Frühjahrstagung  Dresden 2011

Negative differential conductance with symmetric set-up ?

Andrea Donarini, Abdullah Yar, and Milena Grifoni

Institut für Theoretische Physik

Universität Regensburg

Single electron transistor

Small System size+ weak System‐Lead Tunnelling coupling

Strong e‐einteraction

on the System

Single electroncontrol

Vg

μS μDμN

μN‐1

μN+1

μN = E N ‐ E N‐1The chemical potential of the System with N particles

Vb

Source Drain

Gate

System

Vb

Vg

Single electron transistor

Small System size+ weak System‐Lead Tunnelling coupling

Strong e‐einteraction

on the System

Single electroncontrol

Vg

μS μDμN

μN‐1

μN+1

Vb

Source Drain

Gate

System

Vb

Vg

Vg

G

N‐1 N N+1

Single electron transistor

Small System size+ weak System‐Lead Tunnelling coupling

Strong e‐einteraction

on the System

Single electroncontrol

Vg

μS μDμN

μN‐1

μN+1

Vb

Source Drain

Gate

System

Vb

Vg

Vb

IdIdVb

Negative Differential Conductance

μS μDμN

μN‐1

μN+1

Vb

I

Negative differential conductance (NDC) is usually associatedwith a strong asymmetry in the coupling to the leads 

dIdVb

NDC

The free energy formulation

Particle Number

Free

 Ene

rgy

NN‐1 N+1

N‐1 N N+1Vg

Vb

AD, G Begemann, Milena GrifoniPRB, 82, 125451 (2010)

F = H – μ0N    Free energy

Source transition

Drain transition

The free energy formulation

Particle Number

Free

 Ene

rgy

NN‐1 N+1

N‐1 N N+1Vg

Vb

F = H – μ0N    Free energy

Source transition

Drain transition

AD, G Begemann, Milena GrifoniPRB, 82, 125451 (2010)

NDC with symmetric set-upFree

 Ene

rgy

N N+1

N N+1

Vg

Vb

A

B

The dynamics is evaluated with themaster equation: in the stationary limit

PN,0

A

B

PN+1,0 PN+1,1 Current

1/2 1/2 0

1/3 1/3 1/3

Γ01

Γ00

Γ00/2

1/3(Γ00 + Γ01)

NDC  if IB < IA

Γ01 < Γ00/2

Role of the degeneraciesFree

 Ene

rgy

N N+1

N N+1

Vg

Vb

A

BΓ01

Γ00

A

B

PN,0 PN+1,0 PN+1,1 Current

1/3 1/3 0

1/5 1/5 1/5

2Γ00/3

2/5(Γ00 + Γ01)

NDC  if IB < IA

Γ01 < 2Γ00/3

The dynamics is evaluated with themaster equation: in the stationary limit

x 2

x 2

Role of the degeneraciesFree

 Ene

rgy

N N+1

N N+1

Vg

Vb

A

BΓ01

Γ00

A

B

PN,0 PN+1,0 PN+1,1 Current

0 Γ00

(Γ00 + Γ01)

NDC  if IB < IA

Γ01 < Γ00

The dynamics is evaluated with themaster equation: in the stationary limit

n x

x m

x m

n+m1

n+m1

n+2m

1

n+2m

1

n+2m

1n+mnm

n+2mnm n+m

m

An example: a suspended CNTFree

 Ene

rgy

4N 4N+1

x 4

x 4

x 8

1 x

1 x

2 xћω

ћω

The transition rates are proportional to product of Frack‐Condon coefficients

∏=Γn

nij jiFC );,()( λ

A. Yar, AD, S. Koller, and M. Grifoni, arXiv:1101.3892 (2011) 

An example: a suspended CNTFree

 Ene

rgy

4N 4N+1

x 4

x 4

x 8

1 x

1 x

2 x

λ= 0.68 λ= 0.83 λ= 1.18

A. Yar, AD, S. Koller, and M. Grifoni, arXiv:1101.3892 (2011) 

Conclusions

Transport through a single electron transistor is conveniently described bytransition between many‐body states in the free energy diagram

The negative differential conductance is the result of a redistribution of probabilities between fast and slow channels thus: 

It is possible also in a completely symmetric set‐up .

It is related to the degeneracies of the many‐body

Suspended carbon nanotube quantum dots exhibits the NDC presentedhere due to the interplay between Franck‐Condon coefficients and spin/pseudospin degeneracies.

Thanks for your attention !

Particle Number

Free

 Ene

rgy

NN‐1 N+1