Upload
duc
View
28
Download
0
Tags:
Embed Size (px)
DESCRIPTION
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. - PowerPoint PPT Presentation
Citation preview
Modeling of electronic excitation and dynamics
in swift heavy ion irradiated semiconductors
Tzveta ApostolovaInstitute for Nuclear Research and Nuclear
Energy
ELI-NP: THE WAY AHEADMarch 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-of-mass 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 impurity- and 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
k
ink
e
k
ink
e
knWnWn
t ))(())(( 1
)(),(),( cphim
]1
[
2.2 22*22
,
))((
Lqqkk
phqqk
Lqqkk
phqqk
LMMq
qphin
k
MEENn
MEENn
mtqeJCW
220
2
0
2 11
2 s
LOq Qq
e
VC
220
2)( )(
sr
im
ZeqU
qkqkkkqkqkk
qk
ccink EEEEnnnqVW
1)(
2 2
,
)())((
VQq
eqV
sr
c22
0
2)( )(
]
[
2.)(22*22
,
)())((
Lqkkqk
Lqkkqk
LMMq
imI
imink
MEEn
MEEn
mtqeJqUnW
D. Huang, P. Alsing, T. Apostolova et. al. Phys. Rev. B 71, 195205 (2005)
•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 instantaneoussource 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
v/c<0.1
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 and the impact parameter b along defining the straight-line trajectory to be
xe
ye
•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
eN
ii
C pP1
ˆˆ
eN
ii
e
C rN
R1
,ˆ1
C
eii P
Npp ˆ1ˆ'ˆ C
ii Rrr
'
*
2
2
)ˆ(ˆmN
PH
e
C
CM
ji ai
impiimp
i iprjirii RrU
rr
Ze
rr
ep
mH
,0
2
0
22
* )(||4||4
ˆ2
1
...ˆˆˆˆ)ˆˆ(
ˆˆˆˆ2
1ˆˆˆˆˆ
,
†).(
20
2
, ,
†.†
',',,''
†
''
†2
0
2
,
†
,
†
k qkqk
rRqi
rk qkqk
Rqiqqq
kk qkkqkqk
rqqqq
kkkkrel
aaeq
ZeaaebbC
aaaaq
ebbaaH
pCC
solve the Schrodinger equation
trtrVtrm
p
t
tri p ,,,
2
),(*
2
)0,,()( btvtr pp
with velocity of projectile pv
)(4
,0
2
trr
ZetrV
p
p
L.Plagne et. al. Phys. Rev. B 61, (2000),
J.C.Wells, et. al. Phys. Rev. B 54, (1996),
)(, tfV
etr
rki
......40
3
6ln)ln(
5
5
3
322
a
tc
a
tc
a
tcacattc
a
tcacattc ln)ln( 22
)ln()ln( 22
)0()(ac
iattc
i
eeftf
)ln(4
)ln(4 0
222
0
2
)0()(a
v
Zeiatt
v
Zei
pp eeftf
pp v
Zec
v
ba
0
2
4;
Looking closely at the problem parameters for justification of the approx.
ac
iattc
itatc
kklnexplnexp)(ˆ)(ˆ 22
The electron annihilation operator in the ion potential is given by:
Boltzmann scattering equation
e
k
ink
e
k
ink
e
knWnWn
t ))(())(( 1
]41
4[
2
02
02
2))((
bZeEENn
bZeEENn
CW
qqkk
phqqk
qqkk
phqqk
phink
)(),(),( cphim
]4
4[
)(
02
02
2)())((
bZeEEn
bZeEEn
qUnW
qkkqk
qkkqk
q
imI
imink
Numerical results
K. Schwartz, C. Trautmann, T. Steckenreiter, O. Geiß, and M. Krämer, Phys. Rev. B 58, 11232–11240 (1998)
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
Calculated electron distribution function for bulk GaAs as a function of electron kinetic energy
T=300K
Average electron kinetic energy as a function of impact parameter
T=300K
Average electron kinetic energy as a function of ion charge Z
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 ofthe Fermi-Dirac function for the same number of electrons.