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Hot electron effects at microwave frequencies in GaAs
Solid State Communications,
Vol. 7, pp. 267—269, 1969.
Pergamon Press.
Printed in Great Britain
HOT ELECTRON EFFECTS AT MICROWAVE FREQUENCIES IN GaAs H.D. Rees Royal Radar Establishment, M alvern, Worcestershire, England
(Received 25 October 1968)
The frequency dependence of the electron mobility in GaAs has been calculated numerically. Heating processes within the <000> valley control the response speed of the negative conductivity and cause the threshold field to rise with frequency.
THE NEGATIVE differential conductivity exhibited by n-type GaAs above a threshold electric field of about 3 kV/cm is well known and is of practical importance, notably for its exploitation in microwave oscillators. The effect is due to the transfer of electrons from the high mobility <000> valley to the low mobility <100> valleys. Theories of the effect have concentrated on deriving the d.c. velocityfield characteristic, although relaxation time approximations have been used to estimate the frequency dependence of the effect.”2 However,
evaluated for a given field and secondly its time response to a small step in field is calculated. The drift velocity and population ratio are derived from the distribution function; the steady state functions yield the d.c. characteristic and the step response is transformed to give the frequency response. The parameters used in the calculations were the same as one set used for Monte Carlo 6 calculations of the velocity-field characteristic (upper valley effective mass of 0.35m 0 and intervalley deformation potential of iO~eV/cm). The calculated static velocity-field characteristic is, of course, the same as in reference 6.
since most current practical interest is in the microwave frequency properties of the effect, it would be desirable to estimate the frequency dependence of the differential conductivity.
The time response calculations show that the high frequency behaviour is dictated by heating within the lower valley. Above about 2 kV/cm a large number of electrons in the lower valley have energies between 0.1 and 0.36 eV. Electrons with energies in this range are weakly scattered by phonons and consequently randomisation and heating processes arevelocity slow.
A numerical method has been recently developed in which the steady state distribution function for electrons in an electric field may be calculated without making a priori assumptions about its form. ~ The method can be extended to derive the time development 5of and the has distribution been function in time varying fields, applied to the case of electrons in GaAs with
By contrast, the intervalley scattering time and the upper valley momentum relaxation time are —~5x 10~ sec and neither scattering between valleys nor within the upper valleys directly introduce effects significant at microwave frequencies. The slow scattering in the lower valley produces a frequency dependence of the differential mobility illustrated in Fig. 1, which
the objects of investigating the processes limiting the response speed of the electrons and of evaluating the frequency dependence of the conductivity, The calculations proceed in two stages. First the steady state distribution function is 267
268
HOT ELECTRON EFFECTS IN GaAs
Vol.7, No.2
2-
x ,o~
Ui
I.-I
0
I
I-. -‘
I
II
I
-2 0
50
100
FREQUENCY
150 (GHz)
FIG. 1. Differential mobility at 5.5 kV/cm bias. p.~is component in phase with field and quadrature. (Positive /L 2 implies capacitative response.)
FIELD
~2
in
(Ky/CM)
FIG. 2. Field dependence of d.c. mobility and differential mobility at zero frequency, 35 and 140 GHz. is for a bias field of 5.5 kV/cm. The differential mobility is negative below 80 GHz, but positive above. The quadrature component of the mobility implies that the electron contribution to the dielectric constant is positive, and for an electron concentration of 1015 cm-s is about +10 for frequencies below 30 GHz. It should be emphasised that the positive differential mobility at high frequencies results from the slow velocity randomisation within the lower valley; the upper
valley population has a component varying in phase with the field even at high frequency. Figure 2 shows the field dependence of the d.c. mobility and the differential mobility at zero frequency, 35 and 140 GHz. The 35 GHz curve illustrates the facts that the heating effects are important above 2 kV/cm, but the frequency dependence of the mobility falls at high fields because the heating processes are
Vol.7, No.2
HOT ELECTRON EFFECTS IN GaAs
269
faster. Consequently the threshold field for negative conductivity rises with frequency, being 4.5 kV/cm at 50 GHz and 6.5 kV/cm at 100 GHz. At 140 GHz the negative conductivity is negligible. These figures show that the negative conductivity deteriorates at rather lower frequencies than previously estimated.’ 2
has been recently observed’ in a determination of the microwave conductivity for bias fields from 1.5 to 2.5 kV/cm. The theoretical and experimental curves are in close agreement except that experimentally the plateau sets in at about 1.8 kV/cm rather than 1.5 kV/cm.
Below 1.5 kV/cm, the differential conductivity at 140 GHz is less than that at low frequencies, directly reflecting a value of WTm of 0.25. Heating effects produce a plateau in the microwave conductivity from 1.5 to 2.5 kV/cm but at
Acknowledgements The author wishes to thank W. Fawcett, C. Hilsum and T.P. McLean for ~ forwarding a preprint of his paper7 and to J.B. Arthur who wrote some special sections of the computer
higher fields the transfer of electrons to the low mobility valleys causes the microwave conductivity to fall. This plateau in the 140 GHz curve
programme. Contributed by permission of the Director of R.R.E. Copyright Controller H.M.S.O.
—
REFERENCES 1.
DAS P. and BHARAT R., Appi. Phys. Lett. 11, 386 (1967).
2.
OHMI T., MURAYAMA K. and KANBE H., Proc. I.E.E.E. (Lett) 56, 747 (1968).
3.
BUDD H., Phys. Rev. 158, 798 (1967).
4.
REES H.D., Phys. Lett. 26A, 416 (1968).
5. 6. 7.
REES H.D., Physics. Chem. Solids, to be published. BOARDMAN A.D., FAWCETT W. and REES H.D., Solid State Commun. 6, 305 (1968). VLAARDINGERBROEK M.T., KUYPERS W. and ACKET G.A., Phys. Lett. to be published.
La dépendance de la mobilité des electrons en GaAs sur la fréquence a été calculé. Les effets de chaleur dans la vallée <000> contrôlent la vitesse de réponse de la conduction negative et font grandir le champ critique avec Ia fréquence.
Vol. 7, pp. 267—269, 1969.
Pergamon Press.
Printed in Great Britain
HOT ELECTRON EFFECTS AT MICROWAVE FREQUENCIES IN GaAs H.D. Rees Royal Radar Establishment, M alvern, Worcestershire, England
(Received 25 October 1968)
The frequency dependence of the electron mobility in GaAs has been calculated numerically. Heating processes within the <000> valley control the response speed of the negative conductivity and cause the threshold field to rise with frequency.
THE NEGATIVE differential conductivity exhibited by n-type GaAs above a threshold electric field of about 3 kV/cm is well known and is of practical importance, notably for its exploitation in microwave oscillators. The effect is due to the transfer of electrons from the high mobility <000> valley to the low mobility <100> valleys. Theories of the effect have concentrated on deriving the d.c. velocityfield characteristic, although relaxation time approximations have been used to estimate the frequency dependence of the effect.”2 However,
evaluated for a given field and secondly its time response to a small step in field is calculated. The drift velocity and population ratio are derived from the distribution function; the steady state functions yield the d.c. characteristic and the step response is transformed to give the frequency response. The parameters used in the calculations were the same as one set used for Monte Carlo 6 calculations of the velocity-field characteristic (upper valley effective mass of 0.35m 0 and intervalley deformation potential of iO~eV/cm). The calculated static velocity-field characteristic is, of course, the same as in reference 6.
since most current practical interest is in the microwave frequency properties of the effect, it would be desirable to estimate the frequency dependence of the differential conductivity.
The time response calculations show that the high frequency behaviour is dictated by heating within the lower valley. Above about 2 kV/cm a large number of electrons in the lower valley have energies between 0.1 and 0.36 eV. Electrons with energies in this range are weakly scattered by phonons and consequently randomisation and heating processes arevelocity slow.
A numerical method has been recently developed in which the steady state distribution function for electrons in an electric field may be calculated without making a priori assumptions about its form. ~ The method can be extended to derive the time development 5of and the has distribution been function in time varying fields, applied to the case of electrons in GaAs with
By contrast, the intervalley scattering time and the upper valley momentum relaxation time are —~5x 10~ sec and neither scattering between valleys nor within the upper valleys directly introduce effects significant at microwave frequencies. The slow scattering in the lower valley produces a frequency dependence of the differential mobility illustrated in Fig. 1, which
the objects of investigating the processes limiting the response speed of the electrons and of evaluating the frequency dependence of the conductivity, The calculations proceed in two stages. First the steady state distribution function is 267
268
HOT ELECTRON EFFECTS IN GaAs
Vol.7, No.2
2-
x ,o~
Ui
I.-I
0
I
I-. -‘
I
II
I
-2 0
50
100
FREQUENCY
150 (GHz)
FIG. 1. Differential mobility at 5.5 kV/cm bias. p.~is component in phase with field and quadrature. (Positive /L 2 implies capacitative response.)
FIELD
~2
in
(Ky/CM)
FIG. 2. Field dependence of d.c. mobility and differential mobility at zero frequency, 35 and 140 GHz. is for a bias field of 5.5 kV/cm. The differential mobility is negative below 80 GHz, but positive above. The quadrature component of the mobility implies that the electron contribution to the dielectric constant is positive, and for an electron concentration of 1015 cm-s is about +10 for frequencies below 30 GHz. It should be emphasised that the positive differential mobility at high frequencies results from the slow velocity randomisation within the lower valley; the upper
valley population has a component varying in phase with the field even at high frequency. Figure 2 shows the field dependence of the d.c. mobility and the differential mobility at zero frequency, 35 and 140 GHz. The 35 GHz curve illustrates the facts that the heating effects are important above 2 kV/cm, but the frequency dependence of the mobility falls at high fields because the heating processes are
Vol.7, No.2
HOT ELECTRON EFFECTS IN GaAs
269
faster. Consequently the threshold field for negative conductivity rises with frequency, being 4.5 kV/cm at 50 GHz and 6.5 kV/cm at 100 GHz. At 140 GHz the negative conductivity is negligible. These figures show that the negative conductivity deteriorates at rather lower frequencies than previously estimated.’ 2
has been recently observed’ in a determination of the microwave conductivity for bias fields from 1.5 to 2.5 kV/cm. The theoretical and experimental curves are in close agreement except that experimentally the plateau sets in at about 1.8 kV/cm rather than 1.5 kV/cm.
Below 1.5 kV/cm, the differential conductivity at 140 GHz is less than that at low frequencies, directly reflecting a value of WTm of 0.25. Heating effects produce a plateau in the microwave conductivity from 1.5 to 2.5 kV/cm but at
Acknowledgements The author wishes to thank W. Fawcett, C. Hilsum and T.P. McLean for ~ forwarding a preprint of his paper7 and to J.B. Arthur who wrote some special sections of the computer
higher fields the transfer of electrons to the low mobility valleys causes the microwave conductivity to fall. This plateau in the 140 GHz curve
programme. Contributed by permission of the Director of R.R.E. Copyright Controller H.M.S.O.
—
REFERENCES 1.
DAS P. and BHARAT R., Appi. Phys. Lett. 11, 386 (1967).
2.
OHMI T., MURAYAMA K. and KANBE H., Proc. I.E.E.E. (Lett) 56, 747 (1968).
3.
BUDD H., Phys. Rev. 158, 798 (1967).
4.
REES H.D., Phys. Lett. 26A, 416 (1968).
5. 6. 7.
REES H.D., Physics. Chem. Solids, to be published. BOARDMAN A.D., FAWCETT W. and REES H.D., Solid State Commun. 6, 305 (1968). VLAARDINGERBROEK M.T., KUYPERS W. and ACKET G.A., Phys. Lett. to be published.
La dépendance de la mobilité des electrons en GaAs sur la fréquence a été calculé. Les effets de chaleur dans la vallée <000> contrôlent la vitesse de réponse de la conduction negative et font grandir le champ critique avec Ia fréquence.