On steady-state currents through nano-devices: a scattering-states numerical renormalization group approach to open quantum systems

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by  Frithjof B. Anders

On steady-state currents through nano-devices: a scattering-states numerical
renormalization group approach to open quantum systems
Frithjof B. Anders1
1Institut fu¨r Theoretische Physik, Universit¨at Bremen, P.O. Box 330 440, D-28334 Bremen, Germany
(Dated: June13, 2008)
Weproposeanumericalrenormalization group(NRG)approachtosteady-statecurrentsthrough
nano-devices. A discretization of the scattering-states continuum ensures the correct boundary
9 condition for an open quantum system. We introduce two degenerate Wilson chains for current
0 carryingleft andright-movingelectronsreflectingtime-reversalsymmetryin theabsenceofafinite
0 bias V. We employ the time-dependent NRG to evolve the known steady-state density operator
2
foranon-interactingjunctionintothedensityoperatorofthefullyinteractingnano-deviceatfinite

n bias. Wecalculate thedifferential conductanceas function of V,T and theexternal magnetic field.
a
J PACSnumbers: 73.21.La,73.63.Rt,72.15.Qm

5
1
Introduction: The description of quantum systems Dissipative steady-state currents only occur in open

] out of equilibrium is one the fundamental challenges in quantum system in which the system size L has been
l
l theoretical physics. Even a simple non-equilibrium situ- sent to L→∞ before t→∞. Transient currents can be
a
ation,the currenttransportthroughaninteractingjunc- calculated on a finite-size tight-binding chain within the
h
- tion at finite bias is not fully understood. The Coulomb TD-NRG as well as the time-dependent DMRG[23, 24].
s
blockade[1] and advent of the experimental realizations However, such transient currents vanish for t → ∞ or
e
m oftheKondoeffectinsuchdevices[2,3]requiresamany- even reverse their sign[24] in those approaches, a con-
body description at low temperatures. sequence of the non-interchangeable limit t → ∞ and
.
at While the equilibrium dynamics is well understood[4], L→∞[18]. We circumvent this problem by discretizing
m thenon-equilibriumsteady-statehasbeenmainlyinvesti- a single-particle scattering states basis. Therefore, those
- gated using perturbative approaches[5, 6, 7, 8] based on states remain current carrying and a faithful representa-
d Keldysh theory[9], the Toulouse point[10] and the flow tion of an open quantum system.
n
equation[11]. Landauer-Buettiker type approaches[12] Theory: Interacting quantum dots (QD), molecular
o
c treat the charging effect only on a mean-field level by junctions or other nano-devices are modelled by the in-
[ mapping the stronglyinteractingquantumproblemonto teracting region H , a set of non-interacting reser-
imp
2 amodelofnon-interactingfictitiousparticles,unsuitable voirs HB and a coupling between both sub-system HI:
v to describe the Coulomb-blockade physics[1]. In weak H=Himp+HB+HI. Throughoutthispaper,werestrict
1 coupling and high temperature, the ac and dc-transport ourselves to a junction with a single spin-degenerate or-
7 throughmolecular wires canbe addressedby a quantum bital d with energy E , subject to an external magnetic
d
3 master equation for the reduced density matrix of the field H and an on-site Coulomb repulsion U. The or-
0
junction[13]. All those approaches have only a limited bital is coupled to a left (L) and a right (R) lead via the
.
2 validity of their parameter regimes. Recently, Han pro- tunneling matrix elements V , and H given by
0 α=L,R
posed an alternative perturbative method[14] based on
8
Hershfield’ssteady-statedensity operator[15,16, 17,18].
0
: Based on similar ideas, a scattering-states Bethe-ansatz H = dǫǫc†ǫσαcǫσα (1)
v
approachtoaninteractingspinlessquantumdothasbeen σα=L,RZ
i X
X implemented[19] for finite bias. 2
U σ U
ar apWpreoacphr[4e]setnotopaennquumanetruicmalsyrsetenmorsmbaalsizeadtioonnscgartoteurp- +σ=±1(cid:20)Ed+ 2 − 2H(cid:21)nˆdσ+ 2 σ nˆdσ−1!
X X
ing states[15]. It combines (i) Wilson chains for single-
+ V dǫ ρ(ǫ) d†c +c† d .
particlescatteringstatesproposedbelow,(ii)Hershfield’s α σ ǫσα ǫσα σ
steady-state density operator[15] for a non-interacting Xασ Z p (cid:8) (cid:9)
junctionsatfinitebiasand(iii)thetime-dependentNRG
(TD-NRG)[20, 21, 22]. Our scattering-states basis will Here nˆdσ = d†σdσ, and c†ǫσα creates a conduction electron
be also useful for Quantum Monte Carlo and density in the lead α of energy ǫ and density of states ρ(ǫ).
matrix renormalization group (DMRG) approaches[23]. This Hamiltonian is commonly used to model ultra-
Withournon-perturbativemethod,steady-statecurrents small quantum dots[2, 5]. In the absence of the local
through interacting nano-devices can be obtained accu- Coulomb repulsion H = U( nˆd −1)2/2, the single
U σ σ
ratelyforarbitrarytemperatures,magneticfieldsandin- particle problem is diagonalized exactly in the contin-
P
teraction strength. uum limit[14, 15, 16, 17, 25, 26] by the scattering states
2
junction chain of a non-interacting resonant level model[4]. Each
fictitiousd -orbitalconsistsofanormalizedlinearcom-
left−moversµ σα
L U right−moversµR bination of scattering states γǫσα: no auxiliary degrees
backscattering of freedom has been introduced into the problem!
We divide Gr (ǫ + iδ) into magnitude and phase,
0σ
Gr (ǫ+iδ)=eiΦσ(ǫ)|Gr (ǫ+iδ)|, andabsorbthe energy
FIG. 1: The local d-orbital is expanded in left-moving and 0σ 0σ
dependent phase Φ (ǫ) into the scattering-states opera-
right-movingscatteringstates. Eachcontributionsdefinesone σ
tors γ by a gauge transformation. Then, the Wilson
fictitious local orbital dσα of the junction of the scattering- ǫσα
chains consist only of purely real tight-binding parame-
states NRG. The Coulomb repulsion introduces backscatter-
ing between left and right-movers. ters. Diagonalizing the proposed scattering-states Wil-
son chains yields a faithful representation of the steady-
state density operator ρˆ for arbitrary bias.
0
operators The current operator expanded in scattering states
γ acquires an additional energy dependence via the
γ† = c† +V ρ(ǫ)Gr (ǫ+iδ) ǫσα
ǫσα ǫσα α 0σ scattering-phase shift Φσ(ǫ). In our model (1), however,
× d† + p dǫ′Vα′ ρ(ǫ′)c† (2) the current remains connected to the spectral function
" σ α′ Z ǫ+piδ−ǫ′ ǫ′σα′# Ad(ω)oftheretardednon-equilibriumGreenfunction[27]
X
G ∞
0
whereV¯ = V2+V2,andtheGreenfunctionGr (z)= I(V) = dω [f(w−µL)−f(w−µR)]
L R 0σ e
z−(E +U/2−σH/2)−V¯2 dǫρ(ǫ)/(z−ǫ) −1. In Xσ Z−∞
d p ×πA (ω)Γ (5)
thelimitofinfinitelylargeleads,thesingle-particlespec- dσ
(cid:2) R (cid:3)
trum remains unaltered, and these scattering states di- in such a scattering-states formulation even for finite
agonalize the Hamiltonian[15] (1) for U =0: U[15, 17, 28]. f(ω) denotes the Fermi function, G =
0
(e2/h)4Γ Γ /(Γ +Γ )2,Γ =r2πV¯2ρ(0),Γ=Γ +Γ
L R L R α α L R
Hi =H(U =0) = dǫǫγ† γ . (3) and πA (ω)=−ℑmGr (ω+iδ).
0 ǫσα ǫσα dσ dσ
α=XL,R;σZ Coulomb interaction: Expanding the operator nˆdσ in
the orbitals d yields two contributions: a density and
σα
Hershfieldhasshownthatthedensityoperatorforsucha a backscattering term: nˆd = nˆ0 + Oˆback, with nˆ0 =
σ σ σ σ
non-interactingcurrent-carryingquantumsystemretains r2d† d . The backscattering term reads
its Boltzmannian form[15, 18] even at finite bias: α α σα σα
P Oˆback = r r d† d +d† d (6)
e−β(Hi0−Yˆ0) σ L R σR σL σL σR
ρˆ0 = Tr e−β(Hi0−Yˆ0) , Yˆ0 = ασ µαZ dǫγǫ†σαγǫσα(4) and describes transitions(cid:16)between left and rig(cid:17)ht-movers.
X
Thistermvanishesinthetunnellingregime,whereeither
h i
The Yˆ operator accounts for the occupation of the left rL or rR vanishes.
0
We will include the full Coulomb interaction into our
andright-movingscatteringstates,andµ forthe differ-
α
theory in two steps. Since H0, defined as H0 =
ent chemical potentials of the leads. U U
Steady-stateNRG: InordertoapplytheNRGtosuch U2 σnˆ0σ−1 2, commutes with Yˆ0, the steady state
anopen quantum systems,the scattering states γ are densityoperatorρˆ evolvedintoρ˜ =exp[−β(Hi−Yˆ )]/Z
ǫασ (cid:0)P (cid:1) 0 0 0
discretizedona logarithmicenergymesh using the NRG with Hi = Hi +H0 proven by the arguments given in
0 U
discretization parameter Λ[4]. In contrary to a closed Ref.[18]. Oˆback canbeneglectedinthetunneling regime
σ
system, however, each of these single-particle states car- where ρˆ → ρ˜ . Then, the steady-state spectra is com-
0
ries a finite current. Even for asymmetric coupling, the pletely determined by a single effective orbital, and the
spectra of the right and left-movers remains symmetric, equilibrium spectral function is recovered.
and the total current vanishes always at zero bias. Hi marks the new starting point of our theory. The
Defining the creation operator for a fictitious left or full Hamiltonian H of the interacting model differs from
right-moving dσα-orbital d†σα = V¯ dǫ ρ(ǫ)[Gr0σ(ǫ + Hi by the additional backscattering terms. H does not
iδ)]∗γ† , the physical d-level can be decomposed into commutewithYˆ ,andtheanalyticalformofsteady-state
ǫσα R p 0
d† = r d† + r d† by inverting Eq. (2) and using density operator of the fully interacting problem is not
σ R σR L σL
r = V /V¯. For U = 0, the Hamiltonian is diagonal explicitlyknown[15,18]. Weobtainasolution[20,21,22]
α α
in the left and right-movers. We use these d -orbitals byevolvingρ˜ withrespecttothefullHamiltonianHinto
σα 0
as starting vector of the Householder transformation[4] its steady-state value ρˆ = lim e−iHtρ˜ eiHt. In the
∞ t→∞ 0
mappingthediscretizedscatteringstatescontinuumonto current-voltagerelation(5),thespectralfunctionA (ω)
dσ
two semi-infinite Wilson chains[4], as depicted in Fig. 1. for U = 0 is replaced by the non-equilibrium spectral
These chains are almost identical to standard Wilson function[28]calculatedwithrespecttoρˆ . Thedetailsof
∞
3
(a) (a) 2
0.3 0.3 VV==00..55**1100--44 R=1
VV==00..0011 1.5 R=10
VV==00..11 R=100
ω) 0.2 ρ(ω) 00..21 VVVVVVVV========01250125..55 G/G0 1 Rρ(=ω1,0V0=00)
ρ(
0 0.5
-1 -0.5 0 0.5 1
0.1 ω/Γ
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
V/Γ
0 (b) 2
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
ω/Γ
H=0.
(b) 1.5 H=0.1
H=0.2
ω) 00..23 ρ(ω) 000...231 VVVVVVVVVV==========0000100001........50155015*1*11100--44 2G in [e/h] 1 H=0.4
ρ( 0 VV==22 0.5
-1 -0.5 0 0.5 1
ω/Γ
0.1
0
-3 -2 -1 0 1 2 3
V/Γ
0 (c)
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1ω0/Γ1 2 3 4 5 6 7 8 9 10 1 S-NRG U=5, T=0.04
[T(V/2) +T(-V/2)]/2
0.8
h] Han,Heary PRL 2007
FIG. 2: (color online) Non-equilibrium spectral function for 2e/0.6
(a)asymmetricjunctionR=1atvariousvaluesoffinitebias n [20.4
voltage V, and (b) for a strongly asymmetric junction R = G i
1000. Theinsetsshowtheevolution oftheKondo-resonance. 0.2
Parameters: U =8, ǫ =−4 and T →0.
f 0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
V/Γ
this algorithm embedding the calculation of equilibrium FIG. 3: (color online) The differential conductance G =
spectral functions[29, 30] are published in Ref. [22]. dI/dV as function of the bias voltage (a) for different
asymmetry factors R, (b) for different magnetic field H =
Results: All energies are measured in units of
0,0.1,0.2,0.4 and R=1. Parameters: as in Fig. 2. (c) Com-
Γ = πV¯2ρ(0); a constant band width[4] of ρ(ω) =
parison between the results for U =5 from Ref. [16] and the
1/(2D)Θ(D−|ω|) was used with D/Γ = 10. The num-
NRGcalculation at T/Γ=0.04 and R=1 using z-averaging
ber of retained NRG states was Ns = 2200; a Λ = 4 over4 z-values[20, 21].
was chosen. The model lacks channel conservation: only
the total charge and z-component of the spin served as
quantum numbers. We defined R = Γ /Γ and always spectral function is less broadened and, therefore, G(V)
L R
keptΓ=Γ +Γ constant. The twochemicalpotentials decreases for large bias voltage. Asymptotically, G ap-
L R
µ were set to µ = −r2V and µ = r2V as function proaches the equilibrium t-matrix which is the exact re-
α L R R L
of the external source-drain voltage V consistent with a sult for R→∞ and T →0.
serial resistor model. Theeffectofanexternalmagneticfieldontothediffer-
Thenon-equilibriumspectralfunctionforasymmetric ential conductance is shown in Fig. 3(b). An increasing
junction is plotted for U = 8 and different bias V in magnetic field splits the zero-bias anomaly which is fur-
Fig. 2(a). Multiple backscattering events cause gain (or thersuppressedbythefinitebiasinasymmetricjunction.
lost) of single-particle excitation energy proportional to Thisfielddependence hasbeenusedinexperiments[2]as
theappliedbias. TheKondoresonanceisdestroyedwith hallmark for the Kondo physics at low temperatures.
increasing bias due to redistribution of spectral weight In Fig. 3(c), the NRG conductance for U = 5 is com-
towards higher energys. An onset of two weak peaks paredtotheresultofRef.[16]. Bothcurvesagreeforlow
in the vicinity of the two chemical potentials remains bias. The NRG result shows a weaker decay of the zero-
for |V| > Γ[16]. For large R ≫ 1 such backscattering bias anomaly with increasing bias with a less pronounce
processes are suppressed. The spectral function remains maximum at large bias. The symmetrized equilibrium
bias-independent. The Kondo resonance remains pinned t-matrix[4] is added for comparison as dashed line.
to µL → 0 as depicted in Fig. 2(b), and we recover the The more genericcaseofanasymmetric junction with
tunneling regime. respect with a relatively large local Coulomb repulsion
The differential conductance is plotted for different is plotted in Fig. 4. The differential conduction reflects
asymmetry ratios R in Fig. 3(a) using the same parame- thelackofsymmetryundersource-drainvoltagereversal.
tersasinFig.2. With increasingR,the non-equilibrium As depicted, the zero-bias peak vanishes with increasing
4
ing out Ref. [17]. This research was supported in parts
0.6
T=2*10-4 by the DFG project AN 275/6-1 and by the National
h] T=0.05
2n [e/0.4 TT==00..14 SWceienacceknFoowulneddgaetiosnupuenrcdoemrpGurtearntsuNpop.ortPbHyY0th5e-51N1I6C4,.
G i0.2 ForschungszentrumJu¨lich P.No. HHB000.
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
V
FIG.4: (coloronline)ThedifferentialconductanceGasfunc-
[1] M. A.Kastner, Rev.Mod. Phys. 64, 849 (1992).
tionofthebiasvoltagefordifferenttemperatures. Parameters
R=10, ǫ =−1.5 and U =12. [2] D. Goldhaber-Gordon, et al. Nature 391, 156 (1998).
f [3] W. G. van der Wiel, et al. Science 289, 2105 (2000).
[4] R.Bulla,T.A.Costi,andT.Pruschke,Rev.Mod.Phys.
80, 395 (2008).
temperature. [5] N. S. Wingreen and Y. Meir, Phys. Rev. B 49, 11040
Conclusion: A powerfulnew approachto the steady-
(1994).
state currents through nano-devices has been presented. [6] J. K¨onig and H. Schoeller, Phys. Rev. Lett. 81, 3511
We have introduced a NRG method based on scatter- (1998).
ing states to incorporate the correct steady state bound- [7] A. Rosch, J. Paaske, J. Kroha, and P. W¨olfle,
ary condition of current carrying systems. The steady- Phys. Rev.Lett. 90, 076804 (2003).
[8] R. Gezzi, T. Pruschke, and V. Meden, Phys.Rev.B 75,
state density operator[15] of a non-interacting junction
045324 (2007).
is evolved into the one of the interacting nano-device
[9] L. V. Keldysh,Sov. Phys.JETP 20, 1018 (1965).
using the TD-NRG[20]. We have established an accu- [10] A. Schiller and S. Hershfield, Phys. Rev. B 51, 12896
rate solution for the strong-coupling regime and calcu- (1995).
lated steady-state currents for arbitrary ratios R at fi- [11] S. Kehrein, Phys.Rev.Lett. 95, 056602 (2005).
nite bias. The tunneling regime is included as an ex- [12] G. Cuniberti, G. Fagas, and K. Richter, eds., Introduc-
act limit. Our approach does not suffer from any cur- ing Molecular Electronics, vol. 680 of Lecture Notes in
Physics (Springer, Berlin and Heidelberg, 2005).
rentreflectioninherenttonumericalsimulationsofclosed
[13] S. Welack, M. Schreiber, and U. Kleinekathoefer, J.
quantum systems[24]. We haveconcentratedonthe low-
Chem. Phys.124, 044712 (2006).
temperaturepropertiesofthenano-device,sincethecom- [14] J. E. Han, Phys.Rev.B 73, 125319 (2006).
bination of arbitrary bias, large Coulomb repulsion and [15] S. Hershfield, Phys.Rev.Lett. 70, 2134 (1993).
finitemagneticfieldremainsthemostdifficultregimefor [16] J. E. Han and R. J. Heary, Phys. Rev. Lett. 99, 236808
all perturbative methods. However, the NRG is equally (2007).
suitable to calculate the crossover from the low to the [17] A. Oguri, Phys. Rev.B 75, 035302 (2007).
[18] B.DoyonandN.Andrei,Phys.Rev.B73,245326(2006).
high-temperature regime as demonstrated in Fig. 4. An
[19] P. Mehta and N. Andrei, Phys. Rev. Lett. 96, 216802
experimentalhallmark[2]forKondophysics,thesplitting
(2006).
of the zero-bias Kondo peak with increasing magnetic [20] F.B.AndersandA.Schiller,Phys.Rev.Lett.95,196801
field,iscorrectlydescribedbyourapproachforarbitrary (2005).
temperature, bias and field strength. [21] F. B. Anders and A. Schiller, Phys. Rev. B 74, 245113
This theory can be extended to more complicated (2006).
multi-orbital models. Eq. (5) must be modified and re- [22] F. B. Anders, J. Phys.: Condens. Matter 20, 195216
(2008).
quires more complex correlation functions. Since single-
[23] U. Schollw¨ock, Rev.Mod. Phys. 77, 259 (2005).
particle scattering states can always be obtained ex-
[24] P. Schmitteckert,Phys. Rev.B 70, 121302(R) (2004).
actly, the construction of the Wilson chain parameters
[25] E.Lebanon,A.Schiller, andF.B.Anders,Phys.Rev.B
is straight forward using the corresponding expansion of 68, 155301 (2003).
the local degrees of freedom and combining it with the [26] T. Enss, et al. Phys. Rev.B 71, 155401 (2005).
transformationusedfornon-constantdensityofstates[4]. [27] T. A.Costi, Phys.Rev.B 55, 3003 (1997).
I acknowledgestimulating discussions with N. Andrei, [28] Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512
(1992).
J.Bauer,G. Czycholl,Th. Costi,M. Jarrell,H. Monien,
[29] R. Peters, T. Pruschke, and F. B. Anders,Phys. Rev. B
A. Millis, J. Kroha, J. Han for providing the data of 74, 245114 (2006).
Ref. [16], Th. Pruschke, A. Schiller, P. Schmitteckert, [30] A. Weichselbaum and J. von Delft, Phys.Rev.Lett. 99,
K. Schoenhammer, A. Weichselbaum, G. Uhrig and the 076402 (2007).
KITP for its hospitality, at which some of the work has
been carried out. I also thank T. Novotny for point-