Динамика туннелирования
Л. Келдыш
ФИАН
Конференция к 100-летию И. Я. Померанчука
ИТЭФ, 5 Июня 2013г.
План
• 1. Нелинейная ионизация атомов:
Многоквантовый фотоэффект
и Туннельный эффект
• 2. Электрофотопоглощение в
кристаллах
Tunneling – Quantum Mechanical
penetration of particles through potential
barriers
U(x)
E
Nuclear α-decay (Gamov, 1928)
0
x
Atom Ionization by electric field
(Openheimer, 1928)
5
2
0
E
V(r)=e /r - eFr
Cold emission of electrons from
metals (Fowler and Nordheim, 1928)
-I0
-5
-5
0
r
5
Quasiclassic tunneling probability
 2 x2

W ~ exp    2m  U ( x)  E   dx
  x1

What is tunneling time? (McColl, 1932)
1957, Esaki – tunnel diod – interband tunneling
in solids. Predicted by Zener (1932)
Problem of operating frequencies range.
Tunneling in superconductors – Giaver,1960,
Josefson, 1962.
Lasers (Meiman, Javan, 1960) – strong, controllable
fields at optical frequencies.
Two ways of atoms’ ionization by the electrc field
Photoeffect – high frequency, linear
5
Ep
Ionization rate
hw
E
0
-5
-10
-I0
-5
WPE
0
5
10
64 F 2
2
  5  v   v 0
3 
 1
r
Tunneling – low frequency, highly nonlinear
Ionization rate
2
0
V(r)=e /r - eFr
E
8
 4 
WT   exp 

F
 3 F 
How do these effects compete at optical fields?
-I0
r
Parameters and scales
Coulomb (atomic) units -   e  m  1
I 0  1 (generally  0.5  Ry )

Field frequency –  
I0
More convenient -
Typically for powerful lasers
.
E
 1
Field strength – F 
Eat
 ~ 0.1  1
2mI 0
Eat 
e
Intensity scale
c
2
I at 
 Eat  1.6  1016 W / cm 2
4
3
Multiphoton processes –
Perturbation Theory
H   eEr
Dipole approximation Fermi Golden Rule  n  2
Wi


n)
H f(
i    f   i  n 
2
For ionization i - bound, f - free.
For single photon process H f i   E d f i 
For 2-photon process
2 
H f i  
j
Ed f  j Ed j i 
 j   i  
etc..
Function exp 1 x  extrmely nonanalitic at
x  0 . Therefore cannot be represented by
any power series in x .
Thus no tunneling in PT.
However, for unbound states
d~
e2  E
m  2
increasing with field
and
d free
d bound
~
eE
m a B
2

F
2
Therefore for
I  F 2  I at   4  I at ~ TW / cm 2
transitions in the continuous spectrum dominate.
Those, however, can be accounted
beyond the Perturbation Theory
Nonlinear nonperturbative theory for
  1, F  1
(LK,1964)
(Perelomov, Popov, Terent’ev, 1966)
(Ritus, Nikishov, 1966)
(Faisal, 1973)
(Reiss, 1980)
Main results:
a). Ionization potential renormalization
2
2
2
e
E
F
I~0  I 0  U U 

2
4m
2 2
b). Many channels, different in the number
of absorbed photons n  nmin


nmin  Integer  1  1
contributeto the total ionization rate.
~
n -th channel switches off as I 0  n  
( nmin renormalization).
Total ionization rate (sum over all channels integrated over all
final states p ) in atomic units
5
~
 I0 
S ,

  
~
 I0

exp 
 f  
 

Here S - structure function accounting for different channels
contributions,

I 0  

Wi  E ,    A 

2
  1   
2
1  2
f    sinh    
1  2 2
1

~
I0
  2mI 0
eE


F
 1 the last factor in the above formula

dominates field and frequency dependences and overall
magnitude of the effect.
Because of
Limiting cases
a). Frequency dominated regime
  1 F  
2
2
I0
 I0
2   e E
W  A
 exp 
1 

2


 
 
 8m I 0

1   2   1
1
 A
 exp

2
  



  4
I

n min  intege r 
 0   1











n min

n min
Multiphoton regime.
b). Field dominated regime:   1 F  
1
 4 

W  A F   exp 
 1 
  2 
10

 3F 
Tunneling regime.
The physical meaning of  -    tun
 tun 
I
barrier width
~ 0 
average electron velocity
eE
m
I0
Conclusions
Tunneling and Multiquantum Photoeffect
are opposite limiting cases of the same
process of the nonlinear ionization.
Tunneling time can be as short as a few
atomic time units, if the field is strong
enough.
In the multiphoton regime contribution of
the lowest channel dominate. In the
3
tunneling regime many channels (~ )
contribute essentially to the total ionization
rate.
Photoelectron Energy Spectrum
Pulse duration - 25 optical cycles
W=0.0625
Photoelectrons
0.2
g=1
0.1
0.0
0.0
0.1
0.2
0.3
0.4
Energy(Ry)
0.5
0.6
0.7
Ultrashort Pulses (USP)
ionization rate (arb. units)
3.0
chirped pulse
2.0
chirp=0.5
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-150
-100
-50
0
50
100
150
time
3.0
ionization rate (arb. units)
av=1/8
max=1.5
2.5
=1/8
max=1
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-150
-100
-50
0
50
time
100
150
Предельно короткие импульсы
1.0
(В. С. Попов, 2000: ЛК, 2000)
0.8
Half-cycle (HCP) pulses:
field
0.6
0.4
0.2
E0
E
cosh  t 2
"Lorentzian"
"soliton"
E0
E
1  t 2
0.0
-6
-4
-2
0
time
2
4
6
derivatives
And OCP-pulses:
field
1.0
soliton
0.5

 1   t 
1
E  E0  
; exp
2
2

 cosh  t 
2





gauss
0.0
-6
-4
-2
0
2
time
-0.5
-1.0
4
6
E t       f   t 
E 1
 
 
pt   p      f t 
No problem to solve numerically for any particular
pulseshape of f(t).
Because of the extreme nonlinearity does
not reduce to the sum of independent
harmonics contributions.
Number and energies of absorbed
photons not defined exactly.
No more separate channels.
Still some general features exist.
Characteristic parameter ranges:
E

1.Lowest fields -      c
 
Linear absorbtion due to highest harmonics
  I
  1
usually exponentially small.
2. Highest fields -
1    1/ 
  E  1
- universal tunneling ionization.

4 
Wi ~ exp 

 3Emax 
Main contribution – "jet" with
 momentum ~  from the narrow
part of the pulse ~ E , close to the field maximum.

If two or more maxima of E - coherent contributions, interference.
For periodic wave – "quantization" of ejected electron energies.
Intermediate (moderate field strength) range –
c    1
- strong pulse shape sensitivity.
No general rule (formula).
Defined predominately by the narrow
vicinity of the f(t) singularity in the
complex t-plane, closest to the real axis
for the Lorentzian pulse shape
 1
F2 
Wi ~ exp   0.822  3 
 
 
for solitonlike

F 

Wi ~ exp   4  2 
 
 
in the “multiphoton” field range
E      1
Intensity
0
-4
-3
-2
-1
0
-2
-4
-1
 =7.5
xp  1 for Lorentzian
-6
-10
-12
"Lorentzian"
-14
-16
Soliton-like
-18
-20
-22
Ionization
-8
xp  1 and xp 

2
for solitonlike pulse
Electroabsorption in
Semiconductors
Under Strong THz Pulses
• Tunneling assisted
electro-photoabsorption
• Dynamic cw electroabsorption
• Electroabsorption under THz pulses
• Tunneling vs multiphoton assisted
processes
Tunneling Assisted
Photoabsorption
- Electric field
Conduction Band
Eg
Tunneling
Photon
Valence Band
Interband electro-photoabsorption

Tunneling through the barrier under the electric field E action
 4 mH 3  (Fowler and Nordheim, 1928)

 
wt H , E  ~ exp  
 3 e E  H – the energy barrier height.


Valence electron becomes capable of absorbing the photon
  Eg ,
while penetrating into the energy gap to the depth ("barrier")
Eg    H
Absorption coefficient


  , E  ~ wt Eg   , E 
Effective absorption threshold red shift under static field
 3
 2
th E  ~ e E  / m
Electroabsorption Spectrum
Absorption
0.6
0.4
0.2
2
2
2
Energy unit - (e E h /m)
1/3
0.0
-2
0
2
Eg-h
Electroabsorption Under Strong
cw Field
 - weak optical field frequency
 - strong FIR of RF field frequency
0.08
2
2
2
Absorption
0.06

f =e E /(mh )=25
0.04
0.02
0.00
-10
-8
-6
-4
(h-Eg)/h
-2
0
Dynamic electro-photoabsorption.
Y. Yacobi, 1968
ac
Jauho and Johnsen, 1996 Nordstrom et al.,1998

E (t )
fields. Ladder of (multi)photon replicas - absorption of
the optical photon assisted by a few fir photons.
Electroabsorption under USP.
(LK, 2005)
Pulse duration T~1psec 10 fsec - THz range.
Half-cycle type –unipolar splash.
The problem completely equivalent to the multiquantum
photoeffect with (I, Eg)Egћω. Then parameters
1
T
   Eg   


2 2 2

e
E T  T
2
 
and f  
 2m  
10
f=5

Absorption Change
5
0
-4
-2
0
(h-Eg)T/h
-5
-10
2
4
  1
The total absorption in the pulse

A , E  ~ exp    
Tunneling for λ>1, i. e.
1/ 2
 2
 eET 

f
and   
 

 2mE g   
Eg   
 2
e E T 
2m
 1   f 2
and "multiphoton" processes for λ<1, i. e.
Eg   
 2
e E T 
2m
  f2
25
f=0.5
20
-Log A
15
10
f=2.5
5
f=12.5
0
0
2
4
6
8
10
12
14

Full lines for solitonlike HC pulse E(t)~cosh-2(t/T), dotted lines for
the static field of the same strength.
For T=30 fsec unit of Δ corresponds in GaAs to ~20 meV, the
unity of f - to ~50 kV/cm field strength (5 MW/cm2 intensity).
Tunneling time
No clear concept until now. Is it still possible and do we
need it?
”This is a field with diversity of viewpoints,
without a clear consensus.” (Landauer and Martin, 1994)
The reason – absence of real measurments.
Wave packets – no propagation inside
barrier.
What mean imaginary momentum, imaginary time?
1
k
exp kx   w p   2
 k  p2
k  2mU  E 
Different definitions
x2
 BL  
x1
x2
m  dx
dx
d lnT
 
 
2U  x   E  x`1 v x 
dE
(Landauer and Buetticker, 1982)
Under electric fields
 tun
2mU B
~
 time neccessary for free electron
eE
to acquire energy of the barrier heigth
“Lucky” electron finds trajectory above the
barrier and probability of barrier penetration
is just the probability of finding such a
trajectory?
Experimental studies of tunneling ionization
seem to be the only systematic experimental
study of the tunneling time problem.