By Supervisor Urbashi Satpathi Dr. Prosenjit Singha De0

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).