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Conductivity and mobility contactless measurements of semiconducting layers by microwave absorption at 35
GHz
E. Coué, J. Chausse, H. Robert, F. Barbarin
To cite this version:
E. Coué, J. Chausse, H. Robert, F. Barbarin. Conductivity and mobility contactless measurements of semiconducting layers by microwave absorption at 35 GHz. Journal de Physique III, EDP Sciences, 1994, 4 (4), pp.707-718. �10.1051/jp3:1994160�. �jpa-00249139�
Classification Physics Abstracts
07.50 72.80 81.70C
Conductivity and mobility contactless measurements of
semiconducting layers by microwave absorption at 35 GHz
E. Court, J. P. Chausse, H. Robert and F. Barbarin
Laboratoire d'Electronique (U.R.A. C-N-R-S. 830), Universitd Blaise Pascal, Clermont-
Ferrand II, 63177 Aubidre Cedex, France
(Receii,ed 8 June 1993, revised 17 Noi<ember 1993, accepted 7 January 1994)
R4sum4. Un dispositif de caract6risation de semiconducteurs par r6flectivit6 micro-onde h 35 GHz est pr6sent6. La mesure du coefficient de r6flexion et sa variation en fonction d'un champ
magndtique statique ext6rieur donnent accbs h la conductivit6 et h la mobilit6 de fines couches conductrices ou d'6chantillons massifs. Un outil de d6pouillement des r6sultats exp6rimentaux,
a~sociant un modble th60rique h des mdthodes numdriques, est proposd pour ddterminer de fapon fiable [es caract6ristiques du mat6riau. L'6chantillon obture totalement la section droite du guide
d'onde. Les mesures sont sans contact, non destructives pour l'dchantillon et rapides. Des r6sultats
exp6rimentaux montrant la validit6 de ce dispo~itif sont expos6s et confront6s h des mesures de Van der Pauw et d'effet Hall.
Abstract. A technique of semiconductor characterization by microwave reflectivity at 35 GHz is described. Measurement of the reflection coefficient, and its change as a function of the extemal static magnetic field, give access to the conductivity and the mobility of thin conducting layers or bulk samples. An analysis tool of experimental data. associating a theoretical model with numerical methods, is proposed for the determination of reliable material parameters. The sample fills completely the waveguide cros~ section. The measurements are contactless, non destructive and fast. Experimental results, demonstrating the validity of this device, are shown and confronted
with Van der Pauw and Hall measurements.
Introduction.
It is attractive to study the electrical behaviour of semiconducting layers by means of non destructive and contact free techniques. Conventionally, bulk semiconductors or epitaxial layers on semiinsulating substrates are analysed by Hall measurements as a function of
temperature. The carrier concentration N, mobility p and the compensation ratio can be extracted. The temperature dependence of the concentration N, gives informations about energy levels of carrier traps. However, this method requires the implantation of low resistive ohmic contacts on the sample. These contacts are sometimes difficult to realize and make the
sample useless.
For such reasons, an attempt to develop new contactless methods appeared in the literature
[1-6]. These methods use often techniques developed for electric and magnetic characterization
of dielectrics [7, 8]. The reflectance of microwaves in air space was applied to the study of
conductivity of semiconductor panels modulated by optical energy [6]. More recently, electric microwave absorption was shown to be a useful tool for the evaluation of active layers : such
techniques, first suggested by Braslau [3], employs the X band f
=
9.5 GHz. The evaluation of the microwave power absorbed by the sample allows the determination of its conductivity, if its real permittivity and its thickeness are known. The carrier mobility is given by a
measurement of magnetoresistance in an extemal magnetic field. This method presents the
advantages of being fast and non destructive.
In this paper, we present a microwave absorption device operating in the K band
f
=
35 GHz. In such a way, the characterization of smaller samples can be achieved. At this
frequency, the sample holder must be realized with care to avoid microwave loss which may disturb the measurements.
In section I, a theoretical model is developed taking into account the case of the waveguide
filled with n different homogeneous media. In section 2, a description of the experimental apparatus is given, followed by the measurement procedure. Section 3 is devoted to the
analysis tool of experimental data, using theoretical model associated with numerical methods.
Finally experimental results obtained by microwave measurements are presented and compared with Hall determination.
1. Theoretical basis.
1,I REFLECTION COEFFICIENT. Only the dominant TE10 mode will be taken into account.
The characteristic impedance for the TE10 mode for a rectangular waveguide filled with
isotropic and non magnetic medium I is given by
JW~IQ
~~~
Y,
where y, is the complex propagation constant in the medium I, w is the frequency of the wave and Ho is the permeability.
y, may be written as
Y, = ", +jP,
where a, and p, are respectively the attenuation and the phase constants per unit length. For a
wave which is propagating inside a waveguide whose walls are perfectly conducting, the
attenuation and the phase constants can be written as follows
~
l~
~
wj 2
~
wj
"> " f °J ~> l m + ", WE> 1~ m
°J °J
~
l~
~ WI ~
~ WI
fl,# W'F, I-fi +W~+WF, 1--
~
°~~ °~~
where E, and ~r, are respectively the real permittivity and conductivity of the medium I in which the wave is propagating, w~ is the cutoff frequency of the guide. If the medium
is air, the characteristic impedance is given by
~~ wj _1/2
zo> " / I m
o W
where e~ is the permittivity of vacuum.
d~ dj d~
%-+ W-+ %-+
1.__
~ Of
_
l~ 12 yj
Adjustable
in-I yn
~~~~~ ~~~~~~~
Pr £I
l~'
Xi X2 Xi-i Xi Xn-i Xn X
Fig. I. Cross ~ection of
a waveguide filled with different homogeneous media.
A cross section of a waveguide filled with different homogeneous media is presented
schematically in figure I. The voltage reflection coefficient r [9] at position -r, is :
r (,I, ) = ~
~~ ~~
Z(x, + Zo
,
z(,r~) is the impedance at position ,r,. The voltage reflection coefficient at x~ and at
xl
= a., + dx, in the same medium are related in a simple way
r (;~ = r (.;,> e~?< ~" ~"~
then
y (~ - )
i
r (r,'
) ' '
finally
=~,r = z~
,
?~~, + ~ ~~ Y .~i' ~
±o + z(x, th (y~(.;~ -,;, )
This expression can be used for the case of a waveguide filled with n different homogeneous
media [10]. We obtain the recurrence formula :
?(x, + =o th y~ d~
~~~' ' ~°'
zo + z(,r, th (y~ d,
where d, is the thickness of medium I.
The waveguide is closed by an adjustable perfect conductor (~r-m) at position
.i = a.,~. Medium n is an air space between the medium n and the metallic short circuit. This medium fi has a length d,~ and the permittivity of free space. The recurrence formula with
=(x,~) = o is used. Finally z(.r,) is calculated and we have access to the power reflection
coefficient at position xi
z(xj) zoj 2 P(xi)
= r(xi) r*(xi)
= z(xj) + zoj
The reflection coefficient is now connected with the physical characteristics of the different media, namely the conductivity, the thickness and the relative permittivity. The reflection coefficient measurement allows to determine the conductivity, if the thickness and the real
permittivity of the sample are previously known.
1.2 DETERMINATION OF MOBILITY BY MAGNETORESISTANCE EFFECT. in a semiconductor whose free carriers are subject to an alternating electric field, the conductivity is described by a
tensor when a static magnetic field is applied. The
~r~~, component, which is the only concerned
here [11] is given by :
~ro(I + jwr)
~~~ (l + jwr)~ + wj~ r~
where
eB Ne~r
~°cy"q "0~ ~%
w~~ is the cyclotron frequency; ~ro is the zero field conductivity for charge carriers.
N is the concentration, r the relaxation time, m * and e are respectively the effective mass and the electron charge.
If the microwave frequency is funher enough from the collision frequency of the carriers so that wr
~ l, then the real part of ~r~,, only should be taken into account. Moreover if w~~ r ~ l (low field domain),
~r~~ reduces for a limited dimension sample to a simple
expression [3] :
~r~
~r~=~ l + p~B~
The value of mobility is then extracted.
In the high field domain (w~~ r ~ l ), this expression is not valid, a « saturation » of the
magnetoresistance effect appears [12].
2. Apparatus and measurement procedure.
The experimental set-up is shown in figure 2. Rectangular waveguides (inside dimensions : 3.5 x 7 mm) are used. The apparatus includes a microwave source, an attenuator, a circulator,
a cell for carrying sample and a variable short circuit. The cell is inserted between the poles of the magnet supplying a magnetic field B up to 0.8 T. B is normal to the sample surface.
Experiments are realized at temperatures above the liquid nitrogen temperature. The entire sample holder, protected by a plastic film, is immersed into liquid nitrogen for low temperature
measurements. The sample holder is constituted by a kneed copper waveguide. A slot allows to introduce the sample into the microwave system. This slot is on the straight guide wall to avoid
microwave loss through current line discontinuity. The sample is held by a suction pump
against waveguide walls. The characteristics of our cell were determined with a network
analyser HP 85107. The Voltage Standing Wave Ratio (V.S.W.R.) of the cell is about 1.084 at
Power Meter
TEio Mode
Circulator
Ajustabie Short circuit
i
Attenuator
Sample
Magnet
Microwave
Source
Fig. 2. Experimental set-up of the microwave absorption apparatus.
35.275 GHz. A diagram of the cell is shown in figure 3. A no ideal short circuit as well as a no
perfectly conducting waveguide behind the sample, lead to measurement perturbations for
semiinsulating samples. For the measurements presented in this paper, this probleme has no consequence. A detector allows to measure the microwave power p reflected by the sample filling the waveguide cross section. The experiment power reflection coefficient may be
written as :
~ ~
P Po
where po is the incident power measured by putting an ideal conductor (~r - m in place of the sample. The experimental value of p is confronted with the theoretical value and gives
access to the conductivity of the semiconductor sample. By varying B, the mobility
p can easily and separately be deduced from the dependence of the resistance sample with the
magnetic field.
A£- /
~
tile sample
Fig. 3. Representation of the sample holder.
3. Analysis methods and experiments.
3.I CONDUCTIVITY MEASUREMENTS.
3.I.I Analysis methods of experimental data, According to the type of semiconductor
(~r,, e~, d,), the electromagnetic wave is transmitted or not through the sample. Two methods for analyzing the experimental results are necessary.
If, on one hand, the electromagnetic wave is not transmitted through the sample filling the
guide, then the short circuit position has no influence on the reflection coefficient.
If, on the other hand, the electromagnetic wave is transmitted through the sample, the short circuit position will notably change the value of the reflection coefficient.
Figure 4 shows the boundary between these two cases. For different sample thicknesses, we
have numerically determined the conductivity value for which the reflection coefficient
variation as a function of d,,, was in the order of 4fG. The boundary is quite enough
represented by a law I/(~r. d)
=
6 n for sample having a thickness lower than 50 ~m. For thicker samples, we note a deviation from this law (Fig. 4). If the thickness is 000 ~m, the boundary is given by II (~r. d
=
20 n.
Figures 5 and 6 show the p evolution with the conductivity for two distinct values of the short circuit position d~. The boundary between the two cases is here represented by the point
A (I/(~r. d ) = 6 n point A Fig. 5 II (~r. d)
=
14 n point A Fig. 6).
If the short circuit position has no influence, the conductivity of the material will be given by only one value of the reflection coefficient. A dichotomic research of the roots is applied for two distinct positions d,, of the short circuit. Whatever the position of the short circuit, only the f;xed root is the good one. In such a case, the higher is the conductivity, the higher is the
uncertainty upon its determination. For bulk samples, the upper measurement limit is about 30 coo (am )~ ' with an uncertainty lower than 15 fG.
Shot-t cii.r.nit tlo~ition ha~ no influrncp on r
j fi
~
~ lo
>~
t
>
C u
~ +
'o +
c +
O +~
tJ +
Stiorl r.irriit pcsilioii tias mfluenrp on rj ~+~
+
i io ioo i coo
d j~m)
Fig. 4. Simulation of the influence boundary of the short circuit position. (+) Conductivity for which the reflection coefficient variation
versus d,, is about 4 fb. (-) Law II («. d)
= 6 Q.
2
- c W U
$Z o U
~ o
z I
I
#~
lE+01 1E+02 1E+03 1E+04 1E+05
Conductivity o- (Qm)~'
Fig. 5. Evolution of the reflection coefficient versus « with d,, as a parameter for a layer
(e~ = 12, d 3 ~m ). Curve : d,~
= Am- Curve 2 : d,~
= A/2.
If the short circuit position has influence, there is a distinct value of the reflection coefficient for each position of the short circuit. The conductivity value is given by the best fit between the
calculation and the experimental measurements realized for many short circuit positions. To fit
the model parameters to the experimental data, a least square program, employing the
C
~ 1 A
/~
li
O U
~ 2
O Z U
@ w-
@ Ct
1E-01 1E+00 1E+01 1E+02 1E+03
Conductivity cn (Qml'~
Fig. 6. Evolution of the reflection coefficient i,eisus « with d,, as a parameter for a bulk sample
(Er = 12, d 400 ~m), Curve I d,, = AM- Curve 2 d~ A/2.
Marquardt algorithm [13, 14] is used. This algorithm is based on a maximum neighborhood
method in which the truncated Taylor series give an adequate representation of the non linear
model. This algorithm is well adapted to our application. We have only one parameter
~r to determine, if the thickness and the real permittivity of the sample are known, and many
experimental values. The degree of freedom of the system (number of measurements minus number of fitted parameters) is high and so the convergence is easier. The fit quality criterion X selected is the root mean square error divided by the sum of experimental data
lM j [y~~,(I y~~, (I)]~
~ ~l
Y~res(1 ~ ~
where M is the number of measurements, P the number of fitted parameters (P
= in our
case), y~~~(II the experimental value and y~~j(I the calculated value. The uncertainty on the
conductivity is given by the square root of its variance.
For characterizing conducting layer (e~j, ~rj,dj) grown on semiinsulating substrates (~~~, ~r~, d~), a multilayer simulation is necessary. The strong effect of insulating substrate parameters on the simulation of the reflection coefficient is shown figure 7. Curve I presents the case of a multilayer sample and curve 2 is related to the active layer without substrate. The
curve shape of the reflection coefficient as a function of short circuit position is strongly
dependent on the thickness d, and the dielectric constant e~~ of the substrate.
3.1.2 Results. Two examples of conductivity determination by the fit of the experimental
curve p = f(d~l, are presented figures 8 and 9.
In the first case (Fig. 8) it concerns an InP bulk sample, with known thickness
(d=400~m) and dielectric constant (~~=l?). The conductivity is fitted to I12 ± 1.12 (flm )~' with a quality criterion X of about 0.003. The conductivity value, given by
Van der Pauw's method, is 123 (am )~
i
fi
©
(z
8
g ~ l
( 2
j
f j / ~
az / I
', J I j
/
, / ' /
-/ '-
2 4 8 lo 12
dn (mm)
Fig. 7. Simulation of the evolution of the reflection coefficient versus short circuit position d~. Curve I (-) Waveguide filled with an epitaxial layer (p~,
= 12, d, = 3 ~m, «, =
900 (Qm)~') on an insulating substrate (e~
= 12, d,
=
450 ~m, «~
= (Qm)~ ~). Curve 2 (---) Waveguide filled with only the epitaxial layer (p~, = 12, di " 3 ~m, ml 900 (Qm )~~) without substrate.
Bulk lnP
~C
f
~y 4Z
f~ MM «M««** ~««
O
C ~
.° j~
~ >#f
@ tC@
O
dn mm)
Fig. 8.
for a InP ample (p~ = 12, d = 400 ~m).(x) experimentalalues. (-) theoretical curve
fit to a conductivityvalue of112 (flm)-1
Epitaxial layer lnP
i
»
O 2 4 6
dn (mm)
Fig. 9. Evolution of the reflection coefficient versus the short circuit position d~ at 77 K for an InP
epitaxial layer on InP semiinsulating. Substrate characteristics: p~~=12, d,=450~m,
«, = 5 x lO~~(Qm)~' Layer characteristics
E~, = 12, d, = 3 ~m, «~ = 857(Qm )~' fitted parameter.
In the second case (Fig. 9) it concerns an InP epitaxial layer on InP semiinsulating substrate.
The substrate characteristics as well as the thickness and the dielectric constant of the layer
were previously knowu- The layer conductivity ~r~ is fitted to 857 ± 2 (am )~ with a quality
criterion x of about 0.005. The conductivity value, given by the Van der Pauw's method is 770 (flm )~
A comparison between conductivity measurements obtained by microwave method and
conductivity measurements obtained by Van der Pauw's method is shown in figure 10. Ten bulk samples are analyzed. Their thickness is included between 250 ~m and 230 ~m. A good agreement is observed between the two methods ; the maximal difference is less than 15 fG for bulk samples.
3.2 MOBILITY MEASUREMENTS. As seen in paragraph 1.2, when a static magnetic field is
applied perpendicularly to the sample plane, the conductivity ~r(B) is a function of B and p. For each value of the magnetic field, ~r (B is obtained from one of the two methods
previously described. This allows the mobility determination. An example of magnetoresist-
ance effect is presented in figure II for an InAs bulk sample. The experimental values of the conductivity as a function of B, induce to determine the mobility by fit using the Marquardt algorithm. The mobility is fitted to 22590±587cm~V~~ s~' with a quality criterion X =
0.007. This value of mobility is close to the value 23 000 cm~ V~ s~ obtained by Hall
measurement. When B increases, the high field domain is reached beyond 0.2 T. This is
evidenced by the occurring of saturation effects which cause ~r(B) to deviate from the
performed fit. Thus the mobility fit is accurate for B
~ 0.2 T. However, if the sample mobility
is too low, no significant magnetoresistance effect are detected. The sensitivity of our apparatus is such that no reliable measurements can be obtained with mobilities less than about 3 000 cm~ V~ s~
Conductivity o- (Qm)"~
W io
)
g@ i
Y
o
~§
~
i o ioo i coo io coo ioo coo
Van der Pauw values
Fig, 10. Comparison of microwave measurements of conductivity to values obtained by Hall
mea~urements at room temperature for bulk samples.
Bulk lnAs
b
~
~£
Z U
~
~
z «
O «
~ «
«
O.05 o-lo 5 O.20
B ~T)
Fig. II. Variation of the conductivity versus the magnetic field B for an InAs bulk sample. (x) experimental values. (-) theoretical curve evaluated by fit inducing to a mobility value of 22 590 cm2 V-' s-'
