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Drift mobility measurements in thin epitaxial semiconductor layers using time-of-flight techniques
Solid-State Electronics, 1974, Vol. 17, pp. 805-812.
Pergamon Press.
Printed in Great Britain
DRIFT MOBILITY MEASUREMENTS IN THIN EPITAXIAL SEMICONDUCTOR LAYERS USING TIME-OF-FLIGHT TECHNIQUES A. G. R.
EVANS
Department of Natural Philosophy, Strathclyde University, Glasgow, Scotland and P.
N.
ROBSON
Department of Electronic and Electrical Engineering, Sheffield University, Sheffield, England (Receiued 17 August 1973) Abstract-The conventional time-of-flight technique for measuring drift velocity in high mobility semiconductors is limited to relatively thick ( > 200 wrn) and high resistivity material. The microwave time-of-flight technique described here allows thin (- 10 pm) low resistivity epitaxial layers to be measured with greater accuracy than is afforded by the conventional technique. The new experimental method is applied to the measurement of electron drift velocity in n-type GaAs at room temperature. The method, and its theory, is described in detail and its advantages and limitations are discussed.
INTRODUCTION
The conventional time of flight technique which was developed initially by Spear [ 11 for measuring drift mobility has been used successfully for
velocity-field (u/F) measurements of carriers in GaAs[2] InSb[3], Si[4-61, Ge[7] ZnSe[8]. In the usual arrangement the sample to be measured has an electric drift field F established between its contacts. One of these contacts is normally a Schottky barrier or suitably blocking contact and the drift voltage V is applied so that the sample is depleted. The average electric field F in a sample with contacts of separation 1, is equal to V/l. A high energy bunch of electrons from an electron gun source bombards the sample through one of the contacts and dissipates its energy by creation of electron-hole pairs (one electron of 10 keV energy creating - 10’ secondaries). The drift of the secondaries to the other contact is accompanied by current flow in the external circuit of the sample. This measured current pulse duration T is the transit time. The velocity v(F) is then calculated simply from u(F)=-
change in field across the sample AF, due to space charge, is small compared with F; AF is given by
where p. l
As the resistivity of the sample is decreased this field taper AF becomes significant, and the transit time is then given by
’ dx ?- = f-IJ u(x) where x is the distance measured from the blocking contact. The average velocity vov is given by l/r and the true v/F characteristic can be extracted from the v.,/F., characteristic as described in the appendix I(a) provided the field taper is not too great. THEORY OF T?IE MICROWAVE TIME-OF-FLIGHT METHOD
1
49
In our experiment a Schottky barrier was bombarded and the effects of electron transits were observed. The uniform field (F = V/l) approximation is good for material of low carrier density where the
net ionized impurity concentration dielectric constant of semiconductor.
In many materials prepared for device applications, however, the doping is sufficiently high that it is only possible to fully deplete relatively thin samples, otherwise avalanche breakdown occurs. Such samples may well be less than 10 pm and the corresponding transit time of carriers, (at least near 805
806
A. G. R. EVANS
velocity saturation) is - 100 psec. Such short transit times are extremely difficult to measure, and perhaps more importantly, the injected bunch of carriers produced by the electron beam has to last for much less than 100 psec. The microwave time of flight technique described here seeks to overcome these two problems. Since it also allows measurements to be made on thin samples, trapping of injected charge is much less important than with the classical time of flight experiment. If the sample is bombarded with an electron beam which is density modulated at a microwave frequency[9] then the RF current which flows in the external circuit of the sample is modulated at the same frequency but there is a phase delay in this signal caused by transit effects. The amplitude or phase of this signal is dependent on the electron mobility and the electron velocity-field characteristic can be extracted from measurement of these quantities as shown below. The method could be used for measuring hole mobility by bombarding from the other contact. Considering all quantities to vary in time as exp (jot) and considering only the one dimensional problem we can write the current continuity equation at any point as _
where 5 is current density and 6 is free carrier density (superscript tilde denotes a quantity varying at the microwave frequency). Using j = p’u in (4) diffusion we obtain _
_
g+jmf=O,
where v is the local drift velocity. The solution of this equation is
i=Lexp(-jw[s), where &, is a constant depending on the electron beam current and energy. The exponential term represents the phase of the RF convection current density. Poisson’s equation is aP fi -=-
ax
P. N. ROBSON
and when substituted
E
into the continuity
equation
$(.I+$)=0
gives (9) g represents the terminal current of the sample. Integrating equation (9) over the length of the sample and assuming the experiment is performed under constant voltage conditions we obtain
(10) using (6) in (10) we get g=$lexp[-
jtiI,‘-&)dx.
(11)
For the case of a uniform field u(x) = const. = 21 then equation (11) reduces to R = jo(exp (-~)}[sin~$~‘)}. A measurement term A where
$+j@=O,
and neglecting
and
(12)
of the phase term 4 or amplitude
A = I&lI*l will yield the velocity v since both the angular frequency w and sample length 1 are known. However only changes in phase or relative amplitude can be measured conveniently, not absolute values so it is necessary to know one point on the v IF curve a priori in order to calculate the absolute value of v as a function of F using this technique. It is not difficult usually to know one or more points on the v/F curve particularly at low fields. The advantage of this technique then is that the measurement of short pulses of induced current and their consequent distortion by the stray reactances of the mount is replaced by a measurement of phase and amplitude of a fixed frequency microwave signal which can be done with great accuracy. As an example the expected change in phase as the velocity varies from 2 X 10’ to 1 x 10’ cm/set (peak to valley for good GaAs material) would be 7~/2 at 10GHz for a 10 pm long sample. Here we
807
Mobility in epitaxial layers
have neglected field taper which results from the finite net ionized impurity density. We consider fully the effect of this field taper in the Appendix I(b). EXPERIMENTAL.
beam energy was measured by noting the beam deflection on a fluorescent screen to be (1.02 & 0.09) x lo-‘cm/V in good agreement with the calculated value of 1.04 X 10m2cm/V. The technique for microwave modulating the electron beam current was to deflect the beam to and fro in front of the sample aperture at a microwave frequency. The deflection system used to do this is shown in Fig. 2(b). It consists of resonant lecher lines A/2 long. The electric microwave stand-
TECHNIQUE
(a) Beam modulation The electron beam system used to bombard the semiconductor sample is shown schematically in Fig. 1. A beam of 12 keV electrons was electrostati-
II ---------------__ l
--------
_-Sample ,ll /
-12
Sample
kV
F
Cathode
Focus
Grid
Deflection
aperture
system
Anode
Fig. 1. Schematic diagram of electron beam system. Deflectnon plates
I -==y_-==
+v/2
_-
=
-
N-: Sample aperture
Sampling scope -
5oL-l Delay cables
-.----Fast
rise
pulse
Trigger
generotor
(a)
Fig. 2. (a) Conventional time-of-flight deflection system.
tally focused to a spot of < 300 I*m dia. at the sample. The deflection systems used are shown in Fig. 2(a) and (b). The system shown in Fig. 2(a) deflected the beam past the sample aperture and produced short pulses of electrons for a conventional time of flight technique. This system is similar to the one used by Sigmon et al. [4]. The deflecting voltage pulses of magnitude 125 V had rise time of < 0.5 nsec and produced a calculated transverse beam velocity of 3.8 x lo’cmlsec at the sample aperture. With a sample aperture of 400 pm dia. this gave pulses of electrons of duration calculated to be < 20 psec. The deflection sensitivity at 12 keV
ing wave pattern has an antinode half way along these lines and the beam passes between them at this point. The alternating electric field here produces a transverse velocity on the beam which results in it oscillating across the sample aperture. Since the beam passes the aperture twice per microwave cycle it is necessary to arrange the oscillator frequency to be at 5 GHz to produce the beam current modulated at 10 GHz. The lecher lines were enclosed in a circular waveguide, cut-off at the resonant fequency of the lines and end plates were fitted to further reduce loss from the system due to radiation. The cavity had a measured unloaded Q-
808
A. G. R.
EVANSand P. N. olotes
Fig. 2. (b) Microwave time-of-flight deflection system. value of approximately 1300 at 5.2 GHz and produced a measured deflection sensitivity at the sample of 0.05 mm/mW input power. (b) Sample structure and its external circuit The samples described here were made from a compensated epitaxial n-layer of GaAs, 10 pm thick grown on an n+ substrate. The material resistivity was 78 fi cm. and the corresponding field taper AF - 2 kV/cm. The slice was diced into 1 mm squares and each was alloyed with tin on to a gold plated molybdenum stub (see Fig. 3(a)). The epilayer surface was etched with a solution of N&OH-HzOrH20 (volume ratios 5 : 20 : 100). A thin coating of gold was evaporated ( - 150 A) from a molybdenum boat to form a Schottky barrier on the surface of the layer. The sample was mounted in a coaxial holder (Fig. 3(b)) and the drift voltage was applied using a bias tee made from a 2 cm section of stripline. The d.c. blocking capacitor was made by overlaying two pieces of the strip separated by a thin layer of melinex. The equivalent circuit of the diode is represented in Fig. 3(c) where C represents the capacitance of the diode and its mount plus any capacitance between the diode and the detecting system, R is the load resistance, nominally 50 0. (c) Measuring technique For the conventional time of flight technique the expected current response is shown in Fig. 3(c) for the two extremes of equivalent circuit time con-
ROBSON
stant. In our case RC * T (the transit time) and so the circuit integrates the current to give the response shape shown. The transit time is still measurable and this is discussed in more detail in the Appendix. In this case 7 was measured as a function of applied bias voltage V and the velocity (1/~) vs field (V/I) characteristic plotted as shown in Fig. 4. The resultant curve can be corrected for field taper and this corrected v/F curve is shown as the solid line in Fig. 4. The error introduced due to neglect of field taper is discussed in the Appendix where it is shown to be < 1 per cent for fields above 4 kV/cm. With microwave density modulation of the electron beam the detector used was a spectrum analyser. For the microwave amplitude measurements the magnitude of the detected signal at 10.4 GHz was recorded as a function of bias voltage. This sensitive detector was necessary because the signal level was - - 50 dBm. This was as expected from calculations; the sample and mount capacitance ( - 6 pF) present a low impedance which tends to short-circuit the signal at this high frequency (10.4 GHz).The dependence of current amplitude on bias field gives only the shape of the v/F curve and the absolute value of the velocity at one point must be known to calibrate the curve. This value could have been obtained from the low field mobility but in this case it was taken from a point on the conventional time of flight curve, the value was that corresponding to a field of 5.96 kV/cm. The points of this v/F curve are in good agreement with the results from the conventional time of flight and show less scatter. Again we have plotted an average velocity derived from equation (12) not taking into account the fact that there is a field taper across the device. The effect of taper is considered in the Appendix and for this case the error produced by neglect of this effect is less than f per cent in the present circumstances. The microwave phase measurements were made using an X-band waveguide bridge arrangement as shown in Fig. 5. The phase of the sample microwave signal was compared with that of the second harmonic output of the microwave oscillator, the latter being used as the reference. A null in the resultant signal from the magic tee was obtained by using the calibrated phase shifter in the reference arm of the bridge. The phase measurement is not absolute and the true phase for one point on the v/F curve was calculated using a point on the v/F curve determined by the conventional time of flight technique. The measured phases were thus calibrated and the v/F curve from these phases was cal-
809
Mobility in epitaxial layers Sold /
layer n-Type
epi - layer
(a) Molybdenum
Sample
stub
structure Stripline ____-----
bias tee
(b)
Coaxial sample mount Sample
I
t
(cl
Q
external
circuit
J
Sample and mount
R- Resistance
and
R
C
TI
I
C-
mounting
copacitonce
or detector
Fig. 3. (a) Sample structure. (b) Sample mounting and external circuit. (c) Equivalent circuit of diode and detecting system and its response to a current pulse resulting from a transit of a bunch of carriers across the sample.
I :
_ : :
i :
I: :
I 2
I 4
I
I
I
I
I
6
5
IO
12
14
i 16
F kV/cm
Fig. 4. o/F characteristic of sample of GaAs (78 S2cm. 10 /.~rnthick layer). 0 Conventional time-of-flight, 0 microwave amplitude time-of-flight, A microwave phase time-of-flight.
Fig. 5. Microwave bridge circuit for phase measurements. culated and is shown in Fig. 4. The agreement with the other results is seen to be good and like the microwave amplitude results the phase results show much less scatter than the conventional time
810
A. G. R. EVANS and P. N. ROBSON
of flight results. The correction that must be applied due to neglect of field taper is considered in the Appendix and it is less than 3 per cent for fields above 4 kVlcm.
8. J. L. Heaton, III, G. H. Hammond and R. B. Goldner, Appl. Phys. Letts 20, 333 (1972). 9. A. G. R. Evans, P. N. Robson and M. G. Stubbs, Electron. Letts 8, 195 (1972). APPENDIX
CONCLUSIONS The new technique has been demonstrated using a 10 pm thick GaAs layer where transit times are becoming difficult to measure accurately by the conventional methods. The results show an improvement already on the conventional measurement and the technique can be used for still higher conductivity material where the smaller depletion depth makes the conventional technique impossible. With the new technique diffusion and trapping effects have been neglected. Diffusion will give an added term to the phase delay and produce an error or order D202/v4, where D is the diffusion constant; this is negligible in the present case. Trapping will only be significant for trapping times
(I)Efects
of field taper on the v/F characteristic. (a) Conventional time of flight. The transit time of a plane of carriers across a sample of thickness 1is given by equation (3)
If F is not uniform as in the case where we have a field taper we can expand v(F) as a Taylor series
Where AE is the change of F across the sample, x0 equals 1/2 and v,, is the value of v at F = F,, (average field). Using (3) and (13) we obtain after expanding
_ I’ &
- x,)]‘$dx.
(14)
Writing 70= J?,dxlv” and noting that the second term in (14) is zero we obtain
L_AE*
a’v
(15)
24~0 aF2’
70
In the material of our experiment the second term in equation (15) is small. The value of a*v/aF*is 0.17 cm3/V2 set at an electric field of 4 kV/cm and the error introduced in the velocity by neglect of this term is less that 1 per cent for fields above 4 kV/cm. The method used to correct the conventional time of flight for field taper is now described. The transit times, r,z and r3., given by equation (3) for two bias voltages differing by AV can be written in the following form
Acknowledgement-We are grateful to Mr. M. Stubbs, who designed the microwave cavity, and to Dr. A. Majerfeld with whom we had many useful discussions. The work was supported by a grant from the U.K. Science Research Council. REFERENCES
[F(x
712=
I
Fz
.=I F4
734=
I
F,
E dF PO v(F) E
dF
PO v m’
Using AF = F, - F, = F4 - F2 = AVII we have
1. W. E. Spear, Proc. Phys. Sot. 78, 826 (1960).
2. J. G. R&h and G. S. Kino, Phys. Rev. 174,921(1%8). 3. A. Neukermans and G. S. Kino, ADDI. __ Phvs. _ Letts 17, 102 (1970). 4. T. W. Sigmon, J. F. Gibbons and C. B. Norris, Appl. Phys. Letts 14, 90 (1969). 5. A. Alberigi Quaranta, C. Canali and G. Otaviani, ibid. 16, 432 (1970). 6. C. B. Norris and J. F. Gibbons, IEEE Trans. Electron Devices ED-14, 38 (1967). 7. D. M. Chang and J. G. Ruth, Appl. Phys. Letts 12, 111 (1968).
now if AF is small
> giving
_=AF
=-=AF
811
Mobility in epitaxial layers Using equation (16) we can calculate u(FJ from a measurement of the slope of a 7 vs V graph if we know u(F2). If it is assumed that the measured transit time is the correct one for the average field across the device in the high field part of the characteristic where u is almost saturated, then u (FJ is known. The remainder of the true u/F curve can be calculated using equation (16) and working backwards from the saturated velocity. (b) Microwave amplitude and phase time-of-flight The expression for the current at the device terminals, derived earlier (equation (1 l)), is g=$lexp[-jw/:&]dx.
-! (a)
;
(11)
Considering a coordinate system with origin at + Z/2, i.e. device terminals now at ?1/2 then we have
(b)
1
F=F,-AEF where F, is the field at the origin and AE is now the total field taper across the device. The integral
-/ T’
I-=
dx 0 u(x)
(17)
f,
u
p
s
-5
now becomes
11
Fig. 6. Response voltage (b) of circuit shown in Fig. 3(c). to a current pulse (a). and is the uniform field result for the terminal current. The correction term for field taper is from equation (19)
when u0 = u at x = 0. On expanding the denominator order terms l+$+AE
.-
X
and neglecting higher
+exp(-~)($-$~~~~~[exp(-~) x x2-; (
I
1 ;+1/2 zzz-
+ 1 au AE (x*- 1142) I&*aF 21 . UO
(18)
(20) The integral in (20) can be accomplished final form of the terminal current is
Using (11) and (8) we have k=joexp(-%.[ (19)
we now expand
the second term of the expansion has its greatest value equal to w/u0 (Au/uO) l/8 which has a maximum value of 0.1 in the experiment undertaken here. The expansion thus need not be taken beyond this term. The first term of equation (19) then is
dx.
>I
by parts and the
sin (w1/2u,) _ jdu/aFAE wl/fuo 01
au/aF AE j +-.P.sin Wl/2VO Wl
-01
( 2UO11 .
cos ($1
(21)
The last term inside the square brackets is Au/&-times the first and in our case this multiplying factor is -0.025. As his term is also in quadrature to the first term it is neglected. The current is therefore approximated by
- jav/aF.AE 01
01 .‘““2v, 1.
The two terms of this expression are in quadrature and for our material the amplitude of the second term is less than 10 per cent of the first. Neglect of the second term produces an overall error in the microwave amplitude of less than $ per cent. The error in the phase by neglect of this term is less than 3 per cent for fields above 4 kV/cm.
812
A. G. R. EVANS and P. N. ROBSON
(II) The response of the sample circuit (Fig. 3(c)) to a current pulse (Fig. 6(a)) in the conventional time-of-flight method has been calculated and is shown in Fig. 6(b). The current pulse (Fig. 6(a)) has ramp leading and falling edges of time t’ due to the finite width of the injected electron
pulses. An estimate (7’) of the apparent transit time r was found to be approximately t’/6 too long using estimated values of 300 psec for RC and 20 psec for t’. This gives a maximum error of 3 per cent in the velocity for the samples used here.
Pergamon Press.
Printed in Great Britain
DRIFT MOBILITY MEASUREMENTS IN THIN EPITAXIAL SEMICONDUCTOR LAYERS USING TIME-OF-FLIGHT TECHNIQUES A. G. R.
EVANS
Department of Natural Philosophy, Strathclyde University, Glasgow, Scotland and P.
N.
ROBSON
Department of Electronic and Electrical Engineering, Sheffield University, Sheffield, England (Receiued 17 August 1973) Abstract-The conventional time-of-flight technique for measuring drift velocity in high mobility semiconductors is limited to relatively thick ( > 200 wrn) and high resistivity material. The microwave time-of-flight technique described here allows thin (- 10 pm) low resistivity epitaxial layers to be measured with greater accuracy than is afforded by the conventional technique. The new experimental method is applied to the measurement of electron drift velocity in n-type GaAs at room temperature. The method, and its theory, is described in detail and its advantages and limitations are discussed.
INTRODUCTION
The conventional time of flight technique which was developed initially by Spear [ 11 for measuring drift mobility has been used successfully for
velocity-field (u/F) measurements of carriers in GaAs[2] InSb[3], Si[4-61, Ge[7] ZnSe[8]. In the usual arrangement the sample to be measured has an electric drift field F established between its contacts. One of these contacts is normally a Schottky barrier or suitably blocking contact and the drift voltage V is applied so that the sample is depleted. The average electric field F in a sample with contacts of separation 1, is equal to V/l. A high energy bunch of electrons from an electron gun source bombards the sample through one of the contacts and dissipates its energy by creation of electron-hole pairs (one electron of 10 keV energy creating - 10’ secondaries). The drift of the secondaries to the other contact is accompanied by current flow in the external circuit of the sample. This measured current pulse duration T is the transit time. The velocity v(F) is then calculated simply from u(F)=-
change in field across the sample AF, due to space charge, is small compared with F; AF is given by
where p. l
As the resistivity of the sample is decreased this field taper AF becomes significant, and the transit time is then given by
’ dx ?- = f-IJ u(x) where x is the distance measured from the blocking contact. The average velocity vov is given by l/r and the true v/F characteristic can be extracted from the v.,/F., characteristic as described in the appendix I(a) provided the field taper is not too great. THEORY OF T?IE MICROWAVE TIME-OF-FLIGHT METHOD
1
49
In our experiment a Schottky barrier was bombarded and the effects of electron transits were observed. The uniform field (F = V/l) approximation is good for material of low carrier density where the
net ionized impurity concentration dielectric constant of semiconductor.
In many materials prepared for device applications, however, the doping is sufficiently high that it is only possible to fully deplete relatively thin samples, otherwise avalanche breakdown occurs. Such samples may well be less than 10 pm and the corresponding transit time of carriers, (at least near 805
806
A. G. R. EVANS
velocity saturation) is - 100 psec. Such short transit times are extremely difficult to measure, and perhaps more importantly, the injected bunch of carriers produced by the electron beam has to last for much less than 100 psec. The microwave time of flight technique described here seeks to overcome these two problems. Since it also allows measurements to be made on thin samples, trapping of injected charge is much less important than with the classical time of flight experiment. If the sample is bombarded with an electron beam which is density modulated at a microwave frequency[9] then the RF current which flows in the external circuit of the sample is modulated at the same frequency but there is a phase delay in this signal caused by transit effects. The amplitude or phase of this signal is dependent on the electron mobility and the electron velocity-field characteristic can be extracted from measurement of these quantities as shown below. The method could be used for measuring hole mobility by bombarding from the other contact. Considering all quantities to vary in time as exp (jot) and considering only the one dimensional problem we can write the current continuity equation at any point as _
where 5 is current density and 6 is free carrier density (superscript tilde denotes a quantity varying at the microwave frequency). Using j = p’u in (4) diffusion we obtain _
_
g+jmf=O,
where v is the local drift velocity. The solution of this equation is
i=Lexp(-jw[s), where &, is a constant depending on the electron beam current and energy. The exponential term represents the phase of the RF convection current density. Poisson’s equation is aP fi -=-
ax
P. N. ROBSON
and when substituted
E
into the continuity
equation
$(.I+$)=0
gives (9) g represents the terminal current of the sample. Integrating equation (9) over the length of the sample and assuming the experiment is performed under constant voltage conditions we obtain
(10) using (6) in (10) we get g=$lexp[-
jtiI,‘-&)dx.
(11)
For the case of a uniform field u(x) = const. = 21 then equation (11) reduces to R = jo(exp (-~)}[sin~$~‘)}. A measurement term A where
$+j@=O,
and neglecting
and
(12)
of the phase term 4 or amplitude
A = I&lI*l will yield the velocity v since both the angular frequency w and sample length 1 are known. However only changes in phase or relative amplitude can be measured conveniently, not absolute values so it is necessary to know one point on the v IF curve a priori in order to calculate the absolute value of v as a function of F using this technique. It is not difficult usually to know one or more points on the v/F curve particularly at low fields. The advantage of this technique then is that the measurement of short pulses of induced current and their consequent distortion by the stray reactances of the mount is replaced by a measurement of phase and amplitude of a fixed frequency microwave signal which can be done with great accuracy. As an example the expected change in phase as the velocity varies from 2 X 10’ to 1 x 10’ cm/set (peak to valley for good GaAs material) would be 7~/2 at 10GHz for a 10 pm long sample. Here we
807
Mobility in epitaxial layers
have neglected field taper which results from the finite net ionized impurity density. We consider fully the effect of this field taper in the Appendix I(b). EXPERIMENTAL.
beam energy was measured by noting the beam deflection on a fluorescent screen to be (1.02 & 0.09) x lo-‘cm/V in good agreement with the calculated value of 1.04 X 10m2cm/V. The technique for microwave modulating the electron beam current was to deflect the beam to and fro in front of the sample aperture at a microwave frequency. The deflection system used to do this is shown in Fig. 2(b). It consists of resonant lecher lines A/2 long. The electric microwave stand-
TECHNIQUE
(a) Beam modulation The electron beam system used to bombard the semiconductor sample is shown schematically in Fig. 1. A beam of 12 keV electrons was electrostati-
II ---------------__ l
--------
_-Sample ,ll /
-12
Sample
kV
F
Cathode
Focus
Grid
Deflection
aperture
system
Anode
Fig. 1. Schematic diagram of electron beam system. Deflectnon plates
I -==y_-==
+v/2
_-
=
-
N-: Sample aperture
Sampling scope -
5oL-l Delay cables
-.----Fast
rise
pulse
Trigger
generotor
(a)
Fig. 2. (a) Conventional time-of-flight deflection system.
tally focused to a spot of < 300 I*m dia. at the sample. The deflection systems used are shown in Fig. 2(a) and (b). The system shown in Fig. 2(a) deflected the beam past the sample aperture and produced short pulses of electrons for a conventional time of flight technique. This system is similar to the one used by Sigmon et al. [4]. The deflecting voltage pulses of magnitude 125 V had rise time of < 0.5 nsec and produced a calculated transverse beam velocity of 3.8 x lo’cmlsec at the sample aperture. With a sample aperture of 400 pm dia. this gave pulses of electrons of duration calculated to be < 20 psec. The deflection sensitivity at 12 keV
ing wave pattern has an antinode half way along these lines and the beam passes between them at this point. The alternating electric field here produces a transverse velocity on the beam which results in it oscillating across the sample aperture. Since the beam passes the aperture twice per microwave cycle it is necessary to arrange the oscillator frequency to be at 5 GHz to produce the beam current modulated at 10 GHz. The lecher lines were enclosed in a circular waveguide, cut-off at the resonant fequency of the lines and end plates were fitted to further reduce loss from the system due to radiation. The cavity had a measured unloaded Q-
808
A. G. R.
EVANSand P. N. olotes
Fig. 2. (b) Microwave time-of-flight deflection system. value of approximately 1300 at 5.2 GHz and produced a measured deflection sensitivity at the sample of 0.05 mm/mW input power. (b) Sample structure and its external circuit The samples described here were made from a compensated epitaxial n-layer of GaAs, 10 pm thick grown on an n+ substrate. The material resistivity was 78 fi cm. and the corresponding field taper AF - 2 kV/cm. The slice was diced into 1 mm squares and each was alloyed with tin on to a gold plated molybdenum stub (see Fig. 3(a)). The epilayer surface was etched with a solution of N&OH-HzOrH20 (volume ratios 5 : 20 : 100). A thin coating of gold was evaporated ( - 150 A) from a molybdenum boat to form a Schottky barrier on the surface of the layer. The sample was mounted in a coaxial holder (Fig. 3(b)) and the drift voltage was applied using a bias tee made from a 2 cm section of stripline. The d.c. blocking capacitor was made by overlaying two pieces of the strip separated by a thin layer of melinex. The equivalent circuit of the diode is represented in Fig. 3(c) where C represents the capacitance of the diode and its mount plus any capacitance between the diode and the detecting system, R is the load resistance, nominally 50 0. (c) Measuring technique For the conventional time of flight technique the expected current response is shown in Fig. 3(c) for the two extremes of equivalent circuit time con-
ROBSON
stant. In our case RC * T (the transit time) and so the circuit integrates the current to give the response shape shown. The transit time is still measurable and this is discussed in more detail in the Appendix. In this case 7 was measured as a function of applied bias voltage V and the velocity (1/~) vs field (V/I) characteristic plotted as shown in Fig. 4. The resultant curve can be corrected for field taper and this corrected v/F curve is shown as the solid line in Fig. 4. The error introduced due to neglect of field taper is discussed in the Appendix where it is shown to be < 1 per cent for fields above 4 kV/cm. With microwave density modulation of the electron beam the detector used was a spectrum analyser. For the microwave amplitude measurements the magnitude of the detected signal at 10.4 GHz was recorded as a function of bias voltage. This sensitive detector was necessary because the signal level was - - 50 dBm. This was as expected from calculations; the sample and mount capacitance ( - 6 pF) present a low impedance which tends to short-circuit the signal at this high frequency (10.4 GHz).The dependence of current amplitude on bias field gives only the shape of the v/F curve and the absolute value of the velocity at one point must be known to calibrate the curve. This value could have been obtained from the low field mobility but in this case it was taken from a point on the conventional time of flight curve, the value was that corresponding to a field of 5.96 kV/cm. The points of this v/F curve are in good agreement with the results from the conventional time of flight and show less scatter. Again we have plotted an average velocity derived from equation (12) not taking into account the fact that there is a field taper across the device. The effect of taper is considered in the Appendix and for this case the error produced by neglect of this effect is less than f per cent in the present circumstances. The microwave phase measurements were made using an X-band waveguide bridge arrangement as shown in Fig. 5. The phase of the sample microwave signal was compared with that of the second harmonic output of the microwave oscillator, the latter being used as the reference. A null in the resultant signal from the magic tee was obtained by using the calibrated phase shifter in the reference arm of the bridge. The phase measurement is not absolute and the true phase for one point on the v/F curve was calculated using a point on the v/F curve determined by the conventional time of flight technique. The measured phases were thus calibrated and the v/F curve from these phases was cal-
809
Mobility in epitaxial layers Sold /
layer n-Type
epi - layer
(a) Molybdenum
Sample
stub
structure Stripline ____-----
bias tee
(b)
Coaxial sample mount Sample
I
t
(cl
Q
external
circuit
J
Sample and mount
R- Resistance
and
R
C
TI
I
C-
mounting
copacitonce
or detector
Fig. 3. (a) Sample structure. (b) Sample mounting and external circuit. (c) Equivalent circuit of diode and detecting system and its response to a current pulse resulting from a transit of a bunch of carriers across the sample.
I :
_ : :
i :
I: :
I 2
I 4
I
I
I
I
I
6
5
IO
12
14
i 16
F kV/cm
Fig. 4. o/F characteristic of sample of GaAs (78 S2cm. 10 /.~rnthick layer). 0 Conventional time-of-flight, 0 microwave amplitude time-of-flight, A microwave phase time-of-flight.
Fig. 5. Microwave bridge circuit for phase measurements. culated and is shown in Fig. 4. The agreement with the other results is seen to be good and like the microwave amplitude results the phase results show much less scatter than the conventional time
810
A. G. R. EVANS and P. N. ROBSON
of flight results. The correction that must be applied due to neglect of field taper is considered in the Appendix and it is less than 3 per cent for fields above 4 kVlcm.
8. J. L. Heaton, III, G. H. Hammond and R. B. Goldner, Appl. Phys. Letts 20, 333 (1972). 9. A. G. R. Evans, P. N. Robson and M. G. Stubbs, Electron. Letts 8, 195 (1972). APPENDIX
CONCLUSIONS The new technique has been demonstrated using a 10 pm thick GaAs layer where transit times are becoming difficult to measure accurately by the conventional methods. The results show an improvement already on the conventional measurement and the technique can be used for still higher conductivity material where the smaller depletion depth makes the conventional technique impossible. With the new technique diffusion and trapping effects have been neglected. Diffusion will give an added term to the phase delay and produce an error or order D202/v4, where D is the diffusion constant; this is negligible in the present case. Trapping will only be significant for trapping times
(I)Efects
of field taper on the v/F characteristic. (a) Conventional time of flight. The transit time of a plane of carriers across a sample of thickness 1is given by equation (3)
If F is not uniform as in the case where we have a field taper we can expand v(F) as a Taylor series
Where AE is the change of F across the sample, x0 equals 1/2 and v,, is the value of v at F = F,, (average field). Using (3) and (13) we obtain after expanding
_ I’ &
- x,)]‘$dx.
(14)
Writing 70= J?,dxlv” and noting that the second term in (14) is zero we obtain
L_AE*
a’v
(15)
24~0 aF2’
70
In the material of our experiment the second term in equation (15) is small. The value of a*v/aF*is 0.17 cm3/V2 set at an electric field of 4 kV/cm and the error introduced in the velocity by neglect of this term is less that 1 per cent for fields above 4 kV/cm. The method used to correct the conventional time of flight for field taper is now described. The transit times, r,z and r3., given by equation (3) for two bias voltages differing by AV can be written in the following form
Acknowledgement-We are grateful to Mr. M. Stubbs, who designed the microwave cavity, and to Dr. A. Majerfeld with whom we had many useful discussions. The work was supported by a grant from the U.K. Science Research Council. REFERENCES
[F(x
712=
I
Fz
.=I F4
734=
I
F,
E dF PO v(F) E
dF
PO v m’
Using AF = F, - F, = F4 - F2 = AVII we have
1. W. E. Spear, Proc. Phys. Sot. 78, 826 (1960).
2. J. G. R&h and G. S. Kino, Phys. Rev. 174,921(1%8). 3. A. Neukermans and G. S. Kino, ADDI. __ Phvs. _ Letts 17, 102 (1970). 4. T. W. Sigmon, J. F. Gibbons and C. B. Norris, Appl. Phys. Letts 14, 90 (1969). 5. A. Alberigi Quaranta, C. Canali and G. Otaviani, ibid. 16, 432 (1970). 6. C. B. Norris and J. F. Gibbons, IEEE Trans. Electron Devices ED-14, 38 (1967). 7. D. M. Chang and J. G. Ruth, Appl. Phys. Letts 12, 111 (1968).
now if AF is small
> giving
_=AF
=-=AF
811
Mobility in epitaxial layers Using equation (16) we can calculate u(FJ from a measurement of the slope of a 7 vs V graph if we know u(F2). If it is assumed that the measured transit time is the correct one for the average field across the device in the high field part of the characteristic where u is almost saturated, then u (FJ is known. The remainder of the true u/F curve can be calculated using equation (16) and working backwards from the saturated velocity. (b) Microwave amplitude and phase time-of-flight The expression for the current at the device terminals, derived earlier (equation (1 l)), is g=$lexp[-jw/:&]dx.
-! (a)
;
(11)
Considering a coordinate system with origin at + Z/2, i.e. device terminals now at ?1/2 then we have
(b)
1
F=F,-AEF where F, is the field at the origin and AE is now the total field taper across the device. The integral
-/ T’
I-=
dx 0 u(x)
(17)
f,
u
p
s
-5
now becomes
11
Fig. 6. Response voltage (b) of circuit shown in Fig. 3(c). to a current pulse (a). and is the uniform field result for the terminal current. The correction term for field taper is from equation (19)
when u0 = u at x = 0. On expanding the denominator order terms l+$+AE
.-
X
and neglecting higher
+exp(-~)($-$~~~~~[exp(-~) x x2-; (
I
1 ;+1/2 zzz-
+ 1 au AE (x*- 1142) I&*aF 21 . UO
(18)
(20) The integral in (20) can be accomplished final form of the terminal current is
Using (11) and (8) we have k=joexp(-%.[ (19)
we now expand
the second term of the expansion has its greatest value equal to w/u0 (Au/uO) l/8 which has a maximum value of 0.1 in the experiment undertaken here. The expansion thus need not be taken beyond this term. The first term of equation (19) then is
dx.
>I
by parts and the
sin (w1/2u,) _ jdu/aFAE wl/fuo 01
au/aF AE j +-.P.sin Wl/2VO Wl
-01
( 2UO11 .
cos ($1
(21)
The last term inside the square brackets is Au/&-times the first and in our case this multiplying factor is -0.025. As his term is also in quadrature to the first term it is neglected. The current is therefore approximated by
- jav/aF.AE 01
01 .‘““2v, 1.
The two terms of this expression are in quadrature and for our material the amplitude of the second term is less than 10 per cent of the first. Neglect of the second term produces an overall error in the microwave amplitude of less than $ per cent. The error in the phase by neglect of this term is less than 3 per cent for fields above 4 kV/cm.
812
A. G. R. EVANS and P. N. ROBSON
(II) The response of the sample circuit (Fig. 3(c)) to a current pulse (Fig. 6(a)) in the conventional time-of-flight method has been calculated and is shown in Fig. 6(b). The current pulse (Fig. 6(a)) has ramp leading and falling edges of time t’ due to the finite width of the injected electron
pulses. An estimate (7’) of the apparent transit time r was found to be approximately t’/6 too long using estimated values of 300 psec for RC and 20 psec for t’. This gives a maximum error of 3 per cent in the velocity for the samples used here.