by supervisor urbashi satpathi dr. prosenjit singha de0
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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).
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