Conductance in systems ofTomonaga-Luttinger Liquid systems with

resistances

Abhiram Soori †

Centre for High Energy Physics,Indian Institute of Science, Bangalore

Work done with: Prof. Diptiman Sen

PRESENTED AT-Workshop on Low Dimensional Quantum Systems

Harish-Chandra Research Institute, Allahabad

† Email: abhiram@cts.iisc.ernet.in

10th Oct. 2011

Outline

1 Introduction

2 Bosonization

3 Dissipation

4 Scattering approach

5 Three wire junctions

6 Parallel combination

7 Coupled wires

Introduction

• Quantum wire is a narrow channel where the motion ofelectrons is restricted to one spatial dimension.

• Generic examples of such systems are nanostructures.

Figure: SEM micrograph of a suspended silicon quantum wire

Introduction . . .

• For noninteracting electrons in 1-D ballistic quantum wires,the conductance per spin is e2/h (van Wees et. al.PRL-1988).

• This remains true even when electron-electron interactionsare introduced in the clean channel that is connected toFermi-liquid-leads (FLL) [Safi-Schulz, PRB-1995 andMaslov-Stone, PRB-1995 ].

• Study of power dissipation in 1-D interacting systems hasbeen of considerable interest.

• For example, Caldeira and Legget (Ann. Phys. - 1983)studied the problem of Quantum tunneling in a dissipativesystem.

Bosonization

• In 1-D, the fermionic Hamiltonian with interactions can becast in terms of Bosonic field φ.

• And the physical quantities of interest can be calculated inthe bosonic language.

• Bosonic Lagrangian density for spinless electrons in aclean wire-L = 12vK (∂tφ)2 − v2K (∂xφ)2, K and v can vary with x .

• K = 1 corresponds to noninteracting electrons while K < 1to repulsive and K > 1 to attractive interactions.

• The electron charge density n and current j are given byn = −e∂xφ/

√π and j = e∂tφ/

√π.

Introducing dissipation in the wire

FLLFLL

• Dissipation is introduced through a Rayleigh dissipationfunction

∫∞−∞ dx F = 12

∫∞−∞ dx r j

2, the resistivity rand current j can vary with x .

• The function F contributes to the equation of motion as:∂t(δL/δ∂tφ) + ∂x(δL/δ∂xφ)− δL/δφ + δF/δ∂tφ = 0.

1vK

∂2t φ − ∂x (vK∂xφ) +

2e2

hr ∂tφ = 0. (1)

Scattering approach to σdc

• To describe a resistance connected to FLL, we consider asimple model with K = 1 (noninteracting electrons) andv = vF everywhere. Resistivity r(x) = 0 in the leads (for|x | > a) and

∫ a−a dx r(x) = R.

Scattering Approach (Safi-Schulz, PRB-1995)• In this approach, a plane wave in φ is incident on the

dissipative region from the left lead and its transmissionamplitude to the right is determined. This transmissionamplitude in some limit gives us the DC conductance.

Scattering approach to σdc

• We write down the scattering solution φk (x , t) = fk (x)e−iωtto Eq. (1): fk = eikx + sk e−ikx for x ≤ −a, and fk = tk eikxfor a ≤ x .

• Solving Eq. (1) for tk and using the relationσdc =

e2h limk→0+ tk gives us the DC conductance:

σdc =1

he2 + R

(2)

• It is clear from the above expression that resistance Radds in series with the contact resistance h/e2.

• This result can also be obtained using the Green’s functionmethod (Maslov-Stone, PRB-1995).

Time evolution of a pulse

• A Gaussian charge-density profile incident on theresistance is time evolved numerically using Eq. (1)

• SHOW the CLIP

Time evolution of a pulse

• A Gaussian charge-density profile incident on theresistance is time evolved numerically using Eq. (1).

• The width of the reflected pulse (= 4a) is equal to twice thelength of the dissipative region(= 2a).

• This implies that the pulse gets reflected from each point ina dissipative region.

• In our model, dissipation happens exactly in the channel.

Three wire junction

r10

r20

r30

FLL

FLL FLL

• Conductance matrix G for a three-wire junction relates thecurrents in the wires to the voltage’s as Ii =

∑

j GijVj .• Incoming and outgoing currents at the junction are related

by M-matrix as Iout = M · I in.

Y-junction

• There is no dissipation at the junction.• The conditions: current conservation, unitarity and

zero-current for zero-bias imply that each row and columnof the M-matrix should add up to 1 and M-matrix has to beorthogonal.

• The possible M matrices are restricted to two classesparameterized by a single parameter θ: det(M1) = 1 anddet(M2) = −1,

M1 =

a b cc a bb c a

and M2 =

b a ca c bc b a

, (3)

where a = (1 + 2 cos θ)/3 andb(c) = (1 − cos θ + (−)

√3 sin θ)/3.

Time reversal symmetry broken at Y-junction

• M1 matrix describes a junction with broken Time reversalsymmetry (a magnetic flux passing through the junctioncan produce this effect).

• Here G depends on KW (in contrast with the single-wirecase), θ and R.

• Gij ’s plotted as a function of KW for θ = 2π/3 and R = 0:

Large Ri limit

• Also, M1-matrix is invariant under the permutation1 → 2 → 3 → 1.

• And in the limit of the resistances Ri → ∞, we get theconductance matrix that agrees with the G obtained for aclassical circuit using Kirchoff’s circuit laws :

G =1

R1R2 + R2R3 + R3R1

×

−R2 − R3 R3 R2R3 −R1 − R3 R1R2 R1 −R1 − R2

.

Time reversal symmetric Y-junction

• M2 matrix describes a junction with Time reversalsymmetry.

• Here G is independent of KW (similar to the single-wirecase).

• However G depends on θ and R:

G = − e2

π

3(I − M2)D

,

where D = 2(̺1 + ̺2 + ̺3) + cos θ(̺1 + ̺2 − 2̺3)−√

3 sin θ(̺1 − ̺2),and ̺i = 1 + (e

2/π)Ri .

• Even in the limit Ri → ∞, G depends on θ.

Power dissipation

• There is no power dissipation exactly at the junction.• Power dissipation occurs only at the resistive patches and

in the leads due to the contact resistance.

• The power dissipation at the contact resistance occurs dueto the energy relaxation of the electrons in the leads(reservoirs).

• For a three-wire junction, we can define the powerdissipated in two equivalent ways as follows

P = −3

∑

i=1

Vi Ii , (4a)

and P =3

∑

i=1

I2i(

Ri +h

2e2

)

. (4b)

Power dissipation

• Dependence of P on KW , θ are similar to that of G.• Zero-bias ⇒ zero-current tells us that power dissipated

should depend on at most two linear combinations of V1,V2 and V3.

• Surprisingly, P depends only on one linear combination ofthe Vi ’s.

Parallel combination of resistances

FLLFLL ML MR

R2

R3

2

3

12

3

1

−L2

L

2

• Effective resistance of the parallel combination:R|| =

1σdc

− he2 behaves in surprising ways.• R|| in general does not go as R2R3/(R2 + R3) - the

classical result.

• Only when both the junctions are described by M1 withθL = −θR, we recover this classical result.

ML(θL) MR(θR) Expression for R||M1(θ) M1(−θ) R2R3/(R2 + R3)θ 6= 0

M1(θL) M1(θR) Depends on θL, θR and KWθL 6= −θRM1(θL = 0) M2(θR) ∞

M1(θL) M2(θR) Depends only on θR = θM2(θ) M2(θ) Depends only on θ

M2(θL 6= θR) M2(θR) ∞Table: Effective resistance R|| for different choices of ML and MR.

Parallel combination ...

• When both the junctions are described by M2 withsymmetry between the arms 2 and 3, we get the effectiveresistance as R|| = (R2 + R3)/4.

• This is a surprising result since classically one wouldexpect that when R3 → ∞ all the current to pass throughR2 making the effective resistance R2.

• We understand this paradoxical observation.

FLLFLLML MRR22

3

1 2

3

1

Coupled wires

φ1(x1)

φ2(x2)

r12

Figure: Two wires coupled by a Rayleigh-dissipation function.

• Two TLL-wires are coupled by Rayleigh dissipationfunction-

F = 12

∫ ∞

−∞dx [r11 j

21 + r22 j

22 + 2r12 j1j2], (5)

non-zero r12 (in the region |x | < L/2) is responsible forfinite transconductance σ12 = j1/V2.

Coupled wires . . .

• Non-zero transconductance is reminiscent of thephenomenon of Coulomb drag between two wires.

• Coulomb drag is a phenomenon where voltage is inducedin one wire when a current flows in another wire coupledby density-density interaction.

• In our model, the Rayleigh dissipation function couples thecurrents in the two wires resulting in finitetransconductance.

Summary

• We have combined Tomonaga-Luttinger liquid theory withthe concept of Rayleigh dissipation function to develop aphenomenological formalism to study the effect of resistiveregions in a quantum wire.

• Using the M-matrix, we have extended the analysis tothree-wire junction and a parallel combination ofresistances.

• Thus we have generalized the well-known results ofSafi-Schulz and Maslov-Stone to include systems withjunctions and resistances.

• Further, we have demonstrated that the concept ofRayleigh dissipation function can be extended to studycoupled two-wire systems.

References and Acknowledgements

Important References

1 A. Soori and D. Sen, EPL, 93, 57007 (2011).

2 A. Soori and D. Sen, PRB, 84, 035422 (2011).

Acknowledgements• Thanks to the organisers for giving me an opprtunity to

speak in this Workshop.

• Thanks to Dr. Sourin Das and Prof. Sumathi Rao forstimulating discussions.

• Thanks to Abhishek Bhat, CGPL, IISc for help with thenumerics and in preparation of the clip.

• Thanks to CSIR and DST for funding.

THANK YOU

Main PartIntroductionBosonizationDissipationScattering approachThree wire junctionsParallel combinationCoupled wires