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On Schockley-Read-Hall Model with Finite Relaxation Time of Traps for Surface Recombination Velocity in
Case of Illumination
C. Flueraru
To cite this version:
C. Flueraru. On Schockley-Read-Hall Model with Finite Relaxation Time of Traps for Surface Re- combination Velocity in Case of Illumination. Journal de Physique III, EDP Sciences, 1995, 5 (1), pp.33-42. �10.1051/jp3:1995108�. �jpa-00249293�
J. Phys III Franc-e 5 (1995) 33-42 JANUARY 199S, PAGE 33
Classification Physics AbsiJ.acis
73.25 73.60F
On Schockley-Read-Hall Model with Finite Relaxation Time of
Traps for Surface Recombination Velocity in Case of Illumination
C. Flueraru (*)
R & D Center for Electronic Devices, Erou Iancu Nicolae 328. Bucharest72996, Romania (Receii>ed J9 July J993, ieiised 26 Ju/» J994, at c~pled J7 Oc%her J994)
R6sumk. Dans ce travail, nous examinons le modble de Schockley-Read-Hall relatif h la vitesse de recombinaison de surface dans le cas d'une illumination par un flux de photons. Le temps de
relaxation des pidges reste fini. Nous comparons [es vitesses de recombinaison de surface du silicium de type n dans toutes [es configurations d~injection de charge. Nous ddterminons aussi
l'influence des divers parambtres sur cette vitesse de recombinaison.
Abstract. In this paper we discuss the Schockley-Read-Hall model for surface recombination velocity in case of illumination with the photon flux density. The traps have a finite time of
relaxation. We estimate for a n-type silicon the surface recombination velocity in all injection charge cases. Finally, we evaluate the importance of different parameters in the value of ~urface
recombination velocity.
1. Introduction
Thermally oxidised single-crystalline silicon plays an important role in the field by
semiconductor devices technology and integrated circuits. The assessment of surface
recombination velocity is very important in design and modelling of semiconductor devices
which use MOS-technology. Recombination in the surface layer can never be completely
eliminated and in some cases their effects will become significant and eventually dominant such as devices like pn junction solar cells and pin diodes.
A real surface contains a rather complex systems as regards both physical and chemical composition. For a native layer (with a thickness about 401) of SiO~, G. Hallinger [I]
suggested a chemical structure model of interface SiO~-Si in which the passage from Si to
SiO~ is carry out by a layer with thickness 3-4 I of SiO, with 0 ~.x. ~4 and layer with
thickness lo-12 1 of SiO~ without bulk properties. SiO~ with the help of traps and fixed-oxide charge can check the surface recombination velocity. The control is carry out by trap
(*) Permanent address. Institute of Microtechnology~ Erou Iancu Nicolae 348, PO Box, 38-160, R-72225~ Bucharest~ Romania.
@Les Editions de Physique 1995
concentration in surface layer (at surface) and surface potential. Something similarly has to be in semiconductor surface. In this paper is used this structure of free surface made up by surface
layer and surface.
Woods and Williams [2] believe that the defect responsible f'or the carrier trap is related to
excess silicon that is present near the interface. Their results show directly the traps are to be
found only near the interface with almost none in the volume of the oxide. The carriers may be also trapped by dangling bonds of silicon atoms [3].
Walkjiensburg [4] make the bond between defect concentrations in oxide layer and oxide traps in interface. He obtained by electrochemical method slightly higher traps concentration than those by Sigmon [5].
In the following we consider surface recombination in the case of illumination with the
photon flux density. The illumination is of photon energy in excess of the silicon energy gap, and provides for the photogeneration from trapping centres and supply of carriers to the semiconductor surface. The energy position of the surface states in the forbidden gap is determinated by short-range atomic process and is thus unaffected by the presence of the external potential orland illumination. The Schockley-Read-Hall (SRH) model is extended to
include finite relaxation time of traps [6].
A theory based on microscopic SRH process which generalized for multiphonon transition,
was outlined in [7].
In this paper a model which is able to evaluate the surface recombination velocity from material parameters is presented.
2. Model
The following simplifying assumptions have been made
. the doping dependent mobility and diffusion constant are replaced by their average values
. the electron and hole gas is non degenerate
. the traffic of electrons and holes through the recombination centres is governed by the thermal capture and thermal and optical emission rates which have the same values in surface layer as well as in bulk
. we assume the surface to have isolated traps at a fixed energy in forbidden gap
. we assume that traps have a finite time of relaxation.
First assumption is questionable because for surface layer in scattering mechanisms appear
new interactions: surface acoustic phonons and optical phonons scattering, Coulomb
scattering due to fixed oxide charge and surface roughness scattering. In the case of high injection level these phenomena become more important and lead to a variation of mobility
with applied electric field. In the first step of approximation the constant value of mobility for surface layer is widely accepted. It is well-known that interband photogeneration is important
in direct semiconductor. For low level of injection Hsieh and Card [8] showed that for photon energies even slightly in excess of the energy gap, photoionozation via defect states may be neglected in comparison with interband photogeneration, even for indirect semiconductor. For
high level of injection in case of photon energy is less than gap energy of semiconductor that
means that the transfer of carriers between bands are i>ia traps and for trap with finite time of
relaxation, the problem is open.
In this paper we consider isotropic silicon but we can keep account of the dependent density
of trap levels upon orientation of the silicon surface from reference [9].
Figure I shows energetic diagram for SRH model. We assume that traps level has two
ground state filled with a electron and trap empty of electron and each trap has only first excited states.
N° I ON SCHOCKLEY-READ-HALL MODEL WITH FINITE RELAXATION TIME 35
~1[ j~~
tic ~ _Z_
~1j j~~
p p
b÷nd 2j ~3
~P ~P
Fig. I. Energetic diagram for Schockley-Read-Hall model with first excited state of traps.
Dhariwal et al. [6] observed that each of these states may have corresponding excited states.
The system (trap and trapped carrier) transition between ground state and excited state is
phonon assisted that means phonons energies are equal to the difference between states.
In this paper the four possible states of trap are considered. The electron (and hole) have an
average time of transition from excited state to ground state which will be denoted by
&t~(&t~). Times for thermal excitation will be &t((&tj). Transition from trap level for electron (hole) to conduction (valence) band pass through excited level. The eight rates of capture and
emission for electrons and holes considered in Figure l~ can be written as follows
U(
= f) N~/&t~ (I)
Ill
=
fl en Nt ~ 4~p al~ Nt fl (2)
Ill
= Cn >i Ni fl (3)
Uj fl' N~/&t( (4)
~j
~ N~/&t~
~ ~ ~~ ~
~°P ~~~P N f(
(5)
P t 'P ff
(6)
~~ ~~ ~~~~~~
j~
where f), fl', f(, fl' represent the occupation fraction for electrons and holes in ground and
excited states and obey the following relation
fl ~ fl' ~ fl ~ fl'
=
(9)
n~p) is the total concentration of electrons (holes) in conduction (valence) band in
nonequilibrium condition in surface layer, e~(e~ and c~(c~) represent the rates of emission and capture of electrons (holes) from and by traps, ~~~ is the photon flux density or intensity
a)P(a(P) is the optical cross section for electrons (holes), N~ is the concentration of traps.
I denoted the emission and capture rates between the ground state of trap with electron (hole) and first excited state of trap with electron (hole) by U((U() and U((U(). Rates of transition,
between excited states and conduction (valence) band are given by U((U() for emission and
U((U() for capture.
In thermal equilibrium we can write
US
= Ulo and Ulo = Ulo lo)
where subscript « o » denotes quantities in their thermal equilibrium and I is n or p for electrons and holes, respectively.
Under steady-state conditions, the following relationship should be satisfied :
~~
=
0 where
= n, n', p, p' (I I)
~t'
From equation (I I) we get relationship between emission and capture rates for electrons and holes. Now the net rate of recombination in steady-state is
U=U)-U) =U)-U) (12)
and with some algebraic manipulation we can write
~
(>i~. p~ ~~ ~~~
~'~~'~j ~'~f pi* s/ s/
~ ~~ ~~ ~ ~/ (Ps ~ PI ) ~ s/ ~~~ ~ ~~~ (~~~
where n~, p~ are density carrier at surface
$ c,N~
s,*
= = j14)
(1 ~ h,*) (1 ~ h~*) h,*
= h, (1 ~ K~) = c~ ii &i (1 ~ cr/P. ~bople<1 ('5)
y = (&t~ ~ &t~)/N~ (16)
ij*
= ii (I ~K~) (17)
where is n, p and
iii = N~ exp(- (e~ Et)/KB T)
~j8~
pi = N~ exp(- (F~ Ev)/KB T)
On the one hand the density current can be expressed in term of the diffusion and drift current components on the other hand it can be express by product between q-electronic charge and U- rate of recombination.
~ dV
~ d>1 ~~
~ ~~
~
'~ dot ~ ~ ~' d,t
This relationship shows variation of carrier density in surface layer. The behaviour of potential
variation in surface layer can be obtained by Poisson equation. The potential V(~;) has been assumed the following values
v (o)
= ~bs v (»>)
=
o. (20)
Where ~bs is the height of the potential barrier and w depend on the nature of surface. The solution of equation (19) can be write
N° I ON SCHOCKLEY-READ-HALL MODEL WITH FINITE RELAXATION TIME 37
Also we can write similar expression for holes. Coupled equations (13) and (21) describe the carrier motion, potential behaviour and recombination velocity in surface layer. This system
can be solved by iteration method: for a surface carrier concentration (equal with bulk
concentration) is calculated the recombination rate, which replaced in equation (21) yield the surface carrier concentration in the next step of iteration.
The model can be easily particularised to well-known cases so: K,
= 0, nf, pj* become
nj, pi for dark condition. If one neglect the effect of relaxation time of traps that is
&t~=0, the first term of denominator of equation (13) disappears, h,* vanish and S,* becomes $. In the case for
~fi~~ = 0 and &t~ =
0 one obtains the Schockley-Read-Hall
model.
3. Discussion
In the following for the facilitation of calculation and interpretation we will note :
~~ ~ ~ ~"'~'~~P ~~s) NRD U " 'l
(W) ~ ) jj
eXP ~ / )~ xj
exp (~bs (22a)
p~ = p (w) exp (- ~bs ) P
~~ U
=
p(w>) ~
j~ exp ~ ~ "~ dx exp (- ~bs (22b)
Dn o KB T
K
= Y (ns P~ ni* Pi*) St S/
In Figure 2 we show the behaviour of N~D and P~D for different expression of potential in surface layer. We used
x 2
V (x ~b
s (23)
w
The first term in equation (22) contains the modification of carrier density because of energy
bands diagram in surface layer. The variation between the recombination controlled and
diffusion controlled limits can be made by the second term of the same equation.
2 5
I n~, ~ iiio~
y E
~ ~ , ~
~ / '
~ /
p ~ po
a
, ~
/ z
,
5109
~~ ~
Is ~~ ~~°
Fjg. 2. -Parameters N~~ (solid ljne) and P~o (dotted line) ve>.,uis surface potential with
Nd lo'? cm~~
We will give some numerical assessment and we will study the weight of the new
phenomena taking in account : saturation effect and photoionozation of impurity irradiated with photon flux.
We consider a n-type semiconductor in which the trap level e~is deep and &t
= &t~ ~ &t~ has values between 10-7-10-9 s. In the numerical assessment we will take trap with energy level e~ = e~ 0.57 (eV) and the capture coefficient for electrons and holes from reference [10].
The experimental photoionization cross section have been determinated to be in the range
between 10-16 and 10-24 cm2 for common impurities found in Si [8]. With this values we
obtained the following approximation.
K~WI, h~wl, nj*Wpj* (24)
Therefore in this case the surface recombination velocity for holes is greater than surface recombination velocity for electrons. In Figure 3 we have plotted the ratio Sf/S~ i>eisus
(~b °P a(P) with parameter &t. In this model p~* has the same role like p, in standard SRH model and it is plotted in Figure 4. We must note that S~* has not the same physical meaning as surface
recombination velocity $.
In the following we will discuss the case of low level of injection (no W &n), moderately
level of injection (no ~ &n, &n
~ K/Sf) and high level of injection such that &n W K/Sf where
&n is the density of carrier optical generated.
I) Firstly we discuss the case in which N~D W I/Sf, P~D W I/Sf and K is less than those
terms of denominator that it is generality true for n-type silicon and low level of injection. That
means that diffusion of carriers in surface layer can be neglected for this case. It can be shown
by differentiating (13) with respect to ~bs, that the surface potential for maximum surface recombination velocity is
K~T &n.c~
~bs = ln (25)
2'Q No'~n
It is interesting to observe that the values of potential at which surface recombination velocity
takes the maximum values is influenced neither by relaxation time of trap levels nor by the
I
~ ~i
, a
*aj a , /
U' U' £
'
c c
io2 1o4 io4
~°~~~ (£l ~opjop ~-l
Fig. 3. Fig. 4.
Fig. 3. The ratio Slls~ versus (a(P4~~) with 3t 10~?
s (solid line) and 3t 10~~
s (dotted line).
Fig. 4. The ratio p*/p i~ei.vus (a(P4~~).
N° ON SCHOCKLEY-READ-HALL MODEL WITH FINITE RELAXATION TIME 39
value of optical flux. If surface potential ~
s ~ 0 which means accumulation layer at the surface and with some algebric manipulation of equation (13) taking account of (24) we get
U
= Sf exp (- ~bs) &p (26)
we used U
= S~~~ &p and we obtained
~ctf
" S/ eXp (- ~bS) (27)
The case of intrinsic semiconductor surface corresponds to a potential surface given by
relationship
~b -li~( (w)j
~ 2 n(w>) (28)
We obtained for the effective surface recombination velocity
s * s *
S~tf ~ ~ exp(- ~b~) (29)
(St ~ S/
For inversion and depletion layer we get
s~~~ = St exp (~b
s
30) In Figure 5 we have plotted the effective surface recombination for different potential surface.
. a
~
lb
i
Is
Fig- 5. Effective surface recombination velocity for low level of injection vet.sus surface potential.
it) Under the moderate level of injection so that term with K can be neglected in the
expression (13), but N~D, P~~ are important and they can not neglected. We get for
~bs ~ 0.
Sf exp(- ~bs )
~~~~ II ~ S/ P
~~ ) ~~
and for ~b
s ~ 0
St exp (ds )
~~~~ (32)
(l ~ St NRD
In this case we analysed the influence of trap density N~ doping concentration
N~ and the deep of trap level. We have plotted this in Figure 6. It is interesting to note that values of ratio S~~~/Sf exp(- ~bs) for N~
=
lo ~~ cm~ ~, N~ =
10'~ cm~~
are equal with same ratio for N~
=
lo ~~ cm~ ~, N~ =
10~~ cm~~ for trap energy
e~ = e~ 0.35 eV. In Figure 6 we plotted with dashed line the same ratio for trap energy e~= e~-0.57eV [7] with
N~
=
10'~ cm~ ~ N~
=
10'~ cm~~ and N~
=
10'~ cm~~
1.0
__ --
--~
0.8 l''
'$w
*« 0.6 i~
°' 0.2
5
Fig. 6. Normalized effective surface recombination velocity for moderately level of injection versus
surface potential.
Therefore the decrease of doping density, the proximity of energy level traps by the middle of forbidden gap and obviously the increase of traps density lead to decrease of surface recombination velocity. It is seen from Figure 6, that in the case of moderately level of injection
occurs a new mechanism which increases the surface recombination velocity versus surface
potential. This mechanism is due to diffusion and recombination process in surface layer, but the exponential hides it.
The influence of illumination and relaxation time of traps are included in Sf.
ii) For high injection level ~bs ~ 0 we get
~
Sf exp(- ~bs)
~~~ l ~ S/ (PRD ~ ~n y eXp (-~bS))
~~~~
and for ~bs ~ 0
~
Sf exp (~b
s
~~ l ~ St (N~D ~ &p y exp (~bs)) ~~~~
In this case the influence of relaxation time of traps are included not only in Sf but also in y. In Figure 7 we plotted the ratio S~~~/S~ exp (~bs with following parameters N~
=
lo '~ cm~ ~, N~ =
10'~ cm~ ~,
e~ = e~ 0.57 eV, &n
=
10'~ cm~ ~, &t
= 10~ ~ s, for A and we change successively only one parameter as follows N~ = 10~ '~ cm~~ for B, &t 10~9
s for C.
N° ON SCHOCKLEY-READ-HALL MODEL WITH FINITE RELAXATION TIME 41
io
3 5 7 9 lo
Is
Fig. 7. Normalized effective surface recombination velocity for high level of injection versus surface
potential.
In Figure 7, we have plotted the same ratio like in the previous case, to follow the evolution of this new mechanism. The increase of traps density leads obviously to decrease of surface recombination velocity, and the decrease of relaxation time of traps yields a great increase of surface recombination velocity. In both cases at high surface potential saturation effect can be
seen. We can conclude that at high level of injection the behaviour of surface recombination
velocity versus surface potential is different. The new mechanism which had been seen in the moderately level of injection, become important and changes the behaviour of surface
recombination velocity.
4. Conclusion
The assessment of surface recombination velocity i>eisus material parameter is necessary at
least for two reason optimisation for optoelectronical devices and determination of carrier
lifetime on semiconductor wafer. The surface recombination velocity is particularly important
for silicon solar cell in which most of the illuminated surface is not covered by metal, but
covers with thermal or native SiO~. Arora et al. [I I] analyse the normalized efficiency as a
function of surface recombination velocity. They demonstrated that the decrease of normalized
efficiency is at least lo §b when the surface recombination velocity increase from 104 to 105 cm/s. On the other hand the transit time and short-circuit current are considerable different for variable values of surface recombination velocity and are decisive for optimal structure of
solar cells.
Determination of bulk lifetime of carriers on semiconductor wafer is compromise by surface recombination velocity. These two parameters can not be separate by conventional measure-
ment method and the evaluation of surface recombination velocity leads to a correct
determination of bulk lifetime carriers on semiconductor wafer.
The effective recombination velocity has been studied using Schockley-Read-Hall as a
model taking into account the direct photoionization of defect states and saturation effect of trap levels. The analysis presented brings out the need for a careful evaluation of surface
recombination velocity and its relation with other parameters.
Conditions have been chosen for which analytical expression for surface recombination
velocity have been obtained.
The diffusion and recombination effect for carriers becomes important for moderately and high injection cases.
