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 !!!