By Supervisor Urbashi Satpathi Dr. Prosenjit Singha De0

Experimental background

Motivation

PLDOS, Injectivity, Emissivity, Injectance, Emitance, LDOS, DOS

Paradox and its practical implication

Schematic description of experimental set up

[ R. Schuster, E. Buks, M. Heiblum, D. Mahalu, V. Umansky and Hadas Shtrikman , Nature 385, 417 (1997) ]

Colle

ctor

vol

tage

, VCB

Analyticity

Hilbert Transform relates the amplitude and argument

What information we can get from

phase shift ?

Apparently does not follow Friedel sum rule (FSR)

However if carefully seen w.r.t Fano resonance (FR) can be understood from FSR Besides there is a paradox

at FR that can have tremendous practical implication.

Larmor precision time (LPT)

Injectivity

srUe

srUe

ss

ir

4,,

srUe

srUe

ss

irI

41,

r

,rI

Why injectivity is physical?

Local density of states

Density of states

This is an exact expression.

, 4

1, srUe

srUe

ss

iEr

,

3

41 s

rUes

rUes

srdi

E

In semi-classical limit

Hence ,

, is FSR (semi classical).

dEd

rUerd 3

, 41 s

dEds

dEds

si

E

sdE

ds21

[ M. Büttiker, Pramana Journal of Physics 58, 241 (2002) ]

and,

is semi classical injectance.

sdEds

dEds

si

EI41

sdE

ds2

21

[ C. R. Leavens and G. C. Aers, Phys. Rev. B 39, 1202 (1989), E. H. Hauge, J. P. Falck, and T. A. Fjeldly, Phys. Rev. B 36, 4203 (1987) ]

, i kx tin kx t a e dk

, ti kx k x t tsc kx x t t t k a e dk

Incident wave packet

Scattered wave packet

i.e. in semi classical case, density of states is related to energy derivative of scattering phase shift.

td EdE

Considering no reflected part (E>>V), and no dispersion of wave packet,

is stationary phase approximation.

tkx k x t t K

,2cW

V y for y

0 ,2W

for y

The confinement potential,

The scattering potential,

is symmetric in x-direction

),(),( yxVyxV dd

The Schrödinger equation of motion in the defect region is,

In the no defect region,

,

where,

2 2 2

2 2 , , ,2 c d

e

d dV y V x y x y E x y

m dx dy

2 22 2 2

2 22n

ee

knEmm W

)()()(2 2

22

xcExcdxd

m nnne

2

222

2 Wmn

en

and, , is the energy of incidence.

2sin2)( WyWn

Wyn

yyyVdyd

m nnnce

2

22

2

For symmetric potentials,

1

1n n

xikenm

xiknm

en k

eSexc nn

1

1n n

xikonm

xiknm

on k

eSexc nn

For,

where,

2

22

2

22

24

2 WmE

Wm ee

axforet

axforerexc

xik

xikxik

,

,

1

11

~

11

~

111

2

,,2

1111~

111111

~

11

eoeo SStandSSr

w

axaforyxyxyxon

en

,

2,,,

aikrn

eorn

eomn

aik

mrm

eorm

nm eiFSeiF

1

eoeo iGiarceo eeS 2cot211

where,

and,

At resonance,

eonmn

eocc

nm

eom

eoeo FiFFFG 11

2,2111 1

r

r

i

i

er

eit

cos

sin~

11

~

11

oe

r

oe

q

po

e

r

etr

krm

dEdt

dEdrEI

sin

21

2

22

2

xcxcxcon

en

n

The potential at X,

d jV x y y

1

1,12

2

2

1 1,,k

k

W

W

EEyxdydxEI

1,, yx Internal wave function

1,1 kE Modes of the quantum wire

Injectance from wave function is,

,

where,

, and

e m m

mm

e

ee

nm

mn

mn

ki

kki

Er

221

2

ErEt

ErEt

mmmm

mnmn

1

sin sin2 2mn i i

m W n Wy y

W W

m mik

02

11

e e

ee

Injectance from wave function is,

Semi classical injectance is,

,

,

4

214

3

213

11

1tt

hvEI

4

224

3

223

22

1tt

hvEI

dEd

tdEd

tdEd

rdEd

rEI ttrr 12111211 212

211

212

2111 2

1

dEd

tdEd

tdEd

rdEd

rEI ttrr 22212221 222

221

222

2212 2

1

11

sin21

11r

e

krm

22

sin22

22r

e

krm

EI R1

EI R2

13 and

.45iy W

and15 .45iy W

15 and

.45iy W

.45iy W15 and

15 .45iy Wand

.45iy W15 and

There is a paradox at Fano resonance

The semi classical injectivity gets exact at FR

Useful for experimentalists

1. Leggett's conjecture for a mesoscopic ring P. Singha Deo Phys. Rev. B {\bf 53}, 15447 (1996).

2. Nature of eigenstates in a mesoscopic ring coupled to a side branch. P. A. Sreeram and P. Singha Deo Physica B {\bf 228}, 345(1996.

3. Phase of Aharonov-Bohm oscillation in conductance of mesoscopic systems. P. Singha Deo and A. M. Jayannavar. Mod. Phys. Lett. B {\bf 10}, 787 (1996).

4. Phase of Aharonov-Bohm oscillations: effect of channel mixing and Fano resonances. P. Singha Deo Solid St. Communication {\bf 107}, 69 (1998).

5. Phase slips in Aharonov-Bohm oscillations P. Singha Deo Proceedings of International Workshop on $``$Novelphysics in low dimensional electron systems", organized byMax-Planck-Institut Fur Physik Komplexer Systeme, Germanyin August, 1997.\\Physica E {\bf 1}, 301 (1997).

6. Novel interference effects and a new Quantum phase in mesoscopicsystems P. Singha Deo and A. M. Jayannavar, Pramana Journal of Physics, {\bf 56}, 439 (2001). Proceedings of the Winter Institute on Foundations of Quantum Theoryand Quantum Optics, at S.N. Bose Centre,Calcutta, in January 2000.

7. Electron correlation effects in the presence of non-symmetry dictated nodes P. Singha Deo Pramana Journal of Physics, {\bf 58}, 195 (2002)

8. Scattering phase shifts in quasi-one-dimension P. Singha Deo, Swarnali Bandopadhyay and Sourin Das International Journ. of Mod. Phys. B, {\bf 16}, 2247 (2002)

9. Friedel sum rule for a single-channel quantum wire Swarnali Bandopadhyay and P. Singha Deo Phys. Rev. B {\bf 68} 113301 (2003)

10. Larmor precession time, Wigner delay time and the local density of states in a quantum wire. P. Singha Deo International Journal of Modern Physics B, {\bf 19}, 899 (2005)

11. Charge fluctuations in coupled systems: ring coupled to a wire or ring P. Singha Deo, P. Koskinen, M. Manninen Phys. Rev. B {\bf 72}, 155332 (2005).

12. Importance of individual scattering matrix elements at Fano resonances. P. Singha Deo} and M. Manninen Journal of physics: condensed matter {\bf 18}, 5313 (2006).

13. Nondispersive backscattering in quantum wires P. Singha Deo Phys. Rev. B {\bf 75}, 235330 (2007)

14. Friedel sum rule at Fano resonances P Singha Deo J. Phys.: Condens. Matter {\bf 21} (2009) 285303.

15. Quantum capacitance: a microscopic derivation S. Mukherjee, M. Manninen and P. Singha Deo Physica E (in press).

16. Injectivity and a paradox U. Satpathy and P. Singha Deo International journal of modern physics (in press).