Post on 23-Jan-2016
El Método de la Matrix de Transferencia
Pedro Pereyra PadillaArea de Física Teórica y Materia Condensada
Universidad Autónoma Metropolitana-Azcapotzalco, México D.F.
Resumen
Presentaremos una introducción al Método de la Matriz de Transferencia (MMT) y algunas aplicaciones en la teoría del transporte electrónico cuántico y en la opto-electrónica
INDICEI. Introduccíon
a. La Matriz de Transferencia (MT) y su relación con la matriz Sb. La MT de la barrera rectangular y el pozo cuántico. Efecto tunel y
cuantización
II. La Teoría de Sistemas Periódicos Fínitos. a. Estructura de bandas (eigenvalores y eigenfunciones). Aproximación
de masa efectivab. Semiconductores, dispositivos opto-electrónicos, heteroestructuras,
(láseres).
III. Tiempo de tunelaje. Paquetes Gaussianos en superredes ópticas
IV. Dinámica del spín en superredes magnéticas
V. Conclusiones
El Método de la Matrix de Transferencia
Pedro Pereyra PadillaArea de Física Teórica y Materia Condensada
Universidad Autónoma Metropolitana-Azcapotzalco, México D.F.
OVERVIEW
1. From about 1930 to 1993, the tunneling time has been a controversial issue, because of polemic theoretical results: a) superluminal velocities ? (MacColl 1932)b) Hartman effect (1962). These results (obtained using the phase time) were strongly
questioned because of possible conflict with causality and the special relativity.
As a consequence, besides the phase time, a number of other TT definitions and formulas appeared in the literature, accompanied by intense debate.
2. Since 1993, experiments have shown evidences of superluminality and the striking Hartman effect. It has been shown also that the phase time description agrees extremely well. Nevertheless, old theoretical approaches remain.
Introducción
x
V(x)
0 a
This was not accepted because:
i) Possible violation of Causality Principle
ii) Violation of Special Theory of Relativity
•In THE 50’s Eisenbud -Wigner & Bohm introduced the phase time
t
x
i =|i| eii
ri
titeit i
V(x)
t1 t2
x
i =|i| eii
ri
titeit i
t
V(x)
x1 x2
t1 t2
1 = k x1 + w t1 = 1 +t
t = 2 –1
t = k (x2- x1)- w (t2- t1)
2 1t t t
2 = k x2 + w t2
21
The phase time
x
growing a the tunneling time tends to a limiting value
i
ri
titeit i a
V(x)
other TT definitions and formulas appeared in the literature, accompanied by intense debates.
• the dwelling time (1960)
• the Larmor time Baz (1966): the spin precesion with constant frequency w, would allow, in principle, to measure the spin component x and deduce the time
spent
x
V(x)
zy
xxB
0lim x
w
x
of an oscillating potential
V(t) = V0 + V1cos wt
The incoming particles interchange energy
EE + h
=md/h
E + h
E - h E - hE
V(t)
• the Buttiker-Landauer time defined as the inverse of the characteristic frequency
h/md/
d
experiment suggests superluminal velocities !!!
tx s v = c
300 500 700 900
1
2
3
t
v
nm
[fs]
Phase time
experiment
?????¿¿¿
x
Eil
Hil
kil
Esl
Hsl
ksl
Eil
Hil
kil
nH=2.22 nL=1.41
lH lL
lS
En el experimento de Steinberg et al. utilizaron la estructura H(LH)5 que alterna óxido de titanio (H) con silica (L)
x
Eil
Hil
kil
Esl
Hsl
ksl
Eil
Hil
kil
nH=2.22 nL=1.41
lH lL
lS
En el experimento de Steinberg et al. utilizaron la estructura H(LH)5 que alterna óxido de titanio (H) con silica (L)
¿Cómo son nuestras predicciones?
x
Eil
Hil
kil
Esl
Hsl
ksl
Eil
Hil
kil
nH nL
lH lL
lS
Necesitamos obtener la matriz de transferencia de este sistema
* *
T TT sa n aH
T T
M M M M
tph E
ti
T T e
x
x
Er
El
,
Hr
Hlkl
kr
Er
Hr kr
El
Hlkl
1 jj j j j jr j jl
j j
ka b
H k E E E
j j jr j jla b E E E
1 1 1 1 1 2 2 2 2 2cos cosr l r la E b E a E b E
1 21 1 1 1 2 2 2 2
1 2r l r l
n na E b E a E b E
c c
1 1 2 21 1 1 1
1 1 2 21 1 1 1
1 1 2 2
cos cos cos cosr r
l l
a E a En n n n
b E b E
x
x
Er
El
,
Hr
Hlkl
kr
Er
Hr kr
El
Hlkl
1 2 1
2 2 1 12 1 1 2 2 1 1 2
2 2 1 12 1 1 2 2 1 1 22 2
cos cos cos cos1cos cos cos cos2 cos
r r
l l
a E a En n n n
b E b En n n nn
1 1 2 21 1 2 2
1 1 2 21 1 2 2
1 1 2 2
cos cos cos cosr r
l l
a E a En n n n
b E b E
Con esta información se pueden obtener los coeficientes de Fresnal
x
x
Er
El
,
Hr
Hlkl
kr
Er
Hr kr
El
Hlkl
1 2
1 2
0
1
2 2 1 12 1 2 1
2 2 1 12 1 2 12
1
2r r
l l
a E a En n n n
b E b En n n nn
1 1 2 21 1 2 2
1 1 2 21 1 2 2
1 1 2 2
cos cos cos cosr r
l l
a E a En n n n
b E b E
¿Cuál es la matriz de transferencia de la celda unitaria?
x
E1r
H1rk1r
E1l
H1l
k1l
E4r
H4rk4r
…
nH=2.22 nL=1.41
lH lL
x
E4l
H4l
k4l
En la superred, la matriz de transferencia de una celda es
* *M
La matriz de transferencia del sistema completo es
T sa n aHM M M M
x
E1r
H1rk1r
E1l
H1l
k1l
E4r
H4rk4r
x
E4l
H4l
k4l
x
Eil
Hil
kil
Esl
Hsl
ksl
Eil
Hil
kil…
* *
T TT sa n aH
T T
M M M M
* *
n nn
n n
M
n= pn-1
npn pn-1
pn = Un(r)
x
Eil
Hil
kil
Esl
Hsl
ksl
Eil
Hil
kil…
* *
T TT sa n aH
T T
M M M M
tph E
ti
T T e
phase time
experiment
s
tph E
Efecto Hartman
S
H
E
n1 n2
The purpose is to turn into the fundamental laws of nature, i.e. the Maxwell equations, and let them determine the time evolution of wave packets. We will then see whether the EM wave packet motion is superluminal or not.
Cómo es la evolución de un paquete electromagnético Gaussiano?
For a given Optical Superlattice (OSL),
1000 1500 2000 2500 3000
0,0
0,2
0,4
0,6
0,8
1,0
[nm]
Tra
ns.
Co
eff
.
1000 1500 2000 2500 30000
2
4
6
8
10
12
14
16
18
Ph
ase
Tim
e
[fs]
we have
the space-time evolution of a Gaussian WP depends on o and on the WP width k,
1000 1500 2000 2500 3000
0,0
0,2
0,4
0,6
0,8
1,0
[nm]
Tra
ns.
Co
eff
.
1000 1500 2000 2500 30000
2
4
6
8
10
12
14
16
18
Ph
ase
Tim
e
[fs]
)()( ),(),(2 tzkikk oo ekzEedktz
we will choose in the pbgap and in a resonant region
200 400 600 800 1000 12000
2
4
6
8
10
12
14
16
18
2.32 fs
7.2 fs
n2
n1
phase
tim
e
[fs]
[nm]
vg = c
vg < c
vg > c
)()( ),(),(2 tzkikk oo ekzEedktz
To determine the wave packet at any point and time
we use the Theory of Finite Periodic Systems (TFPS).
For z < 0
zik
T
Tzik eekzE *
*
),(
For z > L
ikz
T
ekzE*
1),(
Here and T are matrix elements of the global Superlattice Transfer matrix
The formalism
nInInRT n
n
n
ni
n
n
n
ni
0
1
1
0
0
1
1
0
22
nInIT n
n
n
ni
n
n
n
ni
0
1
1
0
0
1
1
0
22
)()( 1*
RnRnn UU
)(1 Rnn U
1122
2
1
1
222 sin
2cos likelk
n
n
n
nilk
222
1
1
2 sin2
lkn
n
n
ni
)()( ),(),(2 tzkikk oo ekzEedktz
For 0 < z < L
Replacing these fields into
and integrating, we have the wave packet for any value of z and t.
Phys. Rev. Lett, 80 (1998) 2677, Ann. Phys. 320 (2005) 1, Phys. Rev. E 75 (2007)
)()( ),(),(2 tzkikk oo ekzEedktz
For 0 < z < L
*
*
111 11),(),(
T
Tooj n
n
n
njzAkzE
*
*
11
11),(T
Too
n
n
n
njzB
Replacing these fields into
and integrating, we have the wave packet for any value of z and t.
Phys. Rev. Lett, 80 (1998) 2677, Ann. Phys. 320 (2005) 1, Phys. Rev. E 75 (2007)
-16000 0 16000
-1
0
1
2
-16000 0 16000
-1
0
1
-1
0
1
-2
-1
0
1
2
-8000
tb = 2 z
0 / v
g+
(d) z
0+L
t0 = 0
(a)
-z0=-8140 nm
L = nlc= 1221 nm
n = 6
0 = 735 nm
Re
E z
,t (
arb
itra
ry u
nit
s)
ta = z
0 / v
g
ta = 2.71x10-14 s
(b)
t = 2.94 x 10-14 s
= 2.32 fs t = z0 / v
g+
(c)
When it leaves the OSL?
The WP peak touches the OSL at
ta = zo /c
tl = ta +
tb = ta +
Can we determine exactly how much time needs the electromagnetic wave packet to cross the superlattice?
When it returns to -zo or moves to zo +L ?
-16000 0 16000
-1
0
1
2
-16000 0 16000
-1
0
1
-1
0
1
-2
-1
0
1
2
-8000
tb = 2 z
0 / v
g+
(d) z
0+L
t0 = 0
(a)
-z0=-8140 nm
L = nlc= 1221 nm
n = 6
0 = 735 nm
Re
E z
,t
(arb
itra
ry u
nit
s)
ta = z
0 / v
g
ta = 2.71x10-14 s
(b)
t = 2.94 x 10-14 s
= 2.32 fs t = z0 / v
g+
(c)
Superluminal motion
The WP peak touches the OSL at
ta = z o/c = fs
tl = ta + ?
tb = ta + ?
fs32.2
t
p
If is the phase time
where is at
Where is at
-16000 0 16000
-1
0
1
2
-16000 0 16000
-1
0
1
-1
0
1
-2
-1
0
1
2
-8000
tb = 2 z
0 / v
g+
(d) z
0+L
t0 = 0
(a)
-z0=-8140 nm
L = nlc= 1221 nm
n = 6
0 = 735 nm
Re
E z
,t
(arb
itra
ry u
nit
s)
ta = z
0 / v
g
ta = 2.71x10-14 s
(b)
t = 2.94 x 10-14 s
= 2.32 fs t = z0 / v
g+
(c)
Superluminal motion
-16000 0 16000
-1
0
1
2
8000 10000
vg = L/
pvg = c
z (nm)
tb = 5.65 x 10-14 s
tb = 2 z
0 / v
g+
(e)
fs2.32 fs 4.07 c
Lv
?
t
Where will be the WP if vgis NOT
vg = L/p
but vg = c ?
Since
-100000 0 100000-0.00020
-0.00015
-0.00010
-0.00005
0.00000
0.00005
0.00010
0.00015
vg
x0
n =6
7.2 fs7.2 x 10-15 s
400nmt=0 s
x-100000 0 100000
-0.00020
-0.00015
-0.00010
-0.00005
0.00000
0.00005
0.00010
0.00015
x
-100000 0 100000-0.00020
-0.00015
-0.00010
-0.00005
0.00000
0.00005
0.00010
0.00015
x
-100000 0 100000-0.00020
-0.00015
-0.00010
-0.00005
0.00000
0.00005
0.00010
0.00015
x
-100000 0 100000-0.00020
-0.00015
-0.00010
-0.00005
0.00000
0.00005
0.00010
0.00015
x
-100000 0 100000-0.00020
-0.00015
-0.00010
-0.00005
0.00000
0.00005
0.00010
0.00015
x0
n =6
7.2 fs7.2 x 10-15 s
t=x0 / v
g t = 2.7x10-14s
x
-100000 0 100000-0.00020
-0.00015
-0.00010
-0.00005
0.00000
0.00005
0.00010
0.00015
x0
7.2 fs7.2 x 10-15 s
t=x0 / v
g+/2
x-100000 0 100000
-0.00020
-0.00015
-0.00010
-0.00005
0.00000
0.00005
0.00010
0.00015
x0
n =6
7.2 fs7.2 x 10-15 s
t = 3.43 x 10-14 s t=x0 / v
g+
x-100000 0 100000
-0.00020
-0.00015
-0.00010
-0.00005
0.00000
0.00005
0.00010
0.00015
x
-100000 0 100000-0.00020
-0.00015
-0.00010
-0.00005
0.00000
0.00005
0.00010
0.00015
x
-100000 0 100000-0.00020
-0.00015
-0.00010
-0.00005
0.00000
0.00005
0.00010
0.00015
x
-100000 0 100000-0.00020
-0.00015
-0.00010
-0.00005
0.00000
0.00005
0.00010
0.00015
x
-100000 0 100000-0.00020
-0.00015
-0.00010
-0.00005
0.00000
0.00005
0.00010
0.00015
x0
n =6
7.2 fs7.2 x 10-15 s
t = 6.151 x 10-14s t=2 x0 / v
g+
x
0 10000 20000
-1
0
1
-1
0
1
-1
0
1
-1
0
1
9 = 8846 nm
9 = 2.35331 fs
n = 9 z9 = 1126 nm
z (nm)
7 = 8842 nm
7 = 2.34084 fs
n = 7 z7 = 723 nm
5 = 8826 nm
5 = 2.28697 fs
n = 5 z5 = 322 nm
z3 = -9 nm
3 = 8760 nm
Re
E z
,t
(arb
itra
ry u
nit
s)
3 = 2.06667 fs
n = 3Hartman Effect
if it will move with
vg = c
Does the WP move faster when the number of cells of the OSL increases ?
The WP peak will be at the left arrows
They are at the positions suggested by Spielmann’s experiment !!!