conductance in systems of tomonaga-luttinger liquid ...cmworksh/ldqs/talks/abhiram.pdf · abhiram...
TRANSCRIPT
-
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: [email protected]
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