Transcript Modeling of electronic excitation and dynamics in swift heavy ion

Modeling of electronic excitation
and dynamics
in swift heavy ion irradiated
semiconductors
Tzveta Apostolova
Institute for Nuclear Research and Nuclear
Energy
ELI-NP: THE WAY AHEAD
March 11, 2011, Bucharest-Magurele
•We consider a bulk GaAs semiconductor doped with electron
concentration to form a 3D electron gas.
•We separate the dynamics of a many-electron system into a center-ofmass motion plus a relative motion under both dc and infrared fields.
•The relative motion of electrons is studied by using the Boltzmann
scattering equation including anisotropic scattering of electrons with
phonons and impurities beyond the relaxation-time approximation.
•The coupling of the center-of-mass and relative motions can be seen
from the impurity and phonon parts of the relative Hamiltonian
•When the motion of electrons is separated into center-of mass and
relative motions, the incident electromagnetic field is found to be
coupled only to the center-of-mass motion but not to the relative
motion of electrons
•This will generate an oscillating drift velocity in the center-of mass
motion, but the time-average value of this drift velocity remains zero
•The oscillating drift velocity will, however, affect the electron-phonon
and electron-impurity interactions.
•The thermodynamics of electrons is determined by the relative motion
of electrons This includes the scattering of electrons with impurities,
phonons, and other electrons.
•The effect of an incident optical field is reflected in the impurityand phonon-assisted photon absorption through modifying the
scattering of electrons with impurities and phonons.
•This drives the distribution of electrons away from the thermal
equilibrium distribution to a non-equilibrium one. At the same time,
the electron temperature increases with the strength of the incident
electromagnetic field, creating hot electrons.
Previously- Boltzmann scattering equation – impurity and phonon- assisted
photon absorption and Coulomb electron scattering for a doped GaAs
semiconductor
 e
nk  Wk( in)( ) 1  nke   Wk( in)( ) nke
t
( in)( ph )
k
W
2



C



q ,M

q
2
J
2
M


e q.t 



 [nk  q N qph
  Ek  Ek  q  q  M L


  (im), ( ph), (c)
2m 
*





 nk  q N qph
  1  Ek  Ek  q  q  M L ]
Cq
2
 
  LO
 2V
1
e2
 1
  
2
2
    0   0 q  Qs 
2
L

2
( in)( im)
k
W
 nI  U

 E

q ,M
( im)
2
(q) J
2
M


e q.t 
2m 
*
2
L

2

 M ]
 [nk q Ek  Ek q  M L
 nk  q
U
( im)
 Ek  q
L
Ze2
( q) 
 0 r q2  Qs2 
( in)( c )
k
W

k
2


V
 
k , q
(c)

( q) 1  nk  nk  q nk  q   Ek  Ek   Ek  q  Ek  q
2
e
V ( c ) ( q) 
 0 r q 2  Qs2 V
2

D. Huang, P. Alsing, T. Apostolova et. al. Phys. Rev. B 71, 195205 (2005)
Electron dynamics in ion-semiconductor interaction
v/c<0.1
•The projectile has reached its equilibrium charge state - there will be
only minor fluctuations of its internal state
• It will move with constant velocity along a straight-line trajectory until
deep inside the solid.
•Thus, the projectile ion acts as a well defined and virtually instantaneous
source of strongly localized electronic excitation.
G. Schiwietz et al. / Nucl. Instr. and Meth. in Phys. Res. B 225 (2004) 4–26
Electron dynamics in ion-semiconductor interaction
•After investigating the electron dynamics in semiconductors on a femtosecond
time scale in such a physical processes as irradiation by an intense ultrashort laser
pulse we modify the technique to describe the passage of a highly charged ion
through the solid. Same time scales of interaction
•We consider only constant-velocity v/c < 0.1 , straight-line trajectories for the
projectile.
•In terms of three-dimensional Cartesian coordinates, we define the reaction to

occur in the x-y plane with the beam directed along ex and the impact parameter b

along ey defining the straight-line trajectory to be
•We will establish a Boltzmann scattering equation for an
accurate description of the relative scattering motion of
electrons interacting with a swift heavy ion by including
both the impurity- and phonon-assisted photon absorption
processes as well as the Coulomb scattering between two
electrons.
•We study the thermodynamics of hot electrons by
calculating the effective electron temperature as a
function of impact parameter and charge of the ion.
We use the Hamiltonian

ˆ 2
1
e2
Ze2
imp 
H
p 
  
   U ( ri  Rimp )
*  i
2m i
i  j 4 0 r | ri  rj |
i 4 0 r | rp  ri |
i,a
ˆ C Ne 
P   pˆ i
C
1
R 
Ne
i 1
Ne
ˆ
r
 i,
i 1
  C
ri '  ri  R
ˆ
ˆ
1 ˆ C
pi '  pi 
P
Ne
ˆ C 2
(P )

2 N e m*
Hˆ CM
ˆk† aˆk
Hˆ rel  

a
k

k ,
2
1
e
†
†
  a
  a
ˆ
ˆ
ˆk ' 'aˆk 
  q bˆq† bˆq    
a
2
k

q

k
'

q

'

2 k ,k ', , ' q  0 r q 
q ,
ˆ  bˆ
 
C
(
b
q

q


k , q ,
†

 q
)e
 
iq . R C
aˆ
†
 
k  q
aˆk
  
Ze2
iq .( R C  rp ) †
 
e
aˆk  q aˆk  ...
2


k , q  0 r q 
solve the Schrodinger equation


 ( r , t )
p2



i

 r , t   Vp r , t  r , t 
*
t
2m

V p r , t   
Ze2
 
4 0 r  rp (t )
L.Plagne et. al. Phys. Rev. B 61, (2000),
J.C.Wells, et. al. Phys. Rev. B 54, (1996),

rp (t )  (v p t, b,0)
with velocity of projectile
vp
e

 r , t  
 
ik r
V
f (t )  f (0)e
f (t )
i Ze 2
i Ze 2
2
2
ln( t  t  a ) 
ln( a )
 4 0 v p
 4 0 v p
e
b
Ze2
a ;c
vp
4 0 v p
f (t )  f (0)e
i
c ln(t  t 2  a 2 )

e
i
 c ln( a )

c ln(t 
3
5
t
t
3
t
t 2  a 2 )  c ln a  c  c
c
 ......
3
5
a
6a
40a
c ln(t 
t
t  a )  c ln a  c
a
2
2
Looking closely at the problem
parameters for justification of
the approx.
The electron annihilation operator in the ion potential is given by:

i
cˆk (t )  aˆk (t ) exp c ln t  t 2  a 2

 exp i c lna 
Boltzmann scattering equation
 e
nk  Wk( in)( ) 1  nke   Wk( in)( ) nke
t
( in )( ph )
k
W
2



 C 
  (im), ( ph), (c)
2

q

q
 [nk q N qph  Ek  Ek q  q  Ze2 b4 0



 nk  q N qph  1 Ek  Ek  q  q  Ze2 b4 0 ]
( in )( im )
k
W
 [n

 E
 
k q
 nk  q
 nI  U
( im )

q
 E E

k

k
 
k q
( q)
2

b4 ]
 Ze b4 0 
2
 Ek  q  Ze2
0
Numerical results
K. Schwartz, C. Trautmann, T. Steckenreiter, O. Geiß, and M. Krämer,
Phys. Rev. B 58, 11232–11240 (1998)
Calculated electron distribution function for bulk GaAs as a function of
electron kinetic energy
T=300K
Calculated electron distribution function for bulk GaAs as a function of
electron kinetic energy
T=300K
Calculated electron distribution function for bulk GaAs as a function of
electron kinetic energy
T=77K
Average electron kinetic energy as a function of impact parameter
T=300K
Average electron kinetic energy as a function of ion charge Z
T=300K
Conclusions
• The effect of the potential of the incident ion is
reflected in the phonon and impurity assisted
electron transitions through modifying
(“renormalizing”) the scattering of electrons with
phonons and impurities
• This method can offer unique ability to study the
change in the collision dynamics when a single
projectile characteristic is modified.
• The same numerical code as with the excitation with
a laser field is used.
Thank you for your attention!
•For a general transient or steady-state distribution of electrons, there
is no simple quantum statistical definition for the electron temperature
in all ranges. However, at high electron temperatures we can still
define an effective electron temperature through the Fermi-Dirac
function according with the conservation of the total number of
electrons.
•In the nondegenerate case, the average kinetic energy of electrons is
proportional to the electron temperature. The numerically calculated
distribution of electrons in this paper is not the Fermi-Dirac function.
We only use the Fermi-Dirac function to define an effective electron
temperature in the high temperature range by equating the
numerically calculated average kinetic energy of electrons with that of
the Fermi-Dirac function for the same number of electrons.