory [4,5].As typical of interacting electron systems in reduced dimension,SWNTs weakly coupled to leads exhibit Coulomb blockade at low tem-peratures [6]with characteristic even-odd [7]or four-fold periodicity [8,9,10].In [9]not only the ground state,but also several excited states could be seen in stability dia-grams of closed SWNT quantum dots.Such two-fold and four-fold character can be qualitatively understood from symmetry arguments related to the two-fold band degen-eracy of SWNTs and the inclusion of the spin degree of freedom.So far,a quantitative description has relied on density functional theory calculations [11]or on a mean field description of the Coulomb blockade [12].In par-ticular,the position of the spectral lines in the stability diagram measured in [9]was found to be in quantitative agreement with the predictions in [12].However,a mean field description may be not justified for one-dimensional systems.For example,to describe the spectral lines of the sample with four-fold periodicity (sample C)in [9],a quite peculiar choice of the mean field parameters was made,and a quantum dot length three times shorter than the measured SWNT length was assumed.Moreover,to date no quantitative calculation of the nonlinear current across a SWNT dot has been provided.In this Letter we investigate spectral as well as dynam-ical properties of electrons in metallic SWNT quantum dots at low energies.We start from a microscopic descrip-tion of metallic SWNTs and include Coulomb interaction effects,beyond mean-field,by using bosonization tech-niques [2,3],yielding the spectrum and eigenfunctions of the isolated finite length SWNT.Due to the many-fold degeneracies of the spectrum,the current-voltage char-acteristics is obtained by solving equations of motion forthe reduced density matrix (RDM)including off-diagonal elements.Analytical results for the conductance are pro-vided,which account for the different heights of the con-ductance peaks in [9].Moreover,we can quantitatively reproduce all the spectral lines seen in sample C in [9]by solely using the two ground state addition energies provided in that work.The derived level spacing is in agreement with the measured SWNT length.To start with,we consider the total HamiltonianH =H ⊙+H s +H d +H T +H gate ,(1)where H ⊙is the interacting SWNT Hamiltonian (cf.Eq.(5)below)and H s/d describe the isolated metallic source and drain contacts as a thermal reservoir of non-interacting quasi-particles.Upon absorbing terms pro-portional to external source and drain voltages V s/d ,theyread (l =s,d )H l = σ q ε q ,l c † q σl c q σl ,where c †q σl creates a quasi-particle with spin σand energy ε q ,l =ε q −eV s/d in lead s/d .The transfer of electrons between the leads and the SWNT is taken into account byH T = l =s,d σd 3r T l ( r )Ψ†σ( r )Φσl ( r )+h.c.,(2)where Ψ†σand Φ†σl ( r )= q φ∗ q ( r )c † q σl are electron cre-ation operators in the SWNT and in lead l ,respec-tively,and T l ( r )describes the transparency of the tun-neling contact l .Finally,H gate =−µg N c accounts for a gate voltage capacitively coupled to the SWNT,with N c counting the total electron number in the SWNT.SWNT Hamiltonian .In the following the focus is on arm-chair SWNTs at low energies.Then,if periodic bound-ary conditions are applied,only the gapless energy sub-bands nearby the Fermi points F =± K 0=±K 0ˆe x withˆe x along the nanotube axis,are relevant [2,3].To each Fermi point two different branches r =R/L are asso-ciated to the Bloch waves ϕR/L,F,κ( r )=e iκx ϕR/L,F ( r ),where κmeasures the distance from the Fermi points ±K 0[2](Fig.1a left).In this Letter,however,we are interested in finite size effects.Generalizing [13]to the case of SWNTs we introduce standing waves which fulfill2 open boundary conditions(Fig.1a right):ϕOBC˜R/˜L,κ( r)=12 ϕR/L,K0,κ( r)−ϕL/R,−K0,−κ( r),(3)with quantization conditionκ=π(mκ+∆)/L,mκan integer,and L the SWNT length.The offset parameter∆occurs if K0=πn/L,and is responsible for the energy mismatch between the˜R and˜L branches.Including thespin degree of freedom,the electron operator reads Ψ( r)= ˜r=˜R,˜L κ,σϕOBC˜rκ( r)c˜rσκ=: σΨσ( r),(4)with c˜rσκthe operator which annihilates ϕOBC˜rκ |σ . The interacting SWNT Hamiltonian then readsH⊙= v F ˜rσsgn(˜r) κκc†˜rκσc˜rκσ+(5) 1√Lϕsgn(F)˜r,F( r)ψ˜r Fσ(x),(6)where we used the convention that R/L=±1,˜R/˜L=±1.Upon inserting(6)into(5),integration over the co-ordinates perpendicular to the tube axis yields the inter-acting Hamiltonian expressed in terms of1D operatorsand an effective1D interaction V eff(x,x′).Using stan-dard bosonization techniques[2,3]H⊙can now be diag-onalized when keeping only forward scattering processesassociated to V eff(x,x′).It readsL R L R~~K-Ke ekkFIG.1:Energy spectrum of a SWNT with open boundaryconditions(right)described in terms of left(˜L)and right(˜R)branches.It is constructed from suitable combinationsof travelling waves whose spectrum is shown on the left side.H⊙=12+∆sgn(˜r)N˜rσ+ q>0 j=c,s δ=±εjδq a†jδq a jδq,(7)where thefirst line is the fermionic contribution and rep-resents the energy cost,due to Pauli’s principle and theCoulomb interaction,of adding new electrons to the sys-tem.Specifically,N˜rσ= κc†˜rσκc˜rσκ,is the operatorthat counts the number of electrons in the(˜rσ)-branch,N c= ˜rσN˜rσyields the total electron number,andε0= v FπL2 L0dx L0dx′V eff(x,x′)cos(qx)cos(qx′).Thesecond line of(7)describes bosonic excitations in termsof the bosonic operators a jδq.Four channels are associ-ated to total(jδ=c+,s+)and relative(jδ=c−,s−)(with respect to the occupation of the˜R and˜L branch)charge and spin excitations.Generalized spin-charge sep-aration occurs,since for three of the channels the energydispersion is the same as for the noninteracting system,εjδq= v F q= v Fπ∂t=T r leads H I T(t),W I(t) ,(9)for the reduced density matrix(RDM)ρI=T r leads W Iof the SWNT.Here W I(t)is the density matrix of thewhole system consisting of the leads and the quantumdot,and T r leads indicates the trace over the lead de-grees of freedom.The apex I denotes the interactionrepresentation with H T from(2)as the perturbation.We make the following approximations:i)We assumeweak coupling to the leads,and treat H T up to sec-ond order,i.e.,we consider the leads as reservoirs whichstay in thermal equilibrium and make the factorizationansatz W I(t)=ρI(t)ρsρd=:ρI(t)ρleads whereρs/d=Z−1s/de−β(H s/d−µs/d N s/d),with Z s/d the partition functionandβthe inverse temperature.ii)Being interested inlong time properties,we can make the so called Markovapproximation,where the time evolution of˙ρI(t)is onlylocal in time.iii)Since we know the eigenstates| N, mof H⊙,it is convenient to calculate the time evolutionofρI in this basis.We assume that matrix elements be-tween states representing a different number of electrons(charge states)in the SWNT and with different energiesvanish.Coherences between degenerate states with thesame energy E are retained!Hence we can divideρI(t)into block matricesρI,E Nnm(t),where E,N are the energyand number of particles in the degenerate eigenstates|n ,3|m .We arrive at equations of the Bloch-Redfield form˙ρI,E Nnm (t )=−kk ′R E Nnm kk ′ρI,E N kk ′(t )+E ′ M =N ±1kk ′R EN E ′M nm kk ′ρI,E ′Mkk ′(t ),(10)where k,k ′run over all degenerate states with fixed par-ticle number.The Redfield tensors are given by (l =s,d )R E Nnm kk ′= lE ′,M,jδmk ′Γ(+)E N E ′M l,njjk+δnk Γ(−)E N E ′Ml,k ′jjm ,(11)and R E N E ′M nm kk ′= l,α=±Γ(α)E ′M E Nl,k ′mnk,where the quantities Γ(α)E N E ′Ml,njjkare transition rates from a state with N to a state with M particles.Known the stationary densitymatrix ρI st ,the current (through lead l )follows from I =2ReN,E,E ′ nkjΓ(+)E N E ′N +1l,njjk −Γ(+)E N E ′N −1l,njjkρI,E Nkn,st .(12)iv)We exploit the localized character of the transparen-cies T l ( r )in Eq.(2),and make use of the slowly vary-ing nature of the operator ψ˜r F σ(x )in Eq.(6).This enables us to evaluate the 1D operator at the SWNT contacts and pull it out from the space integrals which enter the definition of the transition rates.It holds r |ψ˜r σF (x =0)|s :=(ψ˜r σ)E N E ′N +1rs ; r |ψ˜r σF (x =L )|s =e −iπsgn(F ){N ˜r σsgn(˜r )+∆}(ψ˜r σ)E N E ′N +1rsfor the matrix ele-ments between the states |r ,|s with energy E ,E ′and particle number N ,N +1,respectively.We thus can introduceΦl ˜r ˜r ′(ε)= d 3r d 3r ′T l ( r )T l ( r ′)q |εφl q ( r )φ∗l q ( r ′)×F F ′sgn(F F ′)ϕsgn(F )˜r ,F ( r )ϕsgn(F ′)˜r ′,F ′( r ′)ηl (∆),to describe the influence of the geometry of a tun-neling contact at the tube end.The term ηl (∆)=e iπsgn(F −F ′)∆(1−δl,d )accounts for the mismatch ∆.As-suming a 3D electron gas in the leads,e.g.of gold,we find that for a realistic range of energies is Φl ˜r ˜r ′(ε)=δ˜r ˜r ′Φl ,i.e.the leads are ”unpolarized”.We thus obtain Γ(±)E N E ′N +1l,rss ′r ′=1(ε−eV l −(E ′−E ))t ′,(13)with ρ⊕l (ε)=ρl (ε)f (ε),where ρl (ε)is the density of en-ergy levels in lead l ,and f (ε)the Fermi function.Alike,Γ(±)E N E ′N −1l,rss ′r ′=1(ε−eV l +(E ′−E ))t ′,(14)with ρ⊖l (ε)=ρl (ε)(1−f (ε)).When are coherences needed?Eqs.(10)with (12)show that coherences (in the energy basis)enter the evalua-tion of the current.In the low bias and temperature regime k B T,eV :=e (V s −V d )≪ε0,however,where only ground states contribute to the current,because ofN, 0|ψ†˜r σψ˜r σ| N ′, 0 =(1/2L )δ N ′, N,only diagonal ele-ments of the RDM contribute.Hence,due to the ”unpo-larized”character of the leads,the commonly used mas-ter equation (CME)with population’s dynamics only is valid.At larger biases coherences should be included [14].In the following we focus on the case ∆≈0,relevant to explain the experimental results for sample C in [9].Low bias regime (CME is valid).At low bias the cur-rent can be obtained by looking to transitions between ground states with N and N +1particles and energies E 0N ,E 0N +1.Then,the matrix element (ψ˜rσ) N ′ N is non zero only if N ′= N −ˆe rσ,with ˆe rσthe unit vector,andN′˜r σ(ψ†˜r σ) NN ′(ψ˜r σ) N ′ N =1l =s,d γl [C N,N +1f (εl )+C N +1,N (1−f (εl ))],(16)where ∆f =|f (εs )−f (εd )|,εl =eV l −∆E ,and∆E =E 0N −E 0N +1.Moreover,γl =(π/L )Φl ρl (0).This expression can be further simplified in the regime |eV |≪kT ≪ε0where the linear conductance G N,N +1is obtained by linearizing ∆f in V ,and by evaluating the remaining quantities in (16)at zero bias.The conduc-tance trace exhibits four-electron periodicity (Fig.2a),with two equal in height central peaks for the transitions N =4m +1→N +1,N =4m +2→N +1,and two4 smaller peaks for N=4m→N+1,N=4m+3→N+1also equal in height.The relative height between centraland outer peaks is G max4m+1,4m+2/G max4m,4m+1=27/(10+4√5[8]W.Liang,M.Bockrath and H.Park,Phys.Rev.Lett.88,126801(2002).[9]S.Sapmaz et al.,Phys.Rev.B71,153402(2005).[10]S.Moriyama et al.,Phys.Rev.Lett.94,186806(2005).[11]S.-H.Ke,H.U.Baranger and W.Yang,Phys.Rev.Lett.91,116803(2003).[12]Y.Oreg,K.Byczuk and B.I.Halperin,Phys.Rev.Lett.85,365(2000).[13]M.Fabrizio, A.O.Gogolin,Phys.Rev.B51,17827(1995).[14]In the case of noninteracting electrons and unpolarizedleads,the CME is always correct if the usual Bloch wave Slater determinants are used as energy eigenstates. [15]F.Cavaliere et al.,Phys.Rev.Lett.93,036803(2004).。
