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High-field autocovariance coefficient, diffusion coefficient and noise in InGaAs at 300 K
0038-I 101/87 $3.00 + 0.00 1987 Pergamon Journals Ltd
Solid-State Electronics Printed
Vol. 30, NO. 2, pp. 2355239, 1987 in Great Britain. All rights reserved
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0
HIGH-FIELD AUTOCOVARIANCE COEFFICIENT, DIFFUSION COEFFICIENT AND NOISE IN InGaAs AT 300 K B. R. NAG,’ S. R. AHMED’ and M. DEB ROY’ ‘Calcutta University, Calcutta, India and ‘Jadavpur University, Calcutta, India (Received 6 December 1985; in revised form 26 March 1986)
Abstract-Electron diffusivity in Ino,,,,Ga(,,,, As is calculated by the Monte-Carlo method. The autocovariance coefficient and correlation time of velocity fluctuations, frequency dependence of diffusion coefficient and figure of merit for noise are studied. The results indicate that the limiting frequency of to those of GaAs diodes. operation and noise in In<,,53,Ga(, 47jAs Gunn diodes would be comparable
1. INTRODUCTION The ternary alloy Ino 5,,Gao47jA~ is receiving attention as a prospective material for the realisation of high-efficiency Gunn diodes[l]. High efficiency is expected as the peak-to-valley ratio for electron velocity is high. Some experimental results have also been reported which support the expectations[2, 31. No results are, however, available on the noise characteristic. Estimate of noise behaviour may be obtained from the knowledge of electron diffusivity at the operating fields. However, the diffusion coefficient in Ino 53)Ga(,,,,, As has not yet been measured or calculated. It may be obtained for low-fields from the mobility data using the Einstein relation[4], but for high fields the relation is not applicable. On the other hand, high-field diffusivity may be calculated with good accuracy by the Monte-Carlo method, provided the physical constants of the material are known. A set of values for the physical constants was recently[5] obtained from the available measurements and pseudopotential calculations. The constants gave very good agreement between the calculated and measured values of electron drift velocity over a wide range of electric fields and lattice temperatures. These constants are used in this paper to calculate the diffusivity. The diffusivity has been obtained from the velocity autocovariance coefficient. The method of calculation of the coefficient is explained in Section 2. The characteristics of the covariance coefficients, diffusion coefficient and its field and frequency dependence are discussed in Section 3. The results are used to obtain figures of merit for noise behaviour and these are also compared with the values of GaAs and InP.
the autocovariance coefficient of velocity fluctuations [7-lo]. The latter method was used as the frequency dependence of the diffusion coefficient and the frequency limitations of the material for application in devices may be studied in addition to the diffusion coefficient. The formulae and the method of calculation for the various functions studied in this paper are given below. (a)
Autocovariance
OF CALCULATION
AND FORMULAE
of
velocity-time
c(s)=E{[v(t)-v(t)][v(t+s)-v(t+s)]),
(1)
where the E and the bar indicate the average over an ensemble of velocity functions v(t), s being the time lag. Electron-electron scattering was not included in the calculations as the effects of such scattering cannot be easily evaluated by the Monte-Carlo method. The e-e scattering is important for very large electron concentrations. For low electron concentrations, lattice scattering is more frequent and electron trajectories are not significantly altered by e-e collisions. The present results may, therefore, be considered applicable strictly to samples with such electron concentrations that lattice scattering is more predominant than e-e scattering. As the e-e scattering was neglected, c(s) was obtained from the time average of the velocity function for a single representation assuming the ergodic theorem[ 111 applies. The formula used for the evaluation of c(s) is: c(s)=(l/T)
2. METHOD
coefficient
The autocovariance coefficient function is defined[ 111 as:
T[v(t)-a][v(t
+s)-a]dt,
(2)
50 where
Electron diffusivity for high fields can be obtained by the Monte-Carlo method[6], either by calculating the variance of the displacements or by calculating
now d is the time average
of v(t), given by
I.
u.(t) dt.
v = (l/T) s II 235
(3)
236
B. R. NAG PI ul.
The velocity function was obtained by the usual Monte-Carlo procedure using for the physical constants, the values given in Ref.[S]. Calculations were performed for a uniform electric field across the sample. The field becomes non-uniform in an actual device when the differential mobility is negative due to the appearance of space charge. However, our objective is to compare the properties of %.,,, Ga,,.,,, As with those of GaAs and InP and the relative merits should not be altered by the nonuniformity of the field. The integrals in equations (2) and (3) were cvaluated by summation using the values of r(t) at intervals of 0.04~s. The simulation was carried out for sufficiently long time T so that the values of c converged to within 0.01% and that of c(s) to within 5%. It should be mentioned that the accuracy of 1: is very important for obtaining reliable values of c(s) as it is given by the difference of nearly equal terms for large values of the electric field. (b)
Diffusion
coejicient
The diffusion coefficient was evaluated frequencies using the formulae[ 121:
for different
cc D(w)
=
c (s)cos ws ds.
s0
(4)
The integral was evaluated from the values of c(s), calculated earlier for values of the time lag .Y at intervals of O.O4ps, using the Simpson formula. (c) Figure of merit for noise Noise in the semiconductor devices originate from different sources. However, for the Gunn diodes the thermal noise is an important component. The index of thermal noise is the noise temperature, which may be evaluated from the diffusivityymobility ratio for fields at which the mobility is positive. But, for operating fields at which the differential mobility is negative, the relation cannot be applied. However, diffusivity is an indicator of the noise, even though it may not give the temperature quantitatively. The ratio of the diffusion coefficient and the drift velocity has been used[l3] to estimate the noise performance of the devices. However, a more useful indicator of noise is the ratio of the diffusion coefficient, D (0) and the magnitude of the negative differential mobility, lpdl[14]. The generated noise is larger for a higher value of D(O), but the noise in the oscillator is reduced for higher values of pd due to the plug-in effect of oscillation. Both the ratios D(O)/6 and D(0)/Ip(dj have, therefore, been calculated for different fields to estimate the noise behaviour. (d)
Correlation
evaluated if the autocovariance coefficient decreases exponentially with a single time constant. However, it will be seen from the results presented in the following section that this is not true for the velocity functions at high fields. An effective correlation time, r,, was hence evaluated using the formula[7]: 7< =
3. RESULTS
D (0)/c (0)
(5)
AND DISCUSSSIONS
Calculations were made for the fields between 1 and 30 kV/cm at 300 K. The coefficients were evaluated for the component of velocity parallel to the applied field and also for the component transverse to the field. The autocovariance coefficients are presented in Fig. 1. The transverse coefficient decreases monotonically with time; the rate of decrease. increases with the field. The parallel coefficient also decreases monotonically with time for fields below the threshold field. On the other hand, for fields above the threshold field, the coefficient decreases, becomes zero, attains a minimum value and then increases to zero. The nature of variation of the autocovariance coefficients with the field may be explained as follows. The fluctuations of the electron velocity increases with the field as the electron system gets heated. At the same time, the collisions become more frequent and the correlation time decreases. The autocovariance coefficient, therefore, increases in magnitude but falls faster with increase in the field. However, at fields above the threshold field, as electrons are transferred from the r valley to the valleys with higher effective mass, the magnitude of velocity fluctuations decreases and the collision time decreases further. The function, therefore, decreases in magnitude and decreases still faster with the time
3.0
m-
2.7
time
The frequency limitation of the devices can be studied from D(w), as it involves the same time constants as are involved in drift mobility. A more quantitative estimate can be obtained from the correlation time. The correlation time can be easily
Fig. 1. Transverse and parallel autocovariance coefficients at 300 K; the number on the curve indicates field in kV cm ‘.
High-field effects in InGaAs at 300 K lag at very high fields. Also, at post-threshold fields, electrons scattered back from the satellite valleys to the F valley may have large velocities. If the velocity is in the opposite direction to the direction of acceleration due to the field, the electron initially looses energy as it is accelerated by the field. The scattering probability of such electrons therefore initially decreases with time. These electrons are scattered as the field finally accelerates them to high energies. The trajectories of such electrons are characterised by long flights in which the velocity continuously increases from the negative value to a positive value. The parallel autocovariance coefficient, therefore, becomes negative for large time lag when such trajectories predominate. The characteristic features of autocovariance coefficient discussed above are also exhibited by GaAs and InP and have been discussed in other literature[7,9, lo]. The field-dependences of the diffusion coefficients are illustrated in Fig. 2. The coefficients are plotted against the ratio of the applied field and the threshold field, so that comparison may be made between different materials. The threshold field was taken to be 3.2 kV/cm’. The computed low-field value of the diffusivity is 355 cm*/s, which agrees with the value calculated from the mobility using the Einstein relation. The diffusion coefficients increase with the field, attain a peak value at a field, a little higher than the threshold field and then decreases with the field. The diffusion coefficient is related to (v*T), where u is the fluctuation in velocity and 7 is the collision time. As discussed in connection with the autocovariance coefficient u* increases but 7 decreases initially with the field. Apparently, the rate of increase of u* is higher than the rate of decrease of 7, so that (0’7) and hence the diffusion coefficient increases. But at post-threshold fields, the transferred electrons have
0.L
0.1
APPLIED
ELECTRIC
1.0
l.0
FIELD!
THRESHOLD
231
lower values of u and u* also decreases with field as more and more electrons are transferred. This decrease in u* and the decrease in the value of 7 cause the diffusion coefficient to decrease rapidly with the field. The values of diffusion coefficients in GaAs and InP are also quoted in Fig. 2 from Ref.[lO]. As values were calculated for three fields only, these are indicated by squares and circles. The threshold fields for GaAs and InP were taken to be 3.2 and 10 kV/cm as accepted in the literature. It is seen that the transverse As are larger diffusion coefficients in In, 53)GaC0.47, than those in GaAs and InP. On the other hand, the parallel diffusion coefficients are nearly equal to those in GaAs but larger than those in InP. The frequency dependence of the diffusion coefficients is illustrated in Fig. 3. The transverse component, D,(w)and the below-threshold-field parallel component, D,(w), are initially constant and then decrease monotonically with increasing frequency. The above-threshold-field parallel components, on the other hand, increase with frequency initially, attain peak values and then decrease with frequency. The nature of the curves may be understood considering that D (0) is the Fourier transform of c(s). Hence, as c,(s) and below-threshold-field cp(s) decrease monotonically with S, D(o) decreases monotonically with frequency. Also, as abovethreshold-field cp(s) changes sign with s, D,(w) shows a peak around a frequency corresponding to the period of change in sign of c,,(s). The same features are also seen in the D(W)-Jcurves of GaAs[lO] and InP[lO, 161. We also find on comparing the characteristics of D (0)-fcurves for GaAs[ lo] and InP[lO] with those in InC,,J,,Gao,,,As t:; the limiting frequency of operation of Ino,,,, C0.47,A~would be comparable to that of GaAs but much lower than that of InP.
10
FIELD
Fig. 2. Transverse and parallel diffusion coefficients at 300 K. Parallel diffusion coefficient-In0,,,,Ga,,4,, As (continuous line), GaAs (open square), InP (open circle). Transverse diffusion coefficient-Irq,,,,,Ga(,,,, As (broken line), GaAs (shaded square), InP (shaded circle).
FREOUENCY
Fig. 3. Frequency dependence of tinuous line-Parallel diffusion Transverse diffusion coefficient. indicates field in
IGHzI
diffusion coefficient. Concoefficient, Broken lineThe number of the curve kV cm- ‘.
B. R. NAG et al
Fig. 4. Correlation time of velocity fluctuations. Correlation time for parallel velocity-In&,,Ga(O,,,As (continuous line), GaAs (open square), InP (open circle). Correlation time for transverse velocity--In(,,,,Ga(,,,,As (broken line), GaAs (shaded square), InP (shaded circle).
The
correlation
times
for different
fields
are illus-
decreases with the field above the threshold field. The time for the parallel velocity decreases faster than that for the transverse velocity. Again, on comparing the correlation times for the parallel velocity, which is a measure of the frequency limitation of the material for Gunn diodes, we find that the times are comparable to those for GaAs but much higher than those for InP. The results on correlation time thus support the conclusion made from D(w)-f curves about the relative frequency limitations of In, s3,Ga,, 47jA~, GaAs and InP. It is well known that the frequency limitation of devices trated
utilising electron-transfer-effect arises mainly from the intervalley transfer time. An approximate formula has been given by Ridley[ 171 for estimating the limiting frequency from such considerations. The values given by this formula are 40, 50 and 120 GHz for Ino 53jGaC0.,,jAs, GaAs and InP respectively. These values support the conclusion made above from the present analysis. The figures of merit for noise performance are illustrated in Fig. 5. The diffusivity-velocity ratio decreases with increasing field but the diffusivitydifferential mobility ratio, D(0)/fpdI increases with the field. The field dependence is again identical to that for GaAs and InP. Both the ratios have values comparable to those for GaAs but higher than those for InP. We may note that the accuracy in the values of D (0)/z? is about 5%, but that for D (0)/jpdl is poorer as pd was estimated from the values of d for fields differing by 1 kV/cm. The figures of merit should, therefore, be considered to give qualitative rather than quantitative measures. Nevertheless, one may conclude from the large differences in values that As devices would be much the noise in Ino,,,Ga,,,,, larger than in InP diodes, but similar to that in GaAs.
in Fig. 4. The value
4. CONCLUSION
We may conclude from the results of our calculations that the highest frequency of operation for In,” s3,Gao,,, As Gunn diodes will be nearly the same as that for GaAs diodes but much lower than that for InP diodes. We may also conclude that the noise in to Ino 53jGao.,rj As Gunn diodes will be comparable that in GaAs diodes but much larger than in InP diodes.
REFERENCES
Fig. 5. Figures of merit for noise. Ratio of parallel diffusion coefficient and drift velocity-Ino,,,Gao4,)As (continuous line), GaAs (open square), InP (open circle). Ratio of parallel diffusion coefficient and magnitude of differential line), GaAs (shaded mobility-I+, ,l,Ga(, 47jAs (broken ~ _ _ , . * , ~. _*_\ square), Inr (snaoea cncie).
1. B. R. Nag, Pramana 23, 411 (1984). 2. Y. Takeda, N. Shikagawa and A. Sasaki, Solid-Sf. Electron. 23, 1003 (1980). 3. W. Kowalskv and A. Schlachetzki, Solid-St. Electron. 28, 299 (1985). 4. B. R. Nag, Electron Transport in Compound Semiconductors, p. 214 and p. 319. Springer, Berlin (1980). 5. S. R. Ahmed, B. R. Nag and M. Deb Roy. Solid-S/. Electron 28, 1193 (1985). 6. W. Fawcett and H. D. Rees, Whys. Let/. 29A, 578 (1969). 7. R. Fauquemberge, J. Zimmermann, A. Kaszynski, E. Constant and Greco Microondes, J. uppl. Phys. 51, 1065 (1980). 8. D. K. Ferry and J. R. Barker, J. appl. Phys. 52, 818 (1981). 9. M. Deb Roy and B. R. Nag, Int. J. Electron. 48, 443-446 (1980). 10. M. Deb Roy and B. R. Nag. Appl. Phys. A 28, 195 (1982). 1 I. C. Chatfield, The Analysis of Time Series: Theory and Practice. DD. 6C-65. Chapman and Hall, London (1975).
High-field effects in InGaAs at 300 K 12. A. Van der Ziel, In: Advances in Electronics and Electron Physics, (Edited by L. Marton and C. Marton), Vol. 50, pp. 351409. Academic Press, New York (1980). 13. B. Kallback, Electron. Left. 9, II-12 (1973). 14. H. W. Thim, Electron. Z&t. 7, 106-108 (1971).
239
15. W. Kowalsky, A. Schlachetzki and H. H. Wehmann, Solid-St. Electron. 27, 187 (1984).
16. G. Hill, P. N. Robson and W. Fawcett, J. appl. Phys. 50, 356 (1979). 17. B. K. Ridley, J. appl. Phys. 48, 754 (1977).
Solid-State Electronics Printed
Vol. 30, NO. 2, pp. 2355239, 1987 in Great Britain. All rights reserved
Copyright
0
HIGH-FIELD AUTOCOVARIANCE COEFFICIENT, DIFFUSION COEFFICIENT AND NOISE IN InGaAs AT 300 K B. R. NAG,’ S. R. AHMED’ and M. DEB ROY’ ‘Calcutta University, Calcutta, India and ‘Jadavpur University, Calcutta, India (Received 6 December 1985; in revised form 26 March 1986)
Abstract-Electron diffusivity in Ino,,,,Ga(,,,, As is calculated by the Monte-Carlo method. The autocovariance coefficient and correlation time of velocity fluctuations, frequency dependence of diffusion coefficient and figure of merit for noise are studied. The results indicate that the limiting frequency of to those of GaAs diodes. operation and noise in In<,,53,Ga(, 47jAs Gunn diodes would be comparable
1. INTRODUCTION The ternary alloy Ino 5,,Gao47jA~ is receiving attention as a prospective material for the realisation of high-efficiency Gunn diodes[l]. High efficiency is expected as the peak-to-valley ratio for electron velocity is high. Some experimental results have also been reported which support the expectations[2, 31. No results are, however, available on the noise characteristic. Estimate of noise behaviour may be obtained from the knowledge of electron diffusivity at the operating fields. However, the diffusion coefficient in Ino 53)Ga(,,,,, As has not yet been measured or calculated. It may be obtained for low-fields from the mobility data using the Einstein relation[4], but for high fields the relation is not applicable. On the other hand, high-field diffusivity may be calculated with good accuracy by the Monte-Carlo method, provided the physical constants of the material are known. A set of values for the physical constants was recently[5] obtained from the available measurements and pseudopotential calculations. The constants gave very good agreement between the calculated and measured values of electron drift velocity over a wide range of electric fields and lattice temperatures. These constants are used in this paper to calculate the diffusivity. The diffusivity has been obtained from the velocity autocovariance coefficient. The method of calculation of the coefficient is explained in Section 2. The characteristics of the covariance coefficients, diffusion coefficient and its field and frequency dependence are discussed in Section 3. The results are used to obtain figures of merit for noise behaviour and these are also compared with the values of GaAs and InP.
the autocovariance coefficient of velocity fluctuations [7-lo]. The latter method was used as the frequency dependence of the diffusion coefficient and the frequency limitations of the material for application in devices may be studied in addition to the diffusion coefficient. The formulae and the method of calculation for the various functions studied in this paper are given below. (a)
Autocovariance
OF CALCULATION
AND FORMULAE
of
velocity-time
c(s)=E{[v(t)-v(t)][v(t+s)-v(t+s)]),
(1)
where the E and the bar indicate the average over an ensemble of velocity functions v(t), s being the time lag. Electron-electron scattering was not included in the calculations as the effects of such scattering cannot be easily evaluated by the Monte-Carlo method. The e-e scattering is important for very large electron concentrations. For low electron concentrations, lattice scattering is more frequent and electron trajectories are not significantly altered by e-e collisions. The present results may, therefore, be considered applicable strictly to samples with such electron concentrations that lattice scattering is more predominant than e-e scattering. As the e-e scattering was neglected, c(s) was obtained from the time average of the velocity function for a single representation assuming the ergodic theorem[ 111 applies. The formula used for the evaluation of c(s) is: c(s)=(l/T)
2. METHOD
coefficient
The autocovariance coefficient function is defined[ 111 as:
T[v(t)-a][v(t
+s)-a]dt,
(2)
50 where
Electron diffusivity for high fields can be obtained by the Monte-Carlo method[6], either by calculating the variance of the displacements or by calculating
now d is the time average
of v(t), given by
I.
u.(t) dt.
v = (l/T) s II 235
(3)
236
B. R. NAG PI ul.
The velocity function was obtained by the usual Monte-Carlo procedure using for the physical constants, the values given in Ref.[S]. Calculations were performed for a uniform electric field across the sample. The field becomes non-uniform in an actual device when the differential mobility is negative due to the appearance of space charge. However, our objective is to compare the properties of %.,,, Ga,,.,,, As with those of GaAs and InP and the relative merits should not be altered by the nonuniformity of the field. The integrals in equations (2) and (3) were cvaluated by summation using the values of r(t) at intervals of 0.04~s. The simulation was carried out for sufficiently long time T so that the values of c converged to within 0.01% and that of c(s) to within 5%. It should be mentioned that the accuracy of 1: is very important for obtaining reliable values of c(s) as it is given by the difference of nearly equal terms for large values of the electric field. (b)
Diffusion
coejicient
The diffusion coefficient was evaluated frequencies using the formulae[ 121:
for different
cc D(w)
=
c (s)cos ws ds.
s0
(4)
The integral was evaluated from the values of c(s), calculated earlier for values of the time lag .Y at intervals of O.O4ps, using the Simpson formula. (c) Figure of merit for noise Noise in the semiconductor devices originate from different sources. However, for the Gunn diodes the thermal noise is an important component. The index of thermal noise is the noise temperature, which may be evaluated from the diffusivityymobility ratio for fields at which the mobility is positive. But, for operating fields at which the differential mobility is negative, the relation cannot be applied. However, diffusivity is an indicator of the noise, even though it may not give the temperature quantitatively. The ratio of the diffusion coefficient and the drift velocity has been used[l3] to estimate the noise performance of the devices. However, a more useful indicator of noise is the ratio of the diffusion coefficient, D (0) and the magnitude of the negative differential mobility, lpdl[14]. The generated noise is larger for a higher value of D(O), but the noise in the oscillator is reduced for higher values of pd due to the plug-in effect of oscillation. Both the ratios D(O)/6 and D(0)/Ip(dj have, therefore, been calculated for different fields to estimate the noise behaviour. (d)
Correlation
evaluated if the autocovariance coefficient decreases exponentially with a single time constant. However, it will be seen from the results presented in the following section that this is not true for the velocity functions at high fields. An effective correlation time, r,, was hence evaluated using the formula[7]: 7< =
3. RESULTS
D (0)/c (0)
(5)
AND DISCUSSSIONS
Calculations were made for the fields between 1 and 30 kV/cm at 300 K. The coefficients were evaluated for the component of velocity parallel to the applied field and also for the component transverse to the field. The autocovariance coefficients are presented in Fig. 1. The transverse coefficient decreases monotonically with time; the rate of decrease. increases with the field. The parallel coefficient also decreases monotonically with time for fields below the threshold field. On the other hand, for fields above the threshold field, the coefficient decreases, becomes zero, attains a minimum value and then increases to zero. The nature of variation of the autocovariance coefficients with the field may be explained as follows. The fluctuations of the electron velocity increases with the field as the electron system gets heated. At the same time, the collisions become more frequent and the correlation time decreases. The autocovariance coefficient, therefore, increases in magnitude but falls faster with increase in the field. However, at fields above the threshold field, as electrons are transferred from the r valley to the valleys with higher effective mass, the magnitude of velocity fluctuations decreases and the collision time decreases further. The function, therefore, decreases in magnitude and decreases still faster with the time
3.0
m-
2.7
time
The frequency limitation of the devices can be studied from D(w), as it involves the same time constants as are involved in drift mobility. A more quantitative estimate can be obtained from the correlation time. The correlation time can be easily
Fig. 1. Transverse and parallel autocovariance coefficients at 300 K; the number on the curve indicates field in kV cm ‘.
High-field effects in InGaAs at 300 K lag at very high fields. Also, at post-threshold fields, electrons scattered back from the satellite valleys to the F valley may have large velocities. If the velocity is in the opposite direction to the direction of acceleration due to the field, the electron initially looses energy as it is accelerated by the field. The scattering probability of such electrons therefore initially decreases with time. These electrons are scattered as the field finally accelerates them to high energies. The trajectories of such electrons are characterised by long flights in which the velocity continuously increases from the negative value to a positive value. The parallel autocovariance coefficient, therefore, becomes negative for large time lag when such trajectories predominate. The characteristic features of autocovariance coefficient discussed above are also exhibited by GaAs and InP and have been discussed in other literature[7,9, lo]. The field-dependences of the diffusion coefficients are illustrated in Fig. 2. The coefficients are plotted against the ratio of the applied field and the threshold field, so that comparison may be made between different materials. The threshold field was taken to be 3.2 kV/cm’. The computed low-field value of the diffusivity is 355 cm*/s, which agrees with the value calculated from the mobility using the Einstein relation. The diffusion coefficients increase with the field, attain a peak value at a field, a little higher than the threshold field and then decreases with the field. The diffusion coefficient is related to (v*T), where u is the fluctuation in velocity and 7 is the collision time. As discussed in connection with the autocovariance coefficient u* increases but 7 decreases initially with the field. Apparently, the rate of increase of u* is higher than the rate of decrease of 7, so that (0’7) and hence the diffusion coefficient increases. But at post-threshold fields, the transferred electrons have
0.L
0.1
APPLIED
ELECTRIC
1.0
l.0
FIELD!
THRESHOLD
231
lower values of u and u* also decreases with field as more and more electrons are transferred. This decrease in u* and the decrease in the value of 7 cause the diffusion coefficient to decrease rapidly with the field. The values of diffusion coefficients in GaAs and InP are also quoted in Fig. 2 from Ref.[lO]. As values were calculated for three fields only, these are indicated by squares and circles. The threshold fields for GaAs and InP were taken to be 3.2 and 10 kV/cm as accepted in the literature. It is seen that the transverse As are larger diffusion coefficients in In, 53)GaC0.47, than those in GaAs and InP. On the other hand, the parallel diffusion coefficients are nearly equal to those in GaAs but larger than those in InP. The frequency dependence of the diffusion coefficients is illustrated in Fig. 3. The transverse component, D,(w)and the below-threshold-field parallel component, D,(w), are initially constant and then decrease monotonically with increasing frequency. The above-threshold-field parallel components, on the other hand, increase with frequency initially, attain peak values and then decrease with frequency. The nature of the curves may be understood considering that D (0) is the Fourier transform of c(s). Hence, as c,(s) and below-threshold-field cp(s) decrease monotonically with S, D(o) decreases monotonically with frequency. Also, as abovethreshold-field cp(s) changes sign with s, D,(w) shows a peak around a frequency corresponding to the period of change in sign of c,,(s). The same features are also seen in the D(W)-Jcurves of GaAs[lO] and InP[lO, 161. We also find on comparing the characteristics of D (0)-fcurves for GaAs[ lo] and InP[lO] with those in InC,,J,,Gao,,,As t:; the limiting frequency of operation of Ino,,,, C0.47,A~would be comparable to that of GaAs but much lower than that of InP.
10
FIELD
Fig. 2. Transverse and parallel diffusion coefficients at 300 K. Parallel diffusion coefficient-In0,,,,Ga,,4,, As (continuous line), GaAs (open square), InP (open circle). Transverse diffusion coefficient-Irq,,,,,Ga(,,,, As (broken line), GaAs (shaded square), InP (shaded circle).
FREOUENCY
Fig. 3. Frequency dependence of tinuous line-Parallel diffusion Transverse diffusion coefficient. indicates field in
IGHzI
diffusion coefficient. Concoefficient, Broken lineThe number of the curve kV cm- ‘.
B. R. NAG et al
Fig. 4. Correlation time of velocity fluctuations. Correlation time for parallel velocity-In&,,Ga(O,,,As (continuous line), GaAs (open square), InP (open circle). Correlation time for transverse velocity--In(,,,,Ga(,,,,As (broken line), GaAs (shaded square), InP (shaded circle).
The
correlation
times
for different
fields
are illus-
decreases with the field above the threshold field. The time for the parallel velocity decreases faster than that for the transverse velocity. Again, on comparing the correlation times for the parallel velocity, which is a measure of the frequency limitation of the material for Gunn diodes, we find that the times are comparable to those for GaAs but much higher than those for InP. The results on correlation time thus support the conclusion made from D(w)-f curves about the relative frequency limitations of In, s3,Ga,, 47jA~, GaAs and InP. It is well known that the frequency limitation of devices trated
utilising electron-transfer-effect arises mainly from the intervalley transfer time. An approximate formula has been given by Ridley[ 171 for estimating the limiting frequency from such considerations. The values given by this formula are 40, 50 and 120 GHz for Ino 53jGaC0.,,jAs, GaAs and InP respectively. These values support the conclusion made above from the present analysis. The figures of merit for noise performance are illustrated in Fig. 5. The diffusivity-velocity ratio decreases with increasing field but the diffusivitydifferential mobility ratio, D(0)/fpdI increases with the field. The field dependence is again identical to that for GaAs and InP. Both the ratios have values comparable to those for GaAs but higher than those for InP. We may note that the accuracy in the values of D (0)/z? is about 5%, but that for D (0)/jpdl is poorer as pd was estimated from the values of d for fields differing by 1 kV/cm. The figures of merit should, therefore, be considered to give qualitative rather than quantitative measures. Nevertheless, one may conclude from the large differences in values that As devices would be much the noise in Ino,,,Ga,,,,, larger than in InP diodes, but similar to that in GaAs.
in Fig. 4. The value
4. CONCLUSION
We may conclude from the results of our calculations that the highest frequency of operation for In,” s3,Gao,,, As Gunn diodes will be nearly the same as that for GaAs diodes but much lower than that for InP diodes. We may also conclude that the noise in to Ino 53jGao.,rj As Gunn diodes will be comparable that in GaAs diodes but much larger than in InP diodes.
REFERENCES
Fig. 5. Figures of merit for noise. Ratio of parallel diffusion coefficient and drift velocity-Ino,,,Gao4,)As (continuous line), GaAs (open square), InP (open circle). Ratio of parallel diffusion coefficient and magnitude of differential line), GaAs (shaded mobility-I+, ,l,Ga(, 47jAs (broken ~ _ _ , . * , ~. _*_\ square), Inr (snaoea cncie).
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