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The determination of free carrier distribution functions by light scattering
PHYSICS
Volume 29A, number 10
THE
DETERMINATION
OF BY
11 August 1969
LETTERS
FREE LIGHT
CARRIER DISTRIBUTION SCATTERING
FUNCTIONS
D. HEALEY and T. P. MCLEAN Royal
Radar
Establishment,
Malvem,
Worcestershire,
UK
Received 12 June 1969
The dependence of the Raman scattering cross-section of free carriers on their distribution function is interpreted geometrically and the relation inverted. The cross-section for a model high-field distribution in GaAs is discussed.
The possibility has been mentioned [1,2] of using the technique of laser scattering from the free carriers in a semiconductor as a means of obtaining information about their distribution. In this letter, we discuss this possibility in more detail. Consider the process in which radiation of frequency Wl and wave vector q1 is scattered by the free carriers into a state with frequency w2 and wave vector 42, transferring energy Aw =A(wI - wg) and momentum Eq = ff(q1 - 42) to the carriers. In the limit q >> qD , where qbl is the Debye screening length of the free carrier distribution (assumed non-degenerate), the scattering can be considered to be due to the individual independent carriers and the differential scattering cross-section can be expressed as
-----_ -
I-
.fd#%k+q)12f(k) X
6(Ek+q
- Ek - kW).
x (I)
Here we have assumed that fiwI < Eg, the energy gap, so that no interband transitions are excited; M(k, k +q) is the effective matrix element for the transfer of a carrier from a state of wave vector k and energy Ek to one of wave vector k + q and energy Ek+q in the same band throughout which the carriers are distributed according to the distribution function f(k). The remaining notation is conventional and possible occupancy of the final state has been neglected. When the range of occupied states is not too great, the band can be considered to be parabolic and variation of M(k, k + q) and k can be neglected. Then &
= const,f
dkf(k)b(k.q-T]
-\
‘\ ‘\
L"6'
\
\
\
‘1 \\
Ln F(k)
\ t km
I
-$-&=;(~~
-.
I -0
I
I -4
I
I
O
I 4
I.
I 8
Fig. 1. The distribution function F(k) and corresponding scattering cross-section o”(rnW/fiq) P d2o/dWda for the case q//kzo. using the fact that z2q2/2m* << liw to neglect a term in q2 in the &function. This formula has a simple and useful geometrical interpretation. The cross-section as a function of w is proportional to the integral of f(k) over a series of planes in k-space normal to the direction of the scattering vector q and distant m* w& from the origin. It is therefore strongly dependent on the form of the distribution, although it is clear that in most circumstances detailed structure in the form off (k) will be smoothed out in its frequency dependence. However in the particular situations in which f(k) is separable into two functions of the components 607
Volume 29A, number 10
PHYSICS
LETTERS
k,, and k, of k parallel and perpendicular to q, the frequency dependence of the cross-section will be a replica of the &-dependence of the distribution. For example, the displaced Maxwellian distribution f(k) - exp[-E2(k-ko)2/2m*&T] which is separable for all directions of Q, yields a scattering cross-section of the form exp[-E’(m*o/& - ko,,)2/ 2m*kBT], ko,, being the component of k. parallel to the direction of q. The relationship (2) can be inverted giving
f(k) = const lm ff2 do Jdaq 0
proximating to that calculated for GaAs in a field of 3000 V/cm [3]. The distribution consists for k < kc (the intervalley transfer wave vector) of a displaced Maxwellian distribution (T = 600°K, k z~ = 1.34 cm-l) matched for k > kc to a Maxwellian of temperature 300°K. The cross-section is seen to have a frequency dependence very similar to that of k(I dependence of the distribution function. The main difference arises at the frequency wc = hq kc/m* where the cross-section has a discontinuity in its second derivative as opposed to the first derivative discontinuity in f(k) at k = kc.
JWdw &x -03
11 August 1969
4 (3)
Contributed by permission R. R. E. Copyright Controller
X exp
which shows how in principle a complete knowledge of the scattering cross-section for all energy and momentum transfers to and from the free carrier system determines the form of the distribution function. As an example of the relationship (2), we show in fig. 1 the differential cross-section resulting from a model distribution function ap-
of the Director H. M. S. 0.
References 1. A. Mooradian, Intern. Conf. on Light scattering in solids, New York (1968), to be published. 2. P. A. Wolf, Phys. Rev. 1’71 (1968) 436. 3. W. Fawcett, A. D. Boardman and S. Swain, to be published.
*****
ELECTRONIC
PROPERTIES
OF
LIQUID
Co-Ge
ALLOYS
G. BUSCH, H. J. GUNTHERQDT and H. U. KUNZI Laboratorium fiir Festkb’rperphysik ETH, Ziirich, Switzerland Received
30 May 1969
The electrical resistivity shows a maximum at a composition of 60 at. % Co. The liquid alloys 10 and ‘75 at. % Co show large negative temperature coefficients of the electrical resistivity.
Experimental work on the electronic structure of the liquid transition metals is just beginning. The main purpose of the work on these liquids is to investigate the behavior of the d-electrons in the liquid state. Experimental difficulties arise from the high melting points and from the reactivity of the transition metals. The electrical resistivity has been measured by a four-probe a.c. potentiometric method. Up to now the investigated alloys can be divided into two groups. The typical behavior is represented by the liquid alloys Co-Ge and Au-Ni. In the following we present the measurements of liquid Co-Ge. The electrical resistivity as a function of temperature (see fig. 1) shows a positive temperature coefficient for pure liquid Ge 608
between
and Co [l]. The curves for 40,60, 66.6 and 50.4 at. % Ge show negative temperature coefficients of the electrical resistivity much as is characteristic for liquid Zn and alloys of the Ag-In group ]2,31. The electrical resistivity at constant temperature of liquid Ge (see fig. 2) at first increases with alloying of Co shows a maximum at a composition of 60 at. % Co and finally diminishes with further increase of the concentration to the value of Co. The magnitude of the maximum resistivity is about twice the value of liquid Hg at room temperature. The resistivity value of pure liquid Co is indicated for the melting temperature which lies above 11OOoC. The liquid alloys between 10 and 75 at. % Co show negative temperature co-
Volume 29A, number 10
THE
DETERMINATION
OF BY
11 August 1969
LETTERS
FREE LIGHT
CARRIER DISTRIBUTION SCATTERING
FUNCTIONS
D. HEALEY and T. P. MCLEAN Royal
Radar
Establishment,
Malvem,
Worcestershire,
UK
Received 12 June 1969
The dependence of the Raman scattering cross-section of free carriers on their distribution function is interpreted geometrically and the relation inverted. The cross-section for a model high-field distribution in GaAs is discussed.
The possibility has been mentioned [1,2] of using the technique of laser scattering from the free carriers in a semiconductor as a means of obtaining information about their distribution. In this letter, we discuss this possibility in more detail. Consider the process in which radiation of frequency Wl and wave vector q1 is scattered by the free carriers into a state with frequency w2 and wave vector 42, transferring energy Aw =A(wI - wg) and momentum Eq = ff(q1 - 42) to the carriers. In the limit q >> qD , where qbl is the Debye screening length of the free carrier distribution (assumed non-degenerate), the scattering can be considered to be due to the individual independent carriers and the differential scattering cross-section can be expressed as
-----_ -
I-
.fd#%k+q)12f(k) X
6(Ek+q
- Ek - kW).
x (I)
Here we have assumed that fiwI < Eg, the energy gap, so that no interband transitions are excited; M(k, k +q) is the effective matrix element for the transfer of a carrier from a state of wave vector k and energy Ek to one of wave vector k + q and energy Ek+q in the same band throughout which the carriers are distributed according to the distribution function f(k). The remaining notation is conventional and possible occupancy of the final state has been neglected. When the range of occupied states is not too great, the band can be considered to be parabolic and variation of M(k, k + q) and k can be neglected. Then &
= const,f
dkf(k)b(k.q-T]
-\
‘\ ‘\
L"6'
\
\
\
‘1 \\
Ln F(k)
\ t km
I
-$-&=;(~~
-.
I -0
I
I -4
I
I
O
I 4
I.
I 8
Fig. 1. The distribution function F(k) and corresponding scattering cross-section o”(rnW/fiq) P d2o/dWda for the case q//kzo. using the fact that z2q2/2m* << liw to neglect a term in q2 in the &function. This formula has a simple and useful geometrical interpretation. The cross-section as a function of w is proportional to the integral of f(k) over a series of planes in k-space normal to the direction of the scattering vector q and distant m* w& from the origin. It is therefore strongly dependent on the form of the distribution, although it is clear that in most circumstances detailed structure in the form off (k) will be smoothed out in its frequency dependence. However in the particular situations in which f(k) is separable into two functions of the components 607
Volume 29A, number 10
PHYSICS
LETTERS
k,, and k, of k parallel and perpendicular to q, the frequency dependence of the cross-section will be a replica of the &-dependence of the distribution. For example, the displaced Maxwellian distribution f(k) - exp[-E2(k-ko)2/2m*&T] which is separable for all directions of Q, yields a scattering cross-section of the form exp[-E’(m*o/& - ko,,)2/ 2m*kBT], ko,, being the component of k. parallel to the direction of q. The relationship (2) can be inverted giving
f(k) = const lm ff2 do Jdaq 0
proximating to that calculated for GaAs in a field of 3000 V/cm [3]. The distribution consists for k < kc (the intervalley transfer wave vector) of a displaced Maxwellian distribution (T = 600°K, k z~ = 1.34 cm-l) matched for k > kc to a Maxwellian of temperature 300°K. The cross-section is seen to have a frequency dependence very similar to that of k(I dependence of the distribution function. The main difference arises at the frequency wc = hq kc/m* where the cross-section has a discontinuity in its second derivative as opposed to the first derivative discontinuity in f(k) at k = kc.
JWdw &x -03
11 August 1969
4 (3)
Contributed by permission R. R. E. Copyright Controller
X exp
which shows how in principle a complete knowledge of the scattering cross-section for all energy and momentum transfers to and from the free carrier system determines the form of the distribution function. As an example of the relationship (2), we show in fig. 1 the differential cross-section resulting from a model distribution function ap-
of the Director H. M. S. 0.
References 1. A. Mooradian, Intern. Conf. on Light scattering in solids, New York (1968), to be published. 2. P. A. Wolf, Phys. Rev. 1’71 (1968) 436. 3. W. Fawcett, A. D. Boardman and S. Swain, to be published.
*****
ELECTRONIC
PROPERTIES
OF
LIQUID
Co-Ge
ALLOYS
G. BUSCH, H. J. GUNTHERQDT and H. U. KUNZI Laboratorium fiir Festkb’rperphysik ETH, Ziirich, Switzerland Received
30 May 1969
The electrical resistivity shows a maximum at a composition of 60 at. % Co. The liquid alloys 10 and ‘75 at. % Co show large negative temperature coefficients of the electrical resistivity.
Experimental work on the electronic structure of the liquid transition metals is just beginning. The main purpose of the work on these liquids is to investigate the behavior of the d-electrons in the liquid state. Experimental difficulties arise from the high melting points and from the reactivity of the transition metals. The electrical resistivity has been measured by a four-probe a.c. potentiometric method. Up to now the investigated alloys can be divided into two groups. The typical behavior is represented by the liquid alloys Co-Ge and Au-Ni. In the following we present the measurements of liquid Co-Ge. The electrical resistivity as a function of temperature (see fig. 1) shows a positive temperature coefficient for pure liquid Ge 608
between
and Co [l]. The curves for 40,60, 66.6 and 50.4 at. % Ge show negative temperature coefficients of the electrical resistivity much as is characteristic for liquid Zn and alloys of the Ag-In group ]2,31. The electrical resistivity at constant temperature of liquid Ge (see fig. 2) at first increases with alloying of Co shows a maximum at a composition of 60 at. % Co and finally diminishes with further increase of the concentration to the value of Co. The magnitude of the maximum resistivity is about twice the value of liquid Hg at room temperature. The resistivity value of pure liquid Co is indicated for the melting temperature which lies above 11OOoC. The liquid alloys between 10 and 75 at. % Co show negative temperature co-