T - reseau femto

Influence of recombination process in
the analysis of silicon by LaAPT
Vella Angela , B. Mazumder, F. Vurpillot, B. Deconihout , G. Martel
Groupe de Physique des Matériaux
Université de Rouen
Carry le Rouet
18-19 mars 2010
Laser assisted Atoms probe tomography (LaAPT)
Leser pulses
X
Y
DC HV
< 10-15°
10KV
m
t2
∝V 2
n
L
(-1
10
)
Position Sensitive
Detector
60°
60°
(-1
01
)
timing
45 cm
)
(011
~ 40
Carry le Rouet
18-19 mars 2010
nm
Physical mechanisms of evaporation with fs laser
U
Φ evaporation
λ
metal
Ua
Q(E)
-eEx
 Q( E ) 
∝ exp −

 k BT 
Activation energy
Typical time of the process:1 ps
surface atom vibration time
Ui
Two ways to field evaporate:
1. Increase the electrical field by optical rectification effect : τ evap
2. Increase the tip temperature by photon absorption :
= τ laser
τ evap = τ cooling
Dominant mechanism
Carry le Rouet
18-19 mars 2010
atom count
Thermal response of a tip: a simple model
Not symmetrical profile:
100000
Fe2+
10000
Si2+
1000
B2+
Some ions are evaporated on
vary long time
thermal effect
B+
Nb+
Cu+
100
10
1
0
10
20
30
40
50
60
70
80
m/n (AMU)
• if we assume: T = T0 + γ I
γ is proportional to the absorption coefficient α of the material
T
Heated zone :
Gaussian width σ
Tip ~wire
Carry le Rouet
18-19 mars 2010
z
Thermal response of a tip: a simple model
• if we assume: T = T0 + γ I
γ is proportional to the absorption coefficient α of the material
T
Heated zone :
Gaussian width σ
Tip ~wire
z
T
T
max
T0
t
~ τph-e < 10 ps
Carry le Rouet
18-19 mars 2010
Cooling process
• After the heating (T=Tmax), the cooling follows the Fourier
low.
∂T
d T
−D 2 =0
dz
∂t
2
Whit D =thermal diffusivity
T (x , t ) = T 0 +
T max
1+ 2
T
D .t
σ
.e
Carry le Rouet
18-19 mars 2010
τcooling
(
)




2
τ Cooling
time

x2
−
 2 σ 2 + 2 at

σ 
= 
D
2
atom count
Increasing of temperature ∆T
100000
Fe
10000
Φ evaporation
2+
Si2+
1000
B2+
B+
Nb+
 Q( E ) 
∝ exp −

k
T
B


Cu+
100
−
10
atom count
100000
1
0
10
20
30
40
50
60
70
80
m/n (AMU)
10000
φtemp(t) = C × e
1000
10
1.00E-07
2.00E-07
3.00E-07
4.00E-07
time (s)
Carry le Rouet
18-19 mars 2010






Trise
k  T0 +

2t 

1+

τ cooling 

Q~0.15 eV
τcooling ~200 ns !!
T rise ~50 K
C= N ν Q τevap
100
1
0.00E+00
Q( E )
Si tof spectra for different wavelength
Log(No of atoms/pulse)
1
28Si2+
Photon energy 1.2 eV (IR) near band gap energy (1.1 eV).
29Si2+
A hump appears with increasing laser energy
7 ns after the main peak
30Si2+
0,1
F = (0.07 ÷ 2)mJ/cm 2
0,01
I = (0.2 ÷ 5) ×109 W / cm 2
1E-3
0
20
40
60
tof (nS)
1
28Si2+
Photon energy 2.45eV (Green) higher than the band gap energy (1.1 eV)
Laser energy ~ 100nJ
0,1
Log N
29Si2+
30Si2+
Non existence of the hump in tof spectrum also
at high laser fluency
0,01
Carry le 1E-3
Rouet
18-19 mars 2010
0
20
tof(ns)
40
60
SiC tof spectra for different wavelengths
Photon energy - 2.45eV (Green)
Log(No of atoms/pulse)
28Si2+
---33nJ
---84.6nJ
---98.5nJ
100
29Si2+ 30 2+
Si
Evidence of hump with photon energy of
near band gap energy 2.36 eV
10
F = (0.3 ÷ 1.5)mJ/cm 2
1
380
400
I = (0.3 ÷ 3) ×109 W/cm 2
420
TOF (nS)
Photon energy - 3.62eV (UV)
28Si2+
80
11.7nJ
21.2nJ
30.5nJ
70
Log N
60
29Si2+
50
No evidence of hump, even by increasing laser
energy; and no variation in mass spectra.
30Si2+
40
30
20
-5
0
5
10
TOF nS
Carry le Rouet
18-19 mars 2010
15
20
CONCLUSION
The hump seems to appear only using photons with
near-band gap energies
Model
S(z)Φ−
dV
• I = I 0 exp(− α ⋅ y )
Y
S(z)+
diameter <<1000 nm Absorption
α~10 cm-1
I/I0~1 Homogeneous absorption
Z
Φ+
Initial conditions:
Localized
injected carrier density
(
N 2 = N 0 exp − z 2 σ 2
2-steps transition
)
Temporal evolution:
Relaxation time τ2
Total energy given
to the lattice 1.2 eV
E2=0.1 eV
injected electron density with a
relaxation time τ2 dN
N
2
Relaxation time τ1
E1=1.1 eV
Carry le Rouet
18-19 mars 2010
N2 (z,t),
dt
=−
2
τ2
N1 (z,t),
thermalised electron density with a
relaxation time τ1
dN1
N1 N 2
=−
+
τ1 τ 2
dt
Simple model:
Φ−
Y
S(z)+
dV
Φ+
Z
S(z)-
spatial evolution
Using simple Fourier equation with a
generation term and an approximation
on time evolution of Cv(T)
Heat generation = storage + exchange
r
r
d
G ( z , t ) ⋅ dV ( z ) = [Cv (T (t )) ⋅ T (t )]⋅ dV ( z ) − K (T )[ S + ( z )∇(T ( z )) − S − ( z )∇(T ( z ))]
dt
Cv = volume specific heat
 N 2 ( z, t )
N1 ( z , t ) 

G ( z , t ) =  E2
+ E1
τ2
τ1 

Evaporation flux calculation
Carry le Rouet
18-19 mars 2010
K (T ) = thermal conductivity
Qn
Φ (t ) = υ0 exp(−
)
k BT (t )
6
Model parameters
2.5
x 10
Cv = Volume specific heat
C(J/(Km3))
2
1.5
K (T ) = thermal conductivity = 30 Js -1m −1 K −1
1
For Si nanowires
σ = size of heated zone = 200 nm
0.5
N 0 = nb of charges = (4 ÷ 40) ×10 20 cm −3
0
0
τ 1 = decay time = 20 ns
τ 2 = decay time = 2 ps
Qn = 0.3 eV
Carry le Rouet
18-19 mars 2010
100
200
300
400
500
600
700
T(K)Estimated from
800
experimental results
From data
Qn
Φ (t ) = υ 0 exp(−
)
k BT (t )
T (x, t ) = T0 +
T0 + Trise
.e
D.t
1+ 2 2

x2
−
 2 σ 2 + 2 at

(
)




σ
A.U
1
Qn = 0.3eV
0,1
Trise = 170 K
σ = 200nm
0,01
0,001
-0,5
0
0,5
Energy conservation:
Carry le Rouet
18-19 mars 2010
1
1,5
2
2,5
t(ns)
3
u = C (T )Trise = N 0 E2
N 0 ≈ 10 20 cm −3
Simulation results
N 0 = (4 ÷ 40)10 20 cm −3 ;
E photon = 1.2eV ;
E photon = 2.4eV
0
0
10
10
τ
-1
10
-1
10
-2
10
-3
10
-2
10
-0.5
00
0.5
101
1.5
202
2.5
t(ns)
3
0
5
10
15
-8
x 10
We fixe the decay time τ1 =20 ns to obtain τ=7 ns as experimentally observed
Carry le Rouet
18-19 mars 2010
20
-9
x 10
Open questions
•
1) Why a so small heated zone
σ
Numeric resolution of Maxwell
= 200 nm < laser waist = 50 µm
FDTD : finite difference on time domain
software used : FDTD solution from Lumerical
equations in 3D
IR radiation (1.2eV): propagation of the e.m. field
Carry le Rouet
18-19 mars 2010
FDTD: Si nano-wire with UV light
UV radiation (3.6eV): propagation of the e.m. field
Carry le Rouet
18-19 mars 2010
FDTD absorption maps
IR light: 1030 nm
UV light: 343nm
No absorption
localization
Absorption at
the apex as
for metallic tip
PRB (81), 125411
Carry le Rouet
18-19 mars 2010
Open questions
• 2) Why a so high Nb of charges
αI
N 0 = 10 cm >>
= 1016 cm −3
hν
α = 10cm −1 at T = 70 K
20
−3
Band bending: Internal electric field F= 3V/nm
hv
Many holes can absorb
photon energy
The absorption is localized
at the surface
Carry le Rouet
18-19 mars 2010
Surface potential (max. band bending) vs. surface field
φS
(in eV)
φS = f (ES)
3,0
yB=-20
2,5
yB= 0
2,0
yB= +20
Es =
18
-3
-N = 7.8x10 cm )
D A
-3
(N -N = 0
cm )
D A
18
-3
(N -N = - 7.8x10
cm )
D A
(N
with εr=11.7 (Si)
εr = 11.9
1,0
Evaccum = Eevaporation
0,5
= 33 V/nm
0,0
-3
10
10
-2
10
-1
Es (in V/Angstrom)
Tsong Surface Science85 (1979)1-18
Carry le Rouet
18-19 mars 2010
εr
For Si:
with εr = 9
1,5
Evacuum
10
0
ES 0.27 V/
Open questions
• 2) Why a so high Nb of charges
• Due to Franz-Keldysh effect
In the presence of external electric field (F=3V/nm) the band structure
change and also the absorption:
B.C
hv
Eg
B.V
Carry le Rouet
18-19 mars 2010
Due to tunneling the wave function
penetrates in the semiconductor gap
Open questions
• Franz-Keldysh effect :
Absorption at β < 0 (i..e. below the gap with F)
is similar to ≈ 2 % of absorption at – β (i.e.
above the gap [ h ω0 = E gap + 1.8eV ] with F = 0)
β = h ω0 − E gap
(in eV)
4
β=
“characteristic
energy
of Keldysh”
3
β
2
h e F
β = 
 2mr
2 2
1.8 eV
1
Carry le Rouet
18-19 mars 2010
2´ 10
9
4´ 10
9
ES 0.27 V/
6´ 10
9
8´ 10
9
1´ 10
10
Surface
Field
(V/m)
2
1/ 3



Open questions:
• 3) Why so long decay time 20 ns
Auger recombination time
for N=1020cm-3 is <1ns
electrons
Due to the band bending e-h are
spatially separated and the
recombination time can be longer
holes
Carry le Rouet
18-19 mars 2010
Conclusion
• Thanks to APT we can study the recombination process
under high electric field
• There are many open questions on our model
• Our simple model doesn’t take into account the charges
diffusion
Carry le Rouet
18-19 mars 2010
Confinement or diffusion of laser-generated carriers N and temperature T in SC ?
Boltzmann transport (1Dim.≡z) equation in the relaxation time appx.
∂N uuurr
Transport equation: •
+ divJ = +G − R
∂t
with :
Heat equation:
r
 ∂N
N ∂E g
N
∂T 
J ≡ −D 
+
+
p
+
p'
+
2
(
) 
∂
z
2k
T
∂
z
2k
T
∂z 
B
B

∂T ∂ 
∂T 
=
D
•
 T
 + Gth
∂t ∂z 
∂z 
with :
Gth =
α '( 1 − Γ )I ( r,t ) ( h ω0 − 3k BT ) −α ' z
e
h ω0
C
DT : thermal diffusivity
Carry le RouetC : volume specific heat
18-19 mars 2010
With the relaxation time appx. assumed
to be due to lattice scattering:
p = p’ = -0.5
∂f Fermi − Dirac
∂t
with
=−
scatter
f FD ( t ) − f FD ( 0 )
τ
τ ∝ ( E − Ec ) ≡ ( E v − E )
p
p'
Confinement or diffusion of laser-generated carriers N and temperature T in SC ?
• Boltzmann transport equation in the relaxation time appx.
2
 ∂2N
∂E g ∂T
∂N
N ∂ Eg
N
= D 2 +
+
s
−
1
+ ... + G − R
)
2
2 (
∂t
2k BT ∂z
2k BT
∂z ∂z
 ∂z
1.16
: ambipolar diffusion coefficient
E g [T( x ),N( x )] = 1.16 − 7.02 x10-4
T2
− 1.5 x10-8 N1/3: band gap
T + 1108
( 1 − Γ )αOPA I ( r,t ) ( 1 − Γ ) β TPA I
+
h ω0
2h ω0
2
G=
R = −γ Auger N 3
2
1.12
100
( r,t )
dI
= −αOPA I − β TPA I 2 − σ FCA N( t )I( t )
dz
Are the generated carriers, N
affecting absorption
of the pulse
?
depends on σFCA value
Carry le Rouet
18-19 mars 2010
1.14
Participate to increase heating at the surface !
Confinement:
 T 
D = D0 

 300 
s = -1 for Si
150
200
250
300
Confinement or diffusion of laser-generated carriers N and temperature T in SC ?
• Boltzmann transport equation in the relaxation time appx.
2
 ∂2N
∂E g ∂T
∂N
N ∂ Eg
N
= D 2 +
+
s
−
1
+ ... + G − R
)
2
2 (
∂t
2k BT ∂z
2k BT
∂z ∂z
 ∂z
T2
E g [T( x ),N( x )] = 1.16 − 7.02 x10
− 1.5 x10-8 N1/3:
T + 1108
-4
:
band-bending
z<0
due to positive applied field +V
to the tip of the sample
in an atom probe
Participate to decrease heating at the surface !
Carry le Rouet
18-19 mars 2010
diffusion
z
Ec ,Ev ≅ exp  
δ ' 
band gap
Confinement or diffusion of laser-generated carriers N and temperature T in SC ?
• Boltzmann transport equation in the relaxation time appx.
2
 ∂2N
∂E g ∂T
∂N
N ∂ Eg
N
= D 2 +
+
s − 1)
+ ... + G − R
2
2 (
∂t
∂
z
2k
T
∂
z
2k
T
∂
z
∂
z
B
B

E g [T( x ),N( x )] = 1.16 − 7.02 x10-4
:
band gap
band-bending
z<0
due to positive applied field +V
to the tip of the sample
in an atom probe
diffusion
z
Ec ,Ev ≅ exp  
δ ' 
T2
− 1.5 x10-8 N1/3:
T + 1108
Participate to decrease heating at the surface !
Band-bending could be higher than band-gap: heavily inverted surface zone !
Carry le Rouet
May increase hole absorption
18-19 mars 2010
and so SURFACE HEATING !
Results of simulation for generated carrier density and
temperature
Si
in
visible
Si
in
Infra-Red
19
6x10
16
t = 0 ps
t = 500 fs (start tail effect)
t = 1 ps
t = 1.5 ps
t = 2 ps (end of the pulse)
t = 10 ps
t = 50 ps
t = 100 ps
19
5x10
t = 0 ps
t = 500 fs (start tail effect)
t = 1 ps
t = 1.5 ps
t = 2 ps (end of the pulse)
t = 10 ps
t = 50 ps
t = 100 ps
19
4x10
19
3x10
-3
Carrier density (cm )
-3
Carrier density (cm )
4,5x10
19
2x10
19
1x10
0
16
4,5x10
16
4,5x10
16
4,4x10
16
4,4x10
16
4,4x10
16
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
1,6
1,8
4,4x10
2,0
0,0
0,4
Depth (µm)
t = 0 ps
t = 500 fs (start tail effect)
t = 1 ps
t = 1.5 ps
t = 2 ps (end of the pulse)
t = 10 ps
t = 50 ps
t = 100 ps
150
140
130
Pulse is centered
@ t = 1.5 ps
(pulse duration = 500 fs)
120
110
100
90
80
0,0 Rouet
0,1
Carry le
18-19 mars 2010
0,2
0,3
Depth (µm)
1,6
2,0
0,4
t = 0 ps
t = 500 fs (start tail effect)
t = 1 ps
t = 1.5 ps
t = 2 ps (end of the pulse)
t = 10 ps
t = 50 ps
t = 100 ps
104
Pulse intensity
Temperature (K)
160
1,2
Depth (µm)
Temperature (K)
170
0,8
100
96
92
88
84
80
0,0
0,5
0
500
1000
1500
Time (fs)
2000
2500
3000
0,2
0,4
0,6
Depth (µm)
0,8
1,0