CARRIERS LIFETIME MEASUREMENT IN POWER SILICON DEVICES BY
TRIBICC
C. Manfredotti, E. Vittone, F. Fizzotti, A. Logiudice, C. Paolini and P. Olivero
Experimental Physics Department, University of Torino, Italy
INFN, Sezione di Torino and INFN, UdR Torino, Via Giuria 1 I-101125 Torino ( Italy )
TRIBICC ( Time Resolved Ion Beam Induced Charge Collection ) represents a further improvement with
respect to more traditional IBICC, since it can supply not only the charge collection efficiency ( and
consequently data on mobility and trapping time of carriers in drift regions ), but also its time behaviour. At
long collection times, it can supply informations about diffusion lenghts and lifetimes of carriers in the
diffusion regions, which are always present in undepleted devices such as power devices. These data are of
paramount importance as inputs for device simulation codes. A more general and more powerful TRIBICC
method is presented in this work by using Gunn's theorem and a particular formulation of the generation
function in order to solve the adjoint of the continuity equation. This method represents a strong improvement
with respect to a previous one, in which lateral IBIC was used. An application of this method to a commercial
power device is presented and discussed. By using microbeams, lifetime mapping by this method could be also
possible.
has not such constraints and which can offer a direct
numerical solution in terms of charge collection
efficiency. The results indicate that TRIBICC is a more
powerful method with respect to IBICC and that it can
be used in more general conditions.
INTRODUCTION
The knowledge of important parameters like minority
carriers lifetime is of paramount importance not only
for semiconductor materials producers, but also for
computer simulation applications in which a first value,
even approximate, of these parameters is needed in
order to carry out device simulations. Lateral IBIC can
be used and it has been used to this purpose since it
probes directly the diffusion region giving very
accurate results. However, in order to be applied on a
finished device, it requires a time-consuming operation
in order to prepare and passivate the device crosssection. TRIBICC is particularly useful in this respect,
since it can be applied to a finished device without any
further operation, it can work in a wide range of
lifetime values ( which is a good opportunity for new
materials ) and its depth of investigation can be pushed
towards 100 µm ( which is a " must " for several kinds
power devices ). From simulation point of view, power
devices display complex geometries, with both drift
and diffusion regions which can move under
subsequent heat treatments and, as a consequence, they
cannot be treated under simplified assumptions.
Moreover, in order to analyse both IBICC and
TRIBICC data, standard Ramo's theorem (1, 2 ) cannot
be a good start, since it is not valid in presence of
diffusion regions or, even worse, of space-charge
regions driven by an external bias. In this work, we use
a different approach based on Gunn's theorem which
THE THEORETICAL MODEL
Let us consider a closed region of volume Vol,
bounded by two infinite electrodes spaced by d and
maintained
at
constant
potentials
(Φ(x=0)=0,Φ(x=d)=V) by an external power supply.
Inside the bounded region, there exists a semiconductor
medium with dielectric constant ε and a volume charge
density distribution ρ(r,t) = ρ2(r) + ρ3(r,t), where ρ2(r)
is a fixed charge distribution and ρ3(r,t) is due to the
mobile carriers generated at time t=0 in a certain
position r=r0. In order to evaluate the charge induced
by the motion of carriers, the Green’s reciprocal
theorem ( 3 ) is considered. If Φ is the potential of the
electromagnetic field due to the volume charge density
ρ(r,t) with both the electrodes grounded and Φ’ is the
potential due to the electrodes with their actual
potentials in the absence of any volume charge
distribution, then
∫
ρ(r,t)Φ′(r)d3r + ∫ σ(r, t)Φ′(r)ds = 0
Vol.
S
where σ is the surface charge density at the electrodes
due to the presence of the volume charge density ρ, S
is the total area of the electrodes bounding Vol. and ds
is the oriented element of S.
CP680, Application of Accelerators in Research and Industry: 17th Int'l. Conference, edited by J. L. Duggan and I. L. Morgan
© 2003 American Institute of Physics 0-7354-0149-7/03/$20.00
351
Since Φ’ is constant at the electrodes and is given by
the bias potentials and since the surface integral of σ is
equal to the total charge QS at the electrodes, one gets
QS = −
electric field E is constant (since E=V/d), the derivative
is reduced to a mere constant (dE/dV=1/d), and we get
again Ramo's theorem. To evaluate the actual induced
current at the electrodes, we have to calculate the
following expression:
1
ρ(r, t)Φ′(r)d 3 r
V ∫ Vol
iS = ∫
and the induced current iS entering the electrode is:
∂ρ 3 (r, t)
dQS
1
Φ′(r)d 3r
=− ∫
iS =
Vol
∂t
dt
V
t
t
∂E(r,V) 3 

Qs (r°, t) = ∫ iSdt = ∫ ∫ j(r,t) ⋅
d rdt
Vol
∂V

0
0
where the position vector r0 indicates the point where
the charge is generated at time t=0.
In order to interpret TRIBICC measurements by means
of this equation, let us consider the generation at t=0 of
Neh electron/hole pairs at x = x°. In the following we
will consider a uni-dimensional model in which the
coordinate x is orthogonal to the electrodes. Neh is
v
iS = −q
d
which is the usual expression of Ramo’s theorem ( 1 ).
As already pointed out by Cavalleri et al. ( 2 ) the fixed
space charge affects only the motion of charge carriers
and is not directly involved in the expression of the
induced current. In effect
equal to the ratio
1
j(r, t) ⋅ E' (r)d3 r + ∫ j(r, t)d 2 s
∫
Vol
S
V
Σ ion
where Σion is the ion energy
w eh
and weh is the electron-hole pair creation energy (3.6
eV in silicon). Because of the short time involved in
the ionization process, the charge carries are
considered created all at one time (5, 6 ).
The current density j flowing in the diode is given by:
where E’ = − ∇Φ ′ is the electric field due to the actual
potential at the electrodes in the absence of any volume
charge distribution. The surface integral in the equation
concerns the current entering the electrode, whereas the
volume integral is relevant to the actual induced
current.
The above equation
is the final result of the
generalized Ramo’s theorem ( 2 ) and it is based on the
assumption that the space charge distribution in the
volume of the detector is independent of the external
bias voltage, i.e. ρ2(r) does not depend from V. Such
an assumption is no more valid in the case of a silicon
p-n junction diode, where the space charge
redistribution plays a key role in the determination of
its rectifying behavior.
Hence, the analysis of the charge collection process
should be based on the application of the generalized
Ramo-Gunn’s theorem ( 4 ). According with this
theorem, the point charge q moving with the velocity v
between two parallel electrodes spaced by d induces, in
the external biased circuit, the following induced
current iS:
iS = − q ⋅ v ⋅
∂E(r) 3
dr
∂V
The induced charge is obviously obtained integrating in
time the induced current evaluated in equation:
If there is a single charge q generated in r0 and moving
with a constant velocity v along the x axis, being
ρ3(r,t) = δ(x-(x0+vt)), where δ is the Dirac’s delta
distribution function and Φ’(r) = (V/d)·x, one obtains
iS =
Vol
j(r, t) ⋅
∂n
∂p 

j = q ⋅ n ⋅ µ n ⋅ E + p ⋅ µ p ⋅ E + D n ⋅ − D p ⋅ 
∂x
∂x 

where q is the elementary charge and the excess
electron (n) and hole (p) density can be evaluated by
solving the relevant transport-diffusion equations with
suitable boundary conditions.
Such a current density is then inserted in the previous
equation to get the time evolution of the induced
charge generated at x0.
It is worth noticing that QS(x0,t) as defined by the
previous equation is the Green's function for the
continuity equations. The CCE profile is then evaluated
by solving a single, time dependent equation with a
significant reduction of computational effort ( 7 ).
An efficient method to evaluate the Green's function
has been recently presented by T.H Prettyman ( 7 ).
Since the excess carrier continuity equation involves
linear operators, an adjoint continuity equation can be
constructed: for instance, for electrons
∂n +
∂n +
∂ 
∂n +  n +
 D n
 −
= −vn ⋅
+
+ G +n
∂t
∂x
∂x 
∂x  τ n
dE (r ,V )
dV
where dE/dV is the derivative of the local electric field
at the point charge with respect to the bias voltage
applied at the electrodes. It is worth noticing that if
there is no space charge distribution in the volume, the
Where the apex "+" indicates the adjoint electron
concentration. The adjoint term for electrons is
G n+ = + vn ⋅
352
∂E ∂ 
∂E 
- D n ⋅
∂V ∂x 
∂V 
and vn,p = µn,p·Etot is the drift velocity for electrons and
holes, respectively.
It has been demonstrated ( 8 ) that the charge induced
at the electrodes from the motion of electrons and holes
generated at point x° at time t is given by:
[
QS (x°, t) = q ⋅ n + ( x°, t ) + p + ( x°, t )
The energy loss profile Γ(x) of the protons within the
sample was evaluated by the SRIM2000 simulation
code ( 5 ). The output of the charge sensitive
preamplifier (Canbera 1004) for each individual ion
strike was digitized using a fast computer controlled
Lecroy WAVERUNNER LT342 digital oscilloscope
(0.5 Gsample/s). The time resolution was about 2 ns
and the transient was stored in 2500 points.
Fig. 2 shows the time resolved ion beam induced
charge signals evaluated at three different reverse bias
voltage.
]
if suitable boundary and initial conditions ( Ohmic
contacts at the ends and charge generation at t= 0 and x
= x° . It is worth noticing that the expression of the
adjoint terms differ from those described in ( 7 ) for
the presence of a diffusion term which is essential for
the calculation of the time evolution of the charge
collection profiles in partially depleted devices.
Finally, in order to compare such a model with
experimental data obtained in experiments where the
carrier are generated by MeV ion probes penetrating
the semiconductor device through the electrode, we
have to consider the contribution of the charges
generated at different positions and weighted by the
Bragg ( ion energy loss ) curve:
RESULTS AND DISCUSSION
The mathematical method described in section was
used to interpret the experimental data shown in Fig. 5.
The electric field and the carrier velocity profiles
evaluated by the PISCESII computer code ( 9 ) were
used as input parameters of the adjoint equations taking
into account the dependence of the mobility from the
doping concentration and electric field ( 10 ). The
behavior of the hole generating functions G+n at
different bias voltages are plotted in Fig. 3.
R
Q s (t ) = ∫ Q s ( x°, t ) ⋅ Γ ( x°) dx°
0
where Γ is the energy loss profile and R is the ion
range in the semiconductor.
Charge collection efficiency
1,0
EXPERIMENTAL SET-UP
NA+ND
The investigated sample was a commercial Mesa
Rectifier Diode with the typical vertical structure (
from the top or from incoming protons direction )
p+/n/n+. The doping profile, as measured by the
spreading resistance method, is shown in Fig. 1
IBICC measurements were performed at the Ruder
Boskovic Institute in Zagreb (HR) using a 4 MeV
proton beam. The proton flux was maintained at about
100 protons/s in order to avoid electronic pile-up,
surface trapping of generated carriers and to reduce the
radiation damage.
10
21
10
20
10
19
10
18
10
17
10
16
10
15
10
14
10
13
10
12
+
p
100 V
0,8
0,7
0,6
0,5
50 V
0,4
0,3
0,2
0,1
0,0
0
1
2
Time (µs)
3
4
Fig. 2 - Experimental time behaviour of charge
collection measurements carried out at different reverse
bias voltage. Open circles: experimental data;
continuous line: fitting curves
+
n
n
200 V
0,9
The best fit of the TRIBICC data of Fig. 2 were
obtained by solving the adjoint equations by means of
a one-dimensional finite difference algorithm,
assuming a hole bulk lifetime τ0 = (5±1) µs and a
dependence of the minority carrier lifetime τ from the
doping concentration given by the following
phenomenological expression ( 10 ) :
τ=
0
25 50 75 100 125 150 175 200 225 250 275 300
Depth (µm)
τ0
NA + ND
1+
N ref
where Nref is equal to 7.1 1015 cm-3 and NA,D are the
acceptor and donor concentrations.
Fig. 1 - Doping profile as evaluated by the spreading
resistance method; NA and ND represent the acceptor
and donor concentration, respectively.
353
Apart from the lifetime measurements evaluated by
means of the numerical solution of the adjoint
equations, some qualitative observations can be drawn
from the analysis of the TRIBICC signals.
The source term G+ is obviously connected with the
extension of the depletion layer (DL) where carriers
experience a very rapid drift. Ramo's and RamoGunn's theorems state that the charge is induced at the
electrode only if the carriers move in presence of the
applied electric field, i.e. only when the carriers move
within the depletion layer. This means that carriers
generated within the DL induce ( in less than a
nanosecond ) a very short current pulse which
corresponds to the sharp and intense increase of the
charge collection signal. The increase of the reverse
bias voltage yields an increase of the depletion layer,
which corresponds to a larger amount of carriers
generated within the DL and, consequently, a higher
charge collected in the first nanosecond. After this
short transient, the collection time is much longer. This
is due to the minority carriers generating in the neutral
region which diffuse towards the DL. For the above
mentioned theorem, charge is induced at the electrodes
only when these carriers move in presence of the
electric field, i.e. only when they enter the DL; this
means that their motion in the neutral region (E=0)
does not yield any charge signal and the long tail in the
TRIBICC signal is relevant to their arrival time at the
boundary of the DL.
TRIBICC is proposed as a standard method in order to
evaluate minority carriers lifetime in finished devices.
In the present TRIBICC version, Gunn's theorem and
the adjoint equation method to solve the continuity
equation is used in order to treat complex situations in
which both drift and diffusion regions are present. The
method introduces a generation function which better
defines the depletion or drift region and gives directly
the time behaviour of charge collection efficiency. The
fit on experimental data is carried out with only one
parameter. Preliminary results obtained on a power
mesa device have been presented here : they can be
shown to offer minority carriers lifetime values in
agreement with expectation ones and suitable for
applications in simulation of device design and
behaviour
Acknowledgments - Authors are particularly grateful
to Dr. M. Jaksic and Z. Pastuovic of Ruder Boskovic
Institute in Zagreb ( Croatia ) for help during the
mesurements and for fruitful discussions.
References
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50 V
9
2,5x10
100 V
9
2,0x10
G+p (s-1)
CONCLUSIONS
200 V
9
1,5x10
9
1,0x10
8
5,0x10
0
25
50
75
100
275
300
Depth (µm)
Fig. 3 - Hole generation function profile at different
reverse bias voltage.
354