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On the modes of acoustoelectric gain in n-InSb
PHYSICS
Volume 28A. number 7
ON THE
MODES
13 January 1969
LETTERS
OF ACOUSTOELECTRIC
GAIN
IN n-InSb
J. GORELIK, B. FISHER, B. PRATT and Z. LUZ Depatimznt
of Physics,
Technion
- Israel
Institute
bf Technology,
Haifa,
Israel
and A. MANY Department
of Physics,
The Hebrew
University.
Jerusalem
t Israel
Received 22 November 1968
The mode I and II of the acoustoelectric effect in n-InSb are attributed to the two regimes in which the effective electron mean-free-path is, respectively, large and small compared to the acoustic wavelength. Evidence is presented in support for this contention.
Several studies [l-3] of the effect of transverse magnetic fields on acoustic amplification in n-InSb at ‘77oK have revealed the presence of two distinctly different modes of operation - the so-called models I and II. While the main features of mode II are fairly well understood [3,4], no explanation has so far been given to the mode I operation. In this communication we propose that mode I belongs to the regime kl > 1 (where k is the acoustic wave number and I the electron mean-free-path for scattering) and as such is a result of the gain mechanism [5-81 operative in this regime. In mode II, on the other hand, because of the presence of the high magnetic fields one has in effect kl < I, so that Steele’s extension [Q] of White’s theory [lo] should be valid. In support for this contention we find that analysis of the data in modes I and II, based in each case on the appropriate expression for the acoustoelectric gain, yields comparable values for the non-electronic attenuation factor @l in the two modes. Furthermore, the ol values so obtained agree quite well with those determined independently from measurements of acoustoelectric after-current decay. The measurements were carried out on n-InSb filaments oriented parallel to the [llO] direction and aligned perpendicularly to the magnetic field direction. The threshold drift velocity “gh) for net acoustic gain was derived from the measured threshold current for the onset of current oscillations signifying acoustic-domain formation. Typical results of ~2~) are plotted in fig. 1 against the applied transverse magnetic field B. The two well known modes of operation are clear-
TRANSVERSE
G (GAUSS)
> 1500
@
+ ).t2B2),
p
is w max = amplitude
gain factor
a is given by [9]
a = W2wD/8%)~@ [l +($Y$J)~ (wD/w
(1) c)]-l
cm”.
The symbols are as defined in ref. 10 and ye E ,Vd/us -1, where Vd is the electron drift velo485
Volume
28A.
number
7
PHYSICS
;m > f -,= 4 _’ f -A
1 cy = 9K2w~(wcwD)Z~y/32v~
2
0
4
6 1000
12 /
16
20
4
_ 1 versus l/GE Fig. 2. Plots of y (th) s v(th)/V = (1 + r.r2B2)-l obtained fdr m:de II on three n-InSb samples. The Cyl values derived from slopes of straight lines and those obtained from measured thresholds at B = 0 (mode I) are listed in the table. Also included are the corresponding CY~values derived from after-current data.
city and us the propagation.
sound
velocity
for
In InSb, K2 = 0.0011,
shear
wave
E = 17~0 and
vs = 2.3 x Ias cm/set. Net amplification occurs when the gain factor cr exceeds the non-electronic attenuation factor ~2. The threshold drift velocity vih)is therefore obtained from the condition (Y = czl which, by the use of eq. (l), yields
(th) ~v~),vs _ 1
=
= (2wChD9) [l
-&JD/wc,t] (l/e)
,
(2)
where UJ e 8v&/K2wD. The y (th) values corresponding to the mode II branch in fig. 1, as well as those measured on two other samples, are re-plotted in fig. 2 against (l/G). In each case the points are seen to lie on a straight line which extrapolates to the origin, in good agreement with the predictions of eq. (2). The values of QI derived from the slopes of the straight lines by the use of eq. (2) are listed in the figure (third line of table). In examining the range of validity of Steele’s analysis [9], one should note that the magnetic field reduces not only the r.f. mobility but also the effective mean-free-path for scattering. For pB > 1, it is reasonable [5] to take for the reduced mean-free-path the cyclotron radius E/pB. Hence the range of validity of eq. (l), which is kl ~1 EorB =0, becomeskl/pBl. 486
1969
The latter range is marked off in fig. 1 and is seen to cover the entire mode II branch. Obviously the low magnetic-field portion of the mode I branch lies outside this range. Consider the case of B = 0, where kl >> 1. In this regime, the appropriate expression for (Yat t!e frequency of maximum gain [wmax = (3wcwD)‘], as derived both classically [5,6] and quantummechnically [6 -81, is
6
Y
13 January
LETTERS
cm
-1
,
(3)
where l = (p/q)(kBTm*)f. Here again, “d(th) is given by the condition (Y = “1, and using the measured thresholds (at B = 0) one obtains for the three samples studied the “1 values listed in fig. 2 above. It is seen that these values match to within better than a factor of two those derived for mode II, lending strong support for the general validity of eq. (3) in the regime kl >> 1. A better correspondence can
hardly
between
be expected
the two in view
sets
of values
of the approximate
nature of eq. (3) when applied to the present case. In InSb the drift-velocity thresholds in mode I are comparable to the thermal velocity so that appreciable heating of the electrons is to be expected, as is indeed evidenced by measurements of current-voltage characteristics. Unless eq. (3) is suitably modified to take into account the shifted electron distribution, there is also not much point in attempting its extension to include the effect of magnetic fields. In its present form, then, eq. (3) should be considered merely as a pointer to the origin of mode I rather than as leading to a quantitative description of its detailed features. An independent determination of uyIfor mode I and II has been obtained from measurements of the after-current decay associated with the acoustic-flux attenuation following the termination of the applied voltage pulse [ll]. The “1 values so derived are listed in fig. 2 (last two lines of table). The good agreement obtained for each mode between the sets of ~1 values derived from the threshold and after-current data provides further support for our proposed interpretation of the two modes of operation.
References 1. M. Kikuchi,
H. Hayakawa and Y. Abe, Japan J. Appl. Phys.5 (1966) 1259. 2. C. W. Turner and J. Crow, Appl. Phys. Letters 11 (1967) 187. 3. G. S. Kino and R. Route, Appl. Phys. Letters 11 (1967) 312.
Volume 28A, number
PHYSICS
7
SUSCEPTIBILITY
IN RANDOMLY
DILUTE
T. G. BLOCKER and F. G. WEST
Texas Instruments
Incorporated,
Received
1969
8. J. Yamashita and K. Nakamura, Prog. Theor. Physics 33 (1965) 1022. 9. M. C.Steele, RCA Review 28 (1967) 58; a similar analysis was performed by C. Hervouet, Phys. Status Solidi 21 (1967) 117. 10.D. L. White, J. Appl. Phys. 33 (1962) 2547. 11.1. Balberg and A. Many, Appl. Phys. Letters 13 (1968) 100.
4. V. Dolat, J. B. Ross and R. Bray, Appl. Phys. Letters 13 (1968) 60. 5. H. N. Spector in Solid State Physics, eds. F. Seitz and D. Turnbull, Vol.19 (Academic Press, 1966) p. 291-361. 6. A.Rose, RCA’Review 28 (1967) 634. 7. E. Conwell. Phys. Letters 13 (1964) 285.
ON MAGNETIC
13 January
LETTERS
Dallas,
4 December
SYSTEMS
USA
Texas,
1968
An extension of molecular field theory to magnetic susceptibility of randomly dilute systems is presented. A useful expression involving the paramagnetic Curie temperature 19of the pure system is obtained.
It is the purpose of this note to point out a simple but useful extension * of molecular field theory to magnetic susceptibility of a crystal containing a dilution of randomly substituted nonmagnetic ions or molecules [e.g., xMnF2 - (l-x) ZnF2 and x Cr203(1-x) Al2O3]. Such systems are of interest for ianumber of reasons; e.g., the addition of sub$titutional nonmagnetic ions may allow one to vary magnetic parameters such as exchange, anisotropy and range of interaction in a systematic fashion; or the “pure” system may be a less suitable candidate for the study at hand than a suitably chosen “impure” system. We encountered an etample of the latter kind in a recent investigation of exchange interactions between oxygen molecules. The interpretation of susceptibility measurements in pure oxygen is difficult because of three solid phase transitions each of which has only a narrow temperature range of stability. On the other hand solid mixtures of oxygen with argon exist in a single hcp phase over the entire temperature range of the solid for 02 concentrations of 20 to 50% in A. The temperature range was sufficient to allow extraction of the fundamental parameter e=2s(S+1)Czic$/3k i describing the pure system by using the statistical formalism described below. Consider a Bravais lattice of N sites characterized by each lattice site having zl nearest
neighbor neighbor
sites at distance sites at distance
R1 ,z2 next nearest
R2, etc. Suppose n of these sites are occupied by magnetic ions (or molecules) characterized by spin S and magnetic interaction J(R) and N-n sites are occupied by nonmagnetic ions. For the “pure” case, i.e., a concentration of magnetic ions c = n/N = 1, the susceptibility in the molecular field approximation is given by the Curie-Weiss law x
=C/(T+B)
,
(1)
where c = Ng2&S(S 0 = 3ZiBi; i
+ 1)/3k ,
Oi = 2S(S + l)Ji(Ri)/Sk
(2) ,
(3)
and T is the absolute temperature. Here g is the gyromagnetic ratio, PB is the Bohr magneton, k is Boltzmann’s constant, and Ji(Ri) is the exchange interaction between ith nearest neighbors separated by distance Ri and is presumed to be negligible for R > Rm . If this lattice is now diluted (c =n/N < 1) with nonmagnetic ions each magnetic ion will no longer have q magnetic nearest neighbors etc., but some statistical distribution of magnetic and nonmagnetic neighbors. For the case of random dilution the probability that a given magnetic ion has k ith neighbors * An extension to arbitrary spin and numbers of interacting neighbors.
Volume 28A. number 7
ON THE
MODES
13 January 1969
LETTERS
OF ACOUSTOELECTRIC
GAIN
IN n-InSb
J. GORELIK, B. FISHER, B. PRATT and Z. LUZ Depatimznt
of Physics,
Technion
- Israel
Institute
bf Technology,
Haifa,
Israel
and A. MANY Department
of Physics,
The Hebrew
University.
Jerusalem
t Israel
Received 22 November 1968
The mode I and II of the acoustoelectric effect in n-InSb are attributed to the two regimes in which the effective electron mean-free-path is, respectively, large and small compared to the acoustic wavelength. Evidence is presented in support for this contention.
Several studies [l-3] of the effect of transverse magnetic fields on acoustic amplification in n-InSb at ‘77oK have revealed the presence of two distinctly different modes of operation - the so-called models I and II. While the main features of mode II are fairly well understood [3,4], no explanation has so far been given to the mode I operation. In this communication we propose that mode I belongs to the regime kl > 1 (where k is the acoustic wave number and I the electron mean-free-path for scattering) and as such is a result of the gain mechanism [5-81 operative in this regime. In mode II, on the other hand, because of the presence of the high magnetic fields one has in effect kl < I, so that Steele’s extension [Q] of White’s theory [lo] should be valid. In support for this contention we find that analysis of the data in modes I and II, based in each case on the appropriate expression for the acoustoelectric gain, yields comparable values for the non-electronic attenuation factor @l in the two modes. Furthermore, the ol values so obtained agree quite well with those determined independently from measurements of acoustoelectric after-current decay. The measurements were carried out on n-InSb filaments oriented parallel to the [llO] direction and aligned perpendicularly to the magnetic field direction. The threshold drift velocity “gh) for net acoustic gain was derived from the measured threshold current for the onset of current oscillations signifying acoustic-domain formation. Typical results of ~2~) are plotted in fig. 1 against the applied transverse magnetic field B. The two well known modes of operation are clear-
TRANSVERSE
G (GAUSS)
> 1500
@
+ ).t2B2),
p
is w max = amplitude
gain factor
a is given by [9]
a = W2wD/8%)~@ [l +($Y$J)~ (wD/w
(1) c)]-l
cm”.
The symbols are as defined in ref. 10 and ye E ,Vd/us -1, where Vd is the electron drift velo485
Volume
28A.
number
7
PHYSICS
;m > f -,= 4 _’ f -A
1 cy = 9K2w~(wcwD)Z~y/32v~
2
0
4
6 1000
12 /
16
20
4
_ 1 versus l/GE Fig. 2. Plots of y (th) s v(th)/V = (1 + r.r2B2)-l obtained fdr m:de II on three n-InSb samples. The Cyl values derived from slopes of straight lines and those obtained from measured thresholds at B = 0 (mode I) are listed in the table. Also included are the corresponding CY~values derived from after-current data.
city and us the propagation.
sound
velocity
for
In InSb, K2 = 0.0011,
shear
wave
E = 17~0 and
vs = 2.3 x Ias cm/set. Net amplification occurs when the gain factor cr exceeds the non-electronic attenuation factor ~2. The threshold drift velocity vih)is therefore obtained from the condition (Y = czl which, by the use of eq. (l), yields
(th) ~v~),vs _ 1
=
= (2wChD9) [l
-&JD/wc,t] (l/e)
,
(2)
where UJ e 8v&/K2wD. The y (th) values corresponding to the mode II branch in fig. 1, as well as those measured on two other samples, are re-plotted in fig. 2 against (l/G). In each case the points are seen to lie on a straight line which extrapolates to the origin, in good agreement with the predictions of eq. (2). The values of QI derived from the slopes of the straight lines by the use of eq. (2) are listed in the figure (third line of table). In examining the range of validity of Steele’s analysis [9], one should note that the magnetic field reduces not only the r.f. mobility but also the effective mean-free-path for scattering. For pB > 1, it is reasonable [5] to take for the reduced mean-free-path the cyclotron radius E/pB. Hence the range of validity of eq. (l), which is kl ~1 EorB =0, becomeskl/pB
1969
The latter range is marked off in fig. 1 and is seen to cover the entire mode II branch. Obviously the low magnetic-field portion of the mode I branch lies outside this range. Consider the case of B = 0, where kl >> 1. In this regime, the appropriate expression for (Yat t!e frequency of maximum gain [wmax = (3wcwD)‘], as derived both classically [5,6] and quantummechnically [6 -81, is
6
Y
13 January
LETTERS
cm
-1
,
(3)
where l = (p/q)(kBTm*)f. Here again, “d(th) is given by the condition (Y = “1, and using the measured thresholds (at B = 0) one obtains for the three samples studied the “1 values listed in fig. 2 above. It is seen that these values match to within better than a factor of two those derived for mode II, lending strong support for the general validity of eq. (3) in the regime kl >> 1. A better correspondence can
hardly
between
be expected
the two in view
sets
of values
of the approximate
nature of eq. (3) when applied to the present case. In InSb the drift-velocity thresholds in mode I are comparable to the thermal velocity so that appreciable heating of the electrons is to be expected, as is indeed evidenced by measurements of current-voltage characteristics. Unless eq. (3) is suitably modified to take into account the shifted electron distribution, there is also not much point in attempting its extension to include the effect of magnetic fields. In its present form, then, eq. (3) should be considered merely as a pointer to the origin of mode I rather than as leading to a quantitative description of its detailed features. An independent determination of uyIfor mode I and II has been obtained from measurements of the after-current decay associated with the acoustic-flux attenuation following the termination of the applied voltage pulse [ll]. The “1 values so derived are listed in fig. 2 (last two lines of table). The good agreement obtained for each mode between the sets of ~1 values derived from the threshold and after-current data provides further support for our proposed interpretation of the two modes of operation.
References 1. M. Kikuchi,
H. Hayakawa and Y. Abe, Japan J. Appl. Phys.5 (1966) 1259. 2. C. W. Turner and J. Crow, Appl. Phys. Letters 11 (1967) 187. 3. G. S. Kino and R. Route, Appl. Phys. Letters 11 (1967) 312.
Volume 28A, number
PHYSICS
7
SUSCEPTIBILITY
IN RANDOMLY
DILUTE
T. G. BLOCKER and F. G. WEST
Texas Instruments
Incorporated,
Received
1969
8. J. Yamashita and K. Nakamura, Prog. Theor. Physics 33 (1965) 1022. 9. M. C.Steele, RCA Review 28 (1967) 58; a similar analysis was performed by C. Hervouet, Phys. Status Solidi 21 (1967) 117. 10.D. L. White, J. Appl. Phys. 33 (1962) 2547. 11.1. Balberg and A. Many, Appl. Phys. Letters 13 (1968) 100.
4. V. Dolat, J. B. Ross and R. Bray, Appl. Phys. Letters 13 (1968) 60. 5. H. N. Spector in Solid State Physics, eds. F. Seitz and D. Turnbull, Vol.19 (Academic Press, 1966) p. 291-361. 6. A.Rose, RCA’Review 28 (1967) 634. 7. E. Conwell. Phys. Letters 13 (1964) 285.
ON MAGNETIC
13 January
LETTERS
Dallas,
4 December
SYSTEMS
USA
Texas,
1968
An extension of molecular field theory to magnetic susceptibility of randomly dilute systems is presented. A useful expression involving the paramagnetic Curie temperature 19of the pure system is obtained.
It is the purpose of this note to point out a simple but useful extension * of molecular field theory to magnetic susceptibility of a crystal containing a dilution of randomly substituted nonmagnetic ions or molecules [e.g., xMnF2 - (l-x) ZnF2 and x Cr203(1-x) Al2O3]. Such systems are of interest for ianumber of reasons; e.g., the addition of sub$titutional nonmagnetic ions may allow one to vary magnetic parameters such as exchange, anisotropy and range of interaction in a systematic fashion; or the “pure” system may be a less suitable candidate for the study at hand than a suitably chosen “impure” system. We encountered an etample of the latter kind in a recent investigation of exchange interactions between oxygen molecules. The interpretation of susceptibility measurements in pure oxygen is difficult because of three solid phase transitions each of which has only a narrow temperature range of stability. On the other hand solid mixtures of oxygen with argon exist in a single hcp phase over the entire temperature range of the solid for 02 concentrations of 20 to 50% in A. The temperature range was sufficient to allow extraction of the fundamental parameter e=2s(S+1)Czic$/3k i describing the pure system by using the statistical formalism described below. Consider a Bravais lattice of N sites characterized by each lattice site having zl nearest
neighbor neighbor
sites at distance sites at distance
R1 ,z2 next nearest
R2, etc. Suppose n of these sites are occupied by magnetic ions (or molecules) characterized by spin S and magnetic interaction J(R) and N-n sites are occupied by nonmagnetic ions. For the “pure” case, i.e., a concentration of magnetic ions c = n/N = 1, the susceptibility in the molecular field approximation is given by the Curie-Weiss law x
=C/(T+B)
,
(1)
where c = Ng2&S(S 0 = 3ZiBi; i
+ 1)/3k ,
Oi = 2S(S + l)Ji(Ri)/Sk
(2) ,
(3)
and T is the absolute temperature. Here g is the gyromagnetic ratio, PB is the Bohr magneton, k is Boltzmann’s constant, and Ji(Ri) is the exchange interaction between ith nearest neighbors separated by distance Ri and is presumed to be negligible for R > Rm . If this lattice is now diluted (c =n/N < 1) with nonmagnetic ions each magnetic ion will no longer have q magnetic nearest neighbors etc., but some statistical distribution of magnetic and nonmagnetic neighbors. For the case of random dilution the probability that a given magnetic ion has k ith neighbors * An extension to arbitrary spin and numbers of interacting neighbors.