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“Cooled” electrons in polar semiconductors
PHYSICS
Volume 24A, number 1
“COOLED”
ELECTRONS
2 January196’7
LETTERS
IN POLAR
SEMICONDUCTORS
K. BMTEKJAER Microwave
Department,
Royal Institute Received
of Technology,
28 November
Sweden
Stockholm,
1966
The possibility of decreasing the temperature of electrons in a semiconductor by applying an electric field is discussed. The theory of Paranjape et al. [l-2] is extended to arbitrarily high field strengths.
When an electric field is applied to a semiconductor, the conduction electrons gain momentum and energy from the electric field. In a steady state, the same amount of momentum and energy must be transferred from the electrons to the crystal lattice through collisions. In order to transfer momentum the electrons acquire a drift velocity, and in order to transfer energy the electron temperature is raised above the lattice temperature. This increase of thermal energy influences the mobility, and gives rise to deviation from Ohm’s law. This is the typical behaviour of “warm” or “hot” electrons. Paranjape et al. [l-2] have shown that this picture is not always correct . Under certain circumstances the electrons may be “cooled” rather than heated. The shifting of the velocity distribution which is required to produce a drift velocity causes not only a transfer of momentum, but also a transfer of energy, which actually may exceed the energy supplied by the field. Thus, there is a net loss of energy from the electrons, and the electron temperature decreases below the lattice temperature. Paranjape et al. [l-2] derived conditions for electron cooling in the limit of weak electric fields. The present letter is concerned with an extension of the theory to arbitrary field strengths. We consider semiconductors in which the exchange of momentum and energy between electrons and the lattice takes place via polar scattering by optical phonons . In many semiconductors this is the dominant scattering process over a wide range of temperatures. It is assumed that electron-electron collisions are so frequent that a displaced Maxwellian velocity distribution is maintained. This model was first treated by Frohlich and Paranjape [3] and by Stratton [4]. The possibility of “cooled” electrons was not disclosed by these authors because of an approximation made in the calculations.
“r
T/B
-
exact result
---
1” approximation
-‘-
2”dapproximation
0
0
0
0.0
0.1
I
I
I
,
0.2
0.3
0.L
0.5
6
E/E, Fig. 1. Electron temperature plotted versus electric field, for a lattice temperature TL = 0.26. The exact results and the two approximations discussed in the text are shown.
The momentum and energy balance for electrons is expressed by the equations neE = F(T, To, P)
(1)
neEp/m = G( T, To, P) .
(2)
The left-hand sides of eqs. (1) and (2) express the rate of change of momentum and energy due 15
Volume 24A, number 1
PHYSICS
2 January 1967
LETTERS
In this approximation,
“cooled
electrons”
with
T < To appear when mG1 > F1. For polar optical phonon scattering this condition is satisfied at temperatures below 0.3758, where k0 is the phonon energy. Eqs. (3) and (5) are correct to second order in to second order in the applied field E. p, i.e., However, this approximation has the highly undesirable property of yielding completely erroneous results for higher field strengths. Actually, the set of eqs. (3) and (4) provides a much better description of the high-field behaviour, as demonstrated by fig. 1, which shows the results of both approximations for a lattice temperature To = = 0.28. It is possible to calculate the general term of the power series expansions of F and G in p. The result is [5] F
=d~neEoe~(-r w2)y% w E(-l)n(n+~)(2r
x$y-l[Ko(y) 0
0.1
0.2
0.3
0.L
0.5
0.6
x
sinh y.
n=o
(2n+3)!
sinh (yo-y) +KI(Y) cash (ye-y)],
(6)
0.8
0.7
E/E, Fig, 2. Electron temperature plotted versus electric field, with lattice temperature as parameter. Results of the exact calculations.
to the electric field E. The electron density is n, their average momentum is p, and e and m are the electronic charge and effective mass, respectively. The right-hand sides of the equations express the rate of momentum and energy transfer from electrons to the lattice through collisions. These quantities are rather complicated integrals involving the average momentum p, the electron temperature T and the lattice temperature To. From eqs. (1) and (2) one can calculate p and T as functions of E and To. In the works referred to above [3,4] the first terms of a power series expansion of F and G in p were calculated, F”
PFl(T,To)
G = Go(T,To)
(3) (4)
where F1 and Go are functions of T and To, but not of p . Since F1 IS always positive, and Go and T - To have the same sign, it follows that in this approximation T - To A 0 whenever E # 0. Paranjape et al. [l-2] went one step further, and replaced (4) by G = Go(T, To) + P2G,(T,To) . 16
(5)
x 5
Y-’ K,(Y)
sinh ho -7)
(7)
where E. is a characteristic electric field which expresses the strength of the electron-phonon interaction [2-41,
Y =9/2T Yo
03)
=0/2To
(9)
and w =p/JzXi
.
(10)
Numerical calculations have been performed, and fig. 1 shows the result for To/0= 0.2,compared with the results of the two approximations discussed above. Fig. 2 shows the electron temperature as a function of the applied field, for various values of To. The average momentum caused by a given electric field is higher than the approximate value obtained from eqs. (3) and (4), but the difference is not as profound as for the temperature. As a result of the higher average momentum and lower temperature, the exact theory predicts a considerably higher ratio of drift velocity to thermal velocity. ,When To/B = 0.2,the maximum value of p(2mkT)-2 is 1.2. This may be of importance for
Volume 24A, number 1
PHYSICS
the properties of the semiconductor plasma as a medium for wave propagation. There are al& some interesting aspects of the negative dT/dE as a source of an instability. This may possibly have some relevance to the observed low-field instability in InSb [6]. The cooling effect disappears if any scattering process other than optical phonon scattering is present. Acoustic phonon and impurity scattering must be kept low. It is suggested that relatively pure n-type indium antimonide at lOOoK should be a suitable material for an experimental verification of the effect.
LETTERS
2 January 1967
References 1. V. V. Paranjape and T. P.Ambrose, Phys. Letters 8 (1964) 223. 2. V.V. Paranjape and E. de Alba, Proc. Phys. Sot. 85 (1965) 945. 3. H. Frijhlich and B. V. Paranjape, Proc. Phys.Soc. B 69 (1956) 21. 4. R.Stratton, Proc.Royal Soc.246 (1958) 406. 5. K.Blotekjaer, Arkiv for Fysik 33 (1967) 105. 6. S. J. Buchsbaum, A.G. Chynoweth and W. L. Feldman, Appl.Phys. Letters 6 (1965) 67.
*****
RESISTIVITY
AND
TUNNELLING ALLOY
PROPERTIES FILMS
OF PARAMAGNETIC
F. T. HEDGCOCK and C. RIZZUTO * Eaton Electronics
Research Laboratory, Montreal, Canada
Received
Evaporated suppression
28 November
McGill University,
1966
films of zinc containing paramagnetic impurities are shown to exhibit a resistance of the critical temperature and zero bias tunnelling anomalies.
We wish to report that it is possible to evaporate thin films of dilute paramagnetic alloys that do indeed show resistive anomalies at low temperatures and suppression of the critical temperature of the same type found in bulk metals [1,2]. Fig. la shows a typical result of the temperature dependence of the resistance of a Zn-Mn film prepared as summarized in the caption of fig. lb. The magnitude of slope of this line would indicate that approximately 0.1 at.% Mn was in solution [1,2] which is lower than the nominal composition of the alloy before the evaporation. That this observed resistance minimum is not due to effects different from a magnetic scattering is demonstrated by the fact that in the more dilute samples the concentration dependence of the suppression of the critical temperature for evaporated zinc manganese is in fair agreement with the observed results in the bulk material (see table 1). Shown in fig. lb is a typical tunnelling ano* National Research
Council Postdoctoral
Fellow.
minimum,
Table 1 Concentration values and suppression of the critical temperature for the Zn-Mn alloys. Nominal concentration (atomic parts per million)
AT, evaporated film* (OK)
6.5
0.23hO.05
8.9
0.13+0.01
0.42
> 0.54
ATc /c ATc/c average for evap- average bulk orated films
(OK/at.%)
215
(OK/at.%)
315
* AT, with respect temperature
to a pure Zn strip whose critical is equal to 0.88 *0.05 OK.
maly observed for a Zn-Mn, Al2O3, Al tunnel junction above the superconducting critical temperature of aluminium. (Details of the junction preparation are included in the figure caption.) These anomalies are similar to those reported for Cr-I-Ag junctions [3]. If the observed effects are due to paramagnetic impurities the theoretical work of Anderson [4] and of Applebaum [5] would predict that the magnitude or
17
Volume 24A, number 1
“COOLED”
ELECTRONS
2 January196’7
LETTERS
IN POLAR
SEMICONDUCTORS
K. BMTEKJAER Microwave
Department,
Royal Institute Received
of Technology,
28 November
Sweden
Stockholm,
1966
The possibility of decreasing the temperature of electrons in a semiconductor by applying an electric field is discussed. The theory of Paranjape et al. [l-2] is extended to arbitrarily high field strengths.
When an electric field is applied to a semiconductor, the conduction electrons gain momentum and energy from the electric field. In a steady state, the same amount of momentum and energy must be transferred from the electrons to the crystal lattice through collisions. In order to transfer momentum the electrons acquire a drift velocity, and in order to transfer energy the electron temperature is raised above the lattice temperature. This increase of thermal energy influences the mobility, and gives rise to deviation from Ohm’s law. This is the typical behaviour of “warm” or “hot” electrons. Paranjape et al. [l-2] have shown that this picture is not always correct . Under certain circumstances the electrons may be “cooled” rather than heated. The shifting of the velocity distribution which is required to produce a drift velocity causes not only a transfer of momentum, but also a transfer of energy, which actually may exceed the energy supplied by the field. Thus, there is a net loss of energy from the electrons, and the electron temperature decreases below the lattice temperature. Paranjape et al. [l-2] derived conditions for electron cooling in the limit of weak electric fields. The present letter is concerned with an extension of the theory to arbitrary field strengths. We consider semiconductors in which the exchange of momentum and energy between electrons and the lattice takes place via polar scattering by optical phonons . In many semiconductors this is the dominant scattering process over a wide range of temperatures. It is assumed that electron-electron collisions are so frequent that a displaced Maxwellian velocity distribution is maintained. This model was first treated by Frohlich and Paranjape [3] and by Stratton [4]. The possibility of “cooled” electrons was not disclosed by these authors because of an approximation made in the calculations.
“r
T/B
-
exact result
---
1” approximation
-‘-
2”dapproximation
0
0
0
0.0
0.1
I
I
I
,
0.2
0.3
0.L
0.5
6
E/E, Fig. 1. Electron temperature plotted versus electric field, for a lattice temperature TL = 0.26. The exact results and the two approximations discussed in the text are shown.
The momentum and energy balance for electrons is expressed by the equations neE = F(T, To, P)
(1)
neEp/m = G( T, To, P) .
(2)
The left-hand sides of eqs. (1) and (2) express the rate of change of momentum and energy due 15
Volume 24A, number 1
PHYSICS
2 January 1967
LETTERS
In this approximation,
“cooled
electrons”
with
T < To appear when mG1 > F1. For polar optical phonon scattering this condition is satisfied at temperatures below 0.3758, where k0 is the phonon energy. Eqs. (3) and (5) are correct to second order in to second order in the applied field E. p, i.e., However, this approximation has the highly undesirable property of yielding completely erroneous results for higher field strengths. Actually, the set of eqs. (3) and (4) provides a much better description of the high-field behaviour, as demonstrated by fig. 1, which shows the results of both approximations for a lattice temperature To = = 0.28. It is possible to calculate the general term of the power series expansions of F and G in p. The result is [5] F
=d~neEoe~(-r w2)y% w E(-l)n(n+~)(2r
x$y-l[Ko(y) 0
0.1
0.2
0.3
0.L
0.5
0.6
x
sinh y.
n=o
(2n+3)!
sinh (yo-y) +KI(Y) cash (ye-y)],
(6)
0.8
0.7
E/E, Fig, 2. Electron temperature plotted versus electric field, with lattice temperature as parameter. Results of the exact calculations.
to the electric field E. The electron density is n, their average momentum is p, and e and m are the electronic charge and effective mass, respectively. The right-hand sides of the equations express the rate of momentum and energy transfer from electrons to the lattice through collisions. These quantities are rather complicated integrals involving the average momentum p, the electron temperature T and the lattice temperature To. From eqs. (1) and (2) one can calculate p and T as functions of E and To. In the works referred to above [3,4] the first terms of a power series expansion of F and G in p were calculated, F”
PFl(T,To)
G = Go(T,To)
(3) (4)
where F1 and Go are functions of T and To, but not of p . Since F1 IS always positive, and Go and T - To have the same sign, it follows that in this approximation T - To A 0 whenever E # 0. Paranjape et al. [l-2] went one step further, and replaced (4) by G = Go(T, To) + P2G,(T,To) . 16
(5)
x 5
Y-’ K,(Y)
sinh ho -7)
(7)
where E. is a characteristic electric field which expresses the strength of the electron-phonon interaction [2-41,
Y =9/2T Yo
03)
=0/2To
(9)
and w =p/JzXi
.
(10)
Numerical calculations have been performed, and fig. 1 shows the result for To/0= 0.2,compared with the results of the two approximations discussed above. Fig. 2 shows the electron temperature as a function of the applied field, for various values of To. The average momentum caused by a given electric field is higher than the approximate value obtained from eqs. (3) and (4), but the difference is not as profound as for the temperature. As a result of the higher average momentum and lower temperature, the exact theory predicts a considerably higher ratio of drift velocity to thermal velocity. ,When To/B = 0.2,the maximum value of p(2mkT)-2 is 1.2. This may be of importance for
Volume 24A, number 1
PHYSICS
the properties of the semiconductor plasma as a medium for wave propagation. There are al& some interesting aspects of the negative dT/dE as a source of an instability. This may possibly have some relevance to the observed low-field instability in InSb [6]. The cooling effect disappears if any scattering process other than optical phonon scattering is present. Acoustic phonon and impurity scattering must be kept low. It is suggested that relatively pure n-type indium antimonide at lOOoK should be a suitable material for an experimental verification of the effect.
LETTERS
2 January 1967
References 1. V. V. Paranjape and T. P.Ambrose, Phys. Letters 8 (1964) 223. 2. V.V. Paranjape and E. de Alba, Proc. Phys. Sot. 85 (1965) 945. 3. H. Frijhlich and B. V. Paranjape, Proc. Phys.Soc. B 69 (1956) 21. 4. R.Stratton, Proc.Royal Soc.246 (1958) 406. 5. K.Blotekjaer, Arkiv for Fysik 33 (1967) 105. 6. S. J. Buchsbaum, A.G. Chynoweth and W. L. Feldman, Appl.Phys. Letters 6 (1965) 67.
*****
RESISTIVITY
AND
TUNNELLING ALLOY
PROPERTIES FILMS
OF PARAMAGNETIC
F. T. HEDGCOCK and C. RIZZUTO * Eaton Electronics
Research Laboratory, Montreal, Canada
Received
Evaporated suppression
28 November
McGill University,
1966
films of zinc containing paramagnetic impurities are shown to exhibit a resistance of the critical temperature and zero bias tunnelling anomalies.
We wish to report that it is possible to evaporate thin films of dilute paramagnetic alloys that do indeed show resistive anomalies at low temperatures and suppression of the critical temperature of the same type found in bulk metals [1,2]. Fig. la shows a typical result of the temperature dependence of the resistance of a Zn-Mn film prepared as summarized in the caption of fig. lb. The magnitude of slope of this line would indicate that approximately 0.1 at.% Mn was in solution [1,2] which is lower than the nominal composition of the alloy before the evaporation. That this observed resistance minimum is not due to effects different from a magnetic scattering is demonstrated by the fact that in the more dilute samples the concentration dependence of the suppression of the critical temperature for evaporated zinc manganese is in fair agreement with the observed results in the bulk material (see table 1). Shown in fig. lb is a typical tunnelling ano* National Research
Council Postdoctoral
Fellow.
minimum,
Table 1 Concentration values and suppression of the critical temperature for the Zn-Mn alloys. Nominal concentration (atomic parts per million)
AT, evaporated film* (OK)
6.5
0.23hO.05
8.9
0.13+0.01
0.42
> 0.54
ATc /c ATc/c average for evap- average bulk orated films
(OK/at.%)
215
(OK/at.%)
315
* AT, with respect temperature
to a pure Zn strip whose critical is equal to 0.88 *0.05 OK.
maly observed for a Zn-Mn, Al2O3, Al tunnel junction above the superconducting critical temperature of aluminium. (Details of the junction preparation are included in the figure caption.) These anomalies are similar to those reported for Cr-I-Ag junctions [3]. If the observed effects are due to paramagnetic impurities the theoretical work of Anderson [4] and of Applebaum [5] would predict that the magnitude or
17