- Email: [email protected]
Waves in Plasma
Chapter II
Waves in Plasma
In a conducting medium, such as plasma, the propagating electric field waves may produce the waves of particle density, which, in their turn, may again induce the electric field waves. By making use of the models of particle motion in electric and magnetic fields, considered in Section 3 of Chapter I, one can construct elementary models of all possible kinds of waves in the semiconductor plasma. When the magnetic field B 0 is absent, only one type of waves, the acoustical ones, can exist in the plasma. The cause of these compression-expansion waves is the thermal motion of the plasma particles; this is similar to the way of formation of sound waves in a gas consisting of neutral particles. In the plasma, however, these waves of particle density are accompanied by a variable electric field. For instance, in the single-component electron plasma the acoustical vibrations of the electron density produce waves of space charge which is negative in the region of the excessive electron concentration and positive where the uncompensated donor charge exists. In an unmagnetised plasma the sound waves are called the electron or hole sounds. In the case of a plasma placed in a magnetic field the wave spectrum is much broader. Consider the perturbation of the particle density produced in the intrinsic plasma by a plane variable transverse wave of the electric field E. Assume that the direction of the wave propagation is parallel to that of the external constant magnetic field: k||B0 (Fig. 2.1.a). According to (3.16) of Chapter I, the plasma particles drift independently of their mass and charge sign the drift direction being normal to both acting fields, the electric E and the magnetic B 0 . When the sign of the wave electric field changes, the drift direction changes its sign too. As a result of this, the propagation of the transverse electric field wave is accompanied by transverse hydrodynamic vibrations of the plasma particles on the whole (Fig. 2.1.a). In their own turn, these displacements of the plasma induce an additional electric field which is caused by the cyclotron rotation of particles with different masses (Fig. 1.3). By the same kind of reasoning we arrive at a conclusion that if kJ_B0, the transverse wave of the electric field is accompanied by the longitudinal wave of the plasma particle density compressionexpansion (see Fig. 2.1.b). In the intermediate case corresponding to an arbitrary angle between the vectors k and B 0 the hybrid waves of the plasma particle density will appear. The above hydrodynamic waves in the plasma are called magnetic sounds. The electric field waves which are accompanied by magnetic sounds, in the intrinsic plasma, are called Alfven waves.
46
Waves in Plasma
Fig. 2.1.
47
The formation of hydrodynamic waves in magnetised intrinsic plasma during propagation of the electric field wave: B 0 corresponds to the transverse hydrodynamic wave (a); o - to the longitudinal hydrodynamic wave (b).
As in the unmagnetised plasma, the longitudinal electric field wave for k || Bo produces electron sound. And if k/jfBo, the longitudinal fields E will excite also the transverse waves of the plasma density, in addition to the longitudinal oscillations. In the single-component plasma the particles with a charge of one sign are motionless, and the plasma does not drift on the whole. The electric and magnetic fields cause only the mobile particles to drift. Therefore, the flux of free charged particles (e.g., electrons) with a drift velocity u (see (3.16), Chapter I) creates an electric current j = qn0u which is uncompensated by particles with the charge of the other sign. This current whose direction is normal to that of the vector E produces, according to (1.2) of Chapter I, an electric field Which is normal to the initial one. The resultant field vector proves to be turned by a certain angle with respect to the initial direction of the vector E. Because of the fact that the direction of the electron drift u always remains normal to that of the acting fields, the rotation of the polarization of the field vector results in the rotation of the drift direction by the same angle. This leads to twisting into a kink of the transverse wave for k||B0 (Fig. 2.2). The kink direction depends on the sign of free charges, namely, it coincides, as will be shown below, with the direction of the cyclotron rotation of particles. This is natural as the direction of the drift u of particles is independent of the charge sign and the
48
Plasma and Current Instabilities in Semiconductors
currents caused by the drift have opposite directions for particles with charges of different signs. In the plasma, owing to kinking of the transverse wave, the waves with circular polarization emerge which are called helicon. The helicon waves experience weak absorption. This property of helicon waves made it possible to use them for the study of the plasma properties in semiconductors and metals, as well as for constructing a number of devices.
J=9«QU
In the present chapter we deduce the dispersion relations for plane monochromatic waves in a plasma in the absence of a Fig. 2.2. Polarisation of the transverse wave constant field E 0 and the drift in a magnetised single-component plasma. of the charge carriers associated with it. From the dispersion relation we obtain the wave propagation characteristics: the amplitude attenuation in space, the wave polarization, the phase velocity and, finally, the ranges of frequencies with which waves can propagate in a plasma. Let us present the main formulas that characterize the propagation of a weakly decaying monochromatic plane wave of the form E = E for the case k||r.
exp[t((jo£ - kr)]
(0.1)
The wave length λ =
2π
(0.2)
The phase velocity of the wave ω'
(0.3)
The attenuation factor for the wave amplitude 1
(0.4)
The refractive index N =
kl2~2"| a
(0.5)
The radiation intensity reflection coefficient for the case of normal incidence 2+ (fe'-M ( fc ") 2 , where ka 2
(k' - ko)
* (k"Y
°
^ .
One can see that all these quantities as well as their dependence on the wave frequency ω' can be found if one solves the dispersion equations for the real frequencies with respect to the complex wave number ?c(u)').
(0.6)
49
Waves in Plasma 1.
WAVES IN A COLD PLASMA IN THE ABSENCE OF MAGNETIC FIELD
In the linear approximation, in the absence of magnetic field B 0 the determinant of the wave equation (2.20) of Chapter I contains only the diagonal components a , a , a . The non-diagonal components are equal to zero. The only direction in^the Isotropie unmagnetized plasma which is singled out is the direction of the wave propagation. We shall assume that this direction is parallel to the 2-axis. The transverse and longitudinal waves in an unbounded isotropic plasma are not related to each other and can be considered separately. The Longitudinal Waves. It is possible to reduce the equation (2.20) of Chapter I for the longitudinal wave which propagates, for instance, along the 2-axis to an equation OL E = 0 (since Εχ - Ey = 0 ) , and the dispersion relation (2.22) of Chapter I reduces to azz = 0. On the basis of (2.21) of Chapter I and taking into account the fact that κχ - ky = 0, we obtain immediately
ω = i ^
(1.1)
As was noted in Section 2 of Chapter I, in a cold plasma σ z z does not depend on k. Therefore, equation (1.1) cannot be called a dispersion relation since it does not contain k and, because of this, does not yield the dependence of k on ω. From the physical point of view this corresponds to the fact that in this case waves do not propagate as such. Two extreme modes can be distinguished: the collisional one when ωτ+ « 1, (1.2) J P and the collisionless for which ωτ+ » 1. P
(1.3)
In the first case the electric conductivity σ 2 2 , according to (3.8) of Chapter I, is a real quantity. Then one obtains from (1.1) that ω' = 0 which corresponds to the absence of periodic oscillations. If σ > 0, the perturbation of charges in the plasma decays exponentially, with the decay constant
zz
(1.4)
The time interval τ ^ is called the dielectric relaxation time or the Maxwell M relaxation time. There is an electrical engineering analog of the Maxwell relaxation time, namely, the time of discharge of a charged capacitor through a resistance τ = RC. Indeed, assume that a certain perturbation results in that the negative charge in a semiconductor is displaced with respect to the positive by a distance 1. The corresponding situation can be imagined as that of a charged capacitor with a capacitance C = Z\{S/l), where S is the semiconductor cross-section, the capacitor being shunted by a resistance R = 1/(So). If one neglects diffusion, the discharge time τ = RC coincides with (1.4). Thus, the charges in the plasma that are separated in space move to each other, but friction (collisions with the lattice) prevent the appearance of plasma oscillations: ω' = 0. In the collisionless plasma (1.3) there is no such friction, and because of
50
Plasma and Current Instabilities in Semiconductors
inertia the motion of charges towards each other results in their new separation. The oscillations arise. Indeed, when the inequality (1.3) is valid, according to (3.8) of Chapter I the electric conductivity σ is an imaginary quantity which is given for the single-component plasma by an equation: *2 3- n0 -J-. m ^ω
σ zz
(1.5)
After substitution of (1.5) into (1.1) we obtain ω' - ω , P
(1.6)
where ω =4 V \y me ι
(1.7)
The frequency ωρ, already familiar to us from the first chapter, is called the plasma or Langmuir frequency. It is equal to
where n0
is expressed in cm" 3 , m* -
m/m0.
The plasma frequency is related to the Maxwell relaxation time through a simple equation + =
£
i 23
=
1 7
(1.8) p
The plasma frequency represents the most important parameter of the plasma. As we have seen, the frequency ω ρ corresponds to the oscillations of the negative charges with respect to the positive. These oscillations are sometimes called the eigen modes of plasma oscillations. The energy quantum of plasma oscillations ?2ωρ is called plasmon. The plasma oscillations in a partially ionized gas which were observed in Tonks' and Langmuir's experiments were characterized by a frequency with the order of magnitude of 100 MHz. The energy of plasmons, corresponding to this frequency, is rather smallftcup~ 10" 5 eV, and the oscillations could be excited by the thermal motion of the gas ions. The frequency of plasma oscillations in metals is of the order of magnitude of 1 0 1 5 Hz which corresponds to the order of magnitude of plasmon energy 10 eV. Therefore, they are practically not excited by the thermal motion of the atoms of the solid. Ruthemann (1) and Lang (2) were the first who excited and observed plasmons in a solid in 1948. In their experiments the plasmons were produced in thin metal films by an electron beam with the energy of several KeV. The plasmon energy was determined on the basis of the analysis of the losses1 spectrum for the electrons that passed through the metal film. The following values were obtained: for Al ?ζω « 14.7 eV, for Sb Τίω^ = 22.6 eV, for Be Τζωρ = 19.0 eV. In semiconductors ?2ü)p ~ 10" 3 eV, and the excitation of plasmons can occur as a result of thermal electron oscillations. The Transverse Wave. In the absence of magnetic field the dispersion equation for a transverse wave propagating in the direction of s-axis [E = 0 , fc = k = 0 ) , y according to (2.20) and (2.22) of Chapter I, has the form:
51
Waves in Plasma (χ
= ot
yy
= 0.
(1.9)
xx
= σ = σ, and from (1.9) and (2.21) of Chapter For an isotropic semiconductor σ xx yy I we obtain k2 - ^ S - + ^ωμ0ο* = 0.
(1.10)
For a collisionless plasma, when the inequality (1.3) is valid, the substitution of (1.5) into (1.10) results in
*».ίομ
1
ω2
(1.11)
If ω < ü)p, we have kz < 0 or kf = 0. This means that the transverse waves whose frequency is smaller than the plasma frequency cannot be excited in the plasma. The waves with frequencies smaller than uu are totally reflected from the plasma since the reflection coefficient R (0.6) equals 1 for kr = 0. In this case the so-called non-transmission of the wave is observed (see Section 1 of Chapter III). The physical meaning of non-transmission is that the electrons in a plasma have enough time to become redistributed in such a way that they screen the penetration of waves into the plasma. For ωί/ω 2 » 1, according to (1.11), the depth of the transition layer a = l/ktf inside which the amplitude of the wave incident on the plasma surface decreases by a factor of e equals to a =
(1.12)
UpSxï
Note that the cause of the wave attenuation is not the energy losses but the formation of the reverse wave. This constitutes the difference between nontransmission and absorption of waves. When ω > ω ρ we have k2 > 0, and undamped waves can propagate through a collisionless plasma (with k' Φ 0, k" = 0). As follows from (1.11), the phase velocity of the penetrating wave is v
f
=
ω' k'
Xi 1 -
(o>'r
(1.13)
The free carriers contribute the negative term into the dielectric constant, and the phase velocity turns out to exceed the velocity of light in a medium with a lattice dielectric constant χχ. When ω = ω , we have v ~ -+ °°, and the group velocity
rp
3ω'
dk'
Xit>
(1.14)
/
tends to zero. The equation ω = au corresponds to the resonance increase of the amplitude of the plasma eigen modes. For ω = ΐύρ the displacement current equals the drift current of free charges. Since the direction of these currents is opposite, the total current in the plasma
52
Plasma and Current Instabilities in Semiconductors jf = qriv - ίωελΪ
= i ^-
(ω2 - ω2)ί
(1.15)
proves to be zero when ω = ω . P In order to compute the attenuation depth a = 1/k" of the penetrating wave, it is necessary to take into account the collisions of electrons with the scattering centres which represent the physical factor responsible for the energy losses in the wave and for the attenuation of its amplitude. In solids τ-»· ~ 10" 13 s, and the inequality (1.2) corresponding to a collisional plasma is sStisfied over the whole microwave range. In this case, according to (3.8) of Chapter I, σ is a real quantity. On the basis of (1.10) we obtain for the imaginary and the real components of the wave vector:
W
^f */*/l + S +
cW
2 W
1.
(1-16)
(1.17) The quantity σ / ^ ω ) Ξ1/(ωτ+) equals the ratio of conduction currents oE to displacement currents ε\ωΕ. For a medium with high conductivity °
£lU)
» 1
(1.18)
and k"
= t/SUS, ■V
k' = k".
(1.19)
Since in this case there are no physical phenomena that can supply the energy to the electromagnetic wave, we choose the negative root [k" < 0) among those of (1.19), which corresponds to attenuation. The depth inside which the wave amplitude decreases by a factor of e is equal to α0 Ξ Jfr
= 1.26·10'*(σω)~'ί cm,
(1.20)
where σ is expressed in ohm"1•cm"1. The power flux S ~ (E x H) attenuates by a factor of e inside a depth δ = 2αο. This depth is called the skin-layer, and the limitation of the electromagnetic wave penetrating into a conducting medium - the skin-effect. When the displacement currents are greater than the conduction currents, 7~-«
1,
(1.21)
and from (1.16) and (1.17) we obtain
" '- i a Vl"' k'= * ^'
k
(1 22)
·
53
Waves in Plasma If σ is expressed in ohm"1·cm"1; 1 = 1 .88σχϊ^ cm" .
k"
(1.23)
For a medium with anisotropic conductivity ( σ ^ ί σ ) the dispersion of waves will also depend on the orientation of the wave polarisation plane with respect In analogy with birefringence in optics, which is to the reference frame x and y. caused by the anisotropy of χχ, in this case we observe birefringence on free carriers (3). 2.
WAVES IN A SINGLE-COMPONENT PLASMA PLACED IN MAGNETIC FIELD
The Dispersion Equation. When a magnetic field is applied, the medium becomes anisotropic and, in the general case, all components of the matrix of equations (2.20) of Chapter I are not equal to zero. In a plasma placed in a magnetic field the directions of vectors k and Bo are singled out. Let us choose the reference frame in such a way that the vectors k and B 0 are in the plane xz, with B0||z and B 0 > 0 (Fig. 2.3.a). Then the wave equation will have the form α xx
α xy
a yx
a
a
'xz xz
X
0
yy 0
1E1
α
a
= 0.
E y
(2.1)
E
zz '
1
z ■ In this expression, for a single-component cold plasma with the isotropic electric conductivity (in the absence of magnetic field) σ = qn0\i the components α.λ IJ' according to (2.21) and (3.28) of Chapter I, are 12
= - K
+
ω
k2 - k2
yy
Z
X
- K + '?T Xi α
= α
(uSo):
'^2 Xl " ^ U C -
X1 -
ϊων
"
1 + (μΒ 0 ) 2 '
^ωμ0σ,
(2.2)
= k k , \iB0
yx
xy
1 + (y#o) 2
Note that k2 = k2 cos29, z
k2 = k2 sin26, x
k k ' = k2 cos Θ sin9, x z
where Θ is the angle between the vectors k and Bo (Fig. 2.3.a).
(2.3)
54
Plasma and Current Instabilities in Semiconductors
y'fu
Fig. 2.3. Reference frames for the magnetized plasma. Sometimes it is convenient to consider waves in the plasma with k||z (Fig. 2.3.b), Then in equation (2.20) of Chapter I components a. ., according to (3.26) of Chapter I, are given by 1 + (yBotf)2 Α ω α = - k22 + 7 Xi - ™μ„σ χ + (μΒο)2
xx
o.„. = - k2 ♦ ^ 2/î/
Xl
- «ομ„σ
1
, ^
a
,
USo„ a#
ιωμοσ VBo„
V
= tü)p ö
°
i ♦ (yBo) 2 '
(2.4)
Waves in Plasma 1
. «zz
α
a
χζ
21/
=
S*
= α
Xl
" ^
°
σ
* (l*o a ) 1 + (μ^ο) 2
tü)p0ö
ζχ
= - α
μ
yz
=
- ^
55 2
i + (ys0)2
0 0
1 + (ρΒο) 2
where Β0
Β 0 sin θ,
= ß 0 cos θ.
(2.5)
The dispersion equation formed by the components otVj that are determined by (2.2) or (2.4) is a quadratic equation with respect to k = (ω2/ο2)Ν2 (Ν is the refractive index): Akk + Bk2 + C = 0,
(2.6)
which means the existence of two types of waves with opposite polarisation. The latter can be found from (2.1). From the equation of the middle row a E + a E = 0 we obtain yx x yy y a JÜL (2.7a) a * yx E
and from the third row equation a XX
E
+a ZZ
X
= 0 it follows that Z
(2.7b) We are going to give now general expressions for cases that occur most frequently in the experiment, namely, when the direction of the wave propagation is parallel to the magnetic field ïc||S0 (Θ = 0) and normal to it tjß0 (θ = π/2); in the first case we are dealing with configuration of Faraday, in the second case - with Voigt's configuration. Assume that B0||k||z for configuration of Faraday and B0||z and k||x for Voigt's configuration. Then for both configurations from (2.1) and (2.2) it can be obtained that a xx a yx 0
1E 1
0
a
X
xy
0
a 0yy
= 0.
y
a
zz '
i
z
(2.8)
it
When the determinant of the system of equations (2.8) is set equal to zero, two dispersion equations result:
o,
Vyy
vw = °-
(2.9) (2.10)
56
Plasma and Current Instabilities in Semiconductors
which correspond to different types of waves, depending on configuration. Equation (2.9) describes the waves the field vector E of which is polarised along the magnetic field B 0 . For Faraday's configuration this is a longitudinal wave, and for the configuration of Voigt it is transverse. Equation (2.10) corresponds to waves that are polarised in the plane xy; in the case of Faraday's configuration it is transverse waves, and for Voigt's configuration - the hybrid ones. Consider the propagation of waves in Faraday's and Voigt's configurations separately. Faraday's Configuration. a
xx
= a
xy
yy
= - a
7,2
=-
ω2 a
For S0||k||z equation (2.2) yields
Helicon Waves.
K
+
ω2 β2
.
Χι
ιωμ„σ
1
1 +
(yflo)2.
- ·£ωμ0σ.
(2.11)
μδο σ yx - " ^ » i V ( p B 0 ) 2 ·
The substitution of (2.11) into equation (2.9) for the longitudinal component of the field E results for ωτ+ » 1 in the solution which is known to us already from Section 1: ω = ω ρ . In other words, for Faraday's geometry no longitudinal waves propagate in the magnetised cold plasma. By substituting (2.11) into (2.10) we obtain for the transverse waves k
l
2
- "TXi
+
^ y 0 a e f f ± = 0,
(2.12)
where °eff±
=
1 + ivB0)2
C1 ± ^ 0 ) ·
<2·13)
When (2.11) is substituted into (2.7a) and (2.12) is taken into account, an equation for the polarisation of the transverse waves results: E χ E y
(2.14)
The factor i means the phase shift by π/2. Therefore, formula (2.14) describes transverse waves with circular polarisation. Thus, in the case under consideration the wave in the plasma can be decomposed into two waves with right and left circular polarisation, which are characterised by different velocities and attenuation. For these waves we have E + = E — (x ± iy)
exp(io)t - k + l ) .
(2.15)
v2
The law of the wave dispersion (2.12) depends both on the direction of the wave polarisation (the sign before the imaginary unit) and on the sign of the product u£ 0 = ω 0 τ , where ooc is the cyclotron frequency. For an observer whose direction of observation is parallel to that of the vector B 0 the cyclotron rotation of electrons occurs in the clockwise direction, and that of holes - in the counterclockwise direction (Fig. 1.2). Accordingly, when So > 0, in the electron plasma we have UBQ < 0, and in the hole plasma - μ5 0 > 0. Finally, the term ±i\iB0 in
57
Waves in Plasma
(2.13) will have a positive sign if the direction of rotation of the wave field vector coincides with the direction of the cyclotron rotation of the plasma charge mobile carrier. Such wave is called extraordinary. The second wave the direction of polarisation of which does not coincide with that of the cyclotron rotation of charges is called ordinary. For an electron plasma the wave with the right polarisation for an observer who looks along the vector B 0 is extraordinary. For a hole plasma, on the contrary, the wave with the left polarisation is extraordinary. This is illustrated by Fig. 2.4.
/?=♦/
7=-/
a Fig. 2.4.
a) The direction of rotation of the field vector in a wave with circular polarisation; b) The directions of the cyclotron rotation of electrons and holes; c) The directions of circular polarisation of the extraordinary wave for the electron (1) and hole (2) plasma.
In order to avoid confusion with the signs, we shall write equation (2.13) as
Vfß
=
i + ok)2
i_ i + β | μ β ο | )
(2.16)
where 3 = η
qBp
(2.17)
η = -1 for the right polarisation of the field vector in the case of observation in the direction of the wave propagation and η = +1 for the left polarisation. The quantity 3 = + 1 for the extraordinary wave and 3 = -1 for the ordinary wave. If one takes into account the fact that |μ£ 0 | = |ω0|ΐ and σ ~ μ ~ τ, where τ = τ+/(1 + ίωτ·+), equation (2.12) can be transformed to
?2
ω
(2.18) ω(3ω
- ω) + ίω
In order to simplify the notations, it is assumed here and in what follows that
58
Plasma and Current Instabilities in Semiconductors
ω 0 Ξ |u)c| and that its sign is determined by the coefficient ß. If (3ω v
c
(2.19)
- ω)τ->- » 1 'p
the collisional term in (2.18) can be neglected and we obtain that k2
- ÜL. v
1 +
ω(Βω
Ά
ο
(2.20)
- ω)
h\\8n
^^^..•^Τ^Κ I
■rf—τ j
{
1
ω
l/l
- ^ /?=*/
Βη*0
w
E
ω F i g . 2 . 5 . 1 F, I I .
59
Waves in Plasma
ω
Fig. 2 . 5 . I l l ç IV
60
Plasma and Current Instabilities in Semiconductors
Fig. 2.5.
Schematic drawings of dependences of N2 on ω (a); the displacement of the regions of non-transmission of waves (shaded areas) by magnetic field (b). I - the single-component plasma in Faraday's configuration (2.20): 1 - β = +1; 2 - 3 = -1; 3 = î0 = 0. II - the single-component plasma in Voigt's configuration: 1 - the extraordinary wave; 2 - B0 = 0.
Ill - the singly charged two-component plasma in Faraday's configuration: 1 - 3 = +l; 2 - 3 = -1; 3 - 5 0 = 0. IV - compensated plasma in Faraday's configuration: 1 - 3 = +1; 2 - 3 = -1 (3.46); 3 - B0 = 0. 2
2
ω ω Pi _ _£2_ ω7 ω_ '1 Pi P2 ω ω c\ V - the plasma in a semiconductor with anisotropic valleys in Faraday's configuration: 1 - 3 = +1; 2 - 3 = -1; 3 - B0 = 0. Here ω,
P
2
In t h i s case ω? = -=■ν (2ω* + ω 2 7 ) , ω =K ίω , ωΓ 7 j κ 3 pt pV o\ et cV o2 The cyclotron resonance occurs also for the ordinary wave.
ot
Figure 2.5.1 presents schematically the dependence of the refractive indices squared N
A c,2 on the wave frequency for a collisionless case (2.20).
As the
magnetic field increases, for the extraordinary wave the transmission band becomes more broad from the side of the lower frequencies up to ω , and the non-transmission band becomes more narrow near wc. For the ordinary wave the transmission band increases because of the decrease of the cut-off frequency ω ς 0 from the value equal to ω ρ down to lower frequencies and to values determined from the condition
61
Waves in Plasma 2
of k
turning into zero: ω
No propagation of
ίω + ω ) 1 ω 2 . co^ c co ; p
(2.21)
the ordinary wave (B = -1) is possible in the plasma if ω < ω
,
(2.22) J
co since in this case k2 nary wave (3 = +1) k2
< 0 and kr = 0, R = 1 (0.6) and vr~ > 0 for all ω < ω
= 0.
For the extraordi-
(2.23)
and the reflection coefficient R < 1. Inequality (2.23) is certainly valid for the whole microwave range since in the fields of the order of magnitude 101* G the cyclotron frequency ω 0 « 1 0 1 2 Hz. Thus, when inequalities (2.22) and (2.23) hold, out of two waves with the left and right circular polarisation the wave with only one polarisation can propagate in the plasma, namely, the extraordinary wave. As the wave with the plane polarisation penetrates into a magnetised plasma, it twists into a helix. This wave is called the helicon; it has been already mentioned in the introduction to this Chapter. When ω = ω 0 , the so-called cyclotron resonance occurs. In the vicinity of this point the dependence of the refractive index on ω is very pronounced. Waves in the plasma whose frequencies are close to o)c are called cyclotron waves. In the cyclotron resonance region the drift current produced by the crossed fields E and B 0 turns out to be co-phasal with the particle cyclotron rotation currents. Thus, when the magnetic field is switched on, the response of the plasma to the electromagnetic wave changes substantially. First, the condition of collision neglect is changed. In the absence of magnetic field the role of such condition was played by the inequality ωτ-> » 1 (1.3) which is practically not valid for the whole microwave range. When a plasma is subject to the action of the magnetic field, inequality (2.19) becomes the condition of collision neglect; this inequality holds for all frequencies ω « ω if
V p Ξ V° *
lf
(2 24)
*
which is quite possible in materials characterised by a high mobility. The plasma for which condition (2.24) is valid is called magnetised. This term, though, is applied also in general to describe a plasma placed in a magnetic field. Second, there is no total reflection of the extraordinary wave which penetrates into the plasma at frequencies below ω , which cannot happen in the absence of the magnetic P field È 0 . As has been already noted, the main physical cause responsible for such striking changes in the plasma response are the drift currents of particles that are normal to the electric and magnetic fields. When there are no collisions, the electric and magnetic fields produced by these currents are normal to the corresponding fields of the incident signal and can change only polarisation but not the amplitude of the latter. Although because of collisions the drift currents are deflected from the normal direction by a certain angle, the value of this angle is smaller for stronger magnetic fields, tg Θ = (yBo)" 1 ; therefore, the energy losses in the wave that are associated with the current component parallel to the electric field decrease as the magnetic field increases. To put it differently, the increase of the magnetic field is accompanied by the increase of magnetic resistance and the decrease of losses.
62
Plasma and Current Instabilities in Semiconductors
Let us estimate the depth of penetration of waves from the microwave range. Assume that ωτ* « 1,
ω « ω ,
ω 2 » ωω .
(2.25)
In t h i s c a s e , from ( 2 . 1 8 ) , we o b t a i n (ω2τ+)Η \ [l + (ω τ-*) 2 ]^ - βωc τ+Ι I JLZ p p| J c P _ PI 1 + (ω ο τ-)
(2.26)
and the penetration depth for the extraordinary helicon wave is
a
H
=
W
a(0)
1 + (ωc τ+) 2 P /l + (ω "τ->)ζ - 'ω~τ^ c p c p
(2.27)
where a(0) is the depth of the skin-layer in the unmagnetised plasma (1.20). The penetration depth increases rapidly with the increase of the magnetic field (Fig. 2.6). For the ordinary wave ($ = -1) the initial decrease of the penetration depth is followed by its slow increase accompanying the growth of the magnetic field.
äffi
μΒ0 Fig. 2.6. The penetration depth for the helicon wave a H and the ordinary wave a 0 as a function of the quantity uB0 = ω τ+ in skin-layer depth units for B 0 = 0. c p Because of the large depth of penetration of the helicon wave into the conducting media it is possible to use it for studying the properties of solid state plasma.
63
Waves in Plasma We shall discuss this in Sections 6 and 8. For very high magnetic fields and sufficiently high frequencies when ($ω -ω)τ-* » 1 and ω(ω + 3ω ) » ω* , c ρ c ρ
(2.28)
instead of (2.18) we have 2 k*3 = Hl v C 2 Χι>
(2.29)
in other words, there is no interaction between the wave and the free carriers, and the wave propagates in the conducting medium as in a dielectric with a dielectric constant χ 1β Since in this case the dispersion law is the same for both waves, no twisting of the wave with plane polarisation into a helix occurs. There is also an essential difference between the phase velocity of the helicon wave and the phase velocity of waves in the plasma for B 0 = 0. For the collisionless plasma _ ω'
i
βΧι
1 +
ω(3ω
(2.30) - ω)
For the ordinary wave when 3 = -1 tl^e phase velocity is always greater than the velocity of light in the medium ol\{2 . In the case of the extraordinary or helicon wave 3 = +1 and if ω < ω 0 the phase velocity is always less than the velocity of light in the medium; it approaches zero when ω = ω (Fig. 2.7). 7f/2
c
Fig. 2.7.
The 1 -
dependence corresponding to (2.30): ω /ω = 1, 3 = +1; 2 - ω /ω = 2 , ω_/ω„ c' ρ c' ρ
= +1;
3 - ω ο /ω ρ = 2, 3
64
Plasma and Current Instabilities in Semiconductors
The physical explanation of the small value of the phase velocity for the helicon wave is that the contribution of free carriers in the dielectric constant in the magnetised plasma turns out to be positive (Fig. 1.3). In the low frequency range and in the vicinity of the cyclotron resonance the small phase velocities of the extraordinary waves correspond to short wave lengths. For instance, in vacuum the frequency ω « 10 7 Hz corresponds to the wave length À « 3 0 m, while in the semiconductor for ω ρ « 1 0 1 2 Hz, ω 0 « 1 0 1 2 Hz and χχ « 10 the wave length λ « 3 cm, i.e., is less by a factor of 1000. One can see from Fig. 2.5.I.b that there are frequency ranges for which both the ordinary and the extraordinary waves can propagate in the plasma at the same time. The difference between the waves' phase velocities will lead to rotation of the long axis of the resultant wave polarisation ellipse as the wave propagates in the medium. This phenomenon is called the Faraday effect or the magnetic Kerr effect. Vo ig t ' s Configurât ion. For this geometry tjßQ, only k φ 0. The coefficients a., are given by X
t\\x,
?ο||Ϊ*
θ = π/2 and in (2.2)
tj
«xx = ^ Xl " ^ y ° G 1 + (U) 2 ' k2,
a = a XX yy a
a
zz
xy
= -
(2.31)
2
k
■+
ω2 ^2 Xi * *-ωρ0σ,
- -v
tU)Po°"
y£o i +"(y£0)2·
Again, as in the case of Faraday geometry, there are two types described by dispersion equations (2.9) and (2.10).
of waves which are
Equation (2.9) in Voigt's configuration describes a transverse wave with plane polarisation E||z and dispersion which does not depend on magnetic field: k2
2
= -J2- Xi - ^ων0σ.
(2.32)
Equation (2.32) coincides with the dispersion equation (1.10) that corresponds to transverse waves in the plasma in the absence of magnetic field. From equation (2.10) and by taking into account that α ^ = αχχ - k2 (see (2.31)) we can obtain for the second type of waves the electric field vectors of which are in the plane xy a dispersion equation of the form -2 _
a2
xx
xy
a
(2.33)
XX
According to (2.31) and (2.7), the field components Εχ and Ey form a wave with elliptical polarisation in the plane xy. The component Εχ is longitudinal; it results from the drift of the plasma charges produced by the Lorentz force which is normal to y and B 0 . If the wave is observed along the vector k, it appears to have a linear polarisation along the y axis. The field of the wave with elliptical polarisation can be written as
65
Waves in Plasma
xy
î-ΐο
x + y
exp[t(u)t - kx)] .
(2.34)
xx Such wave is called the extraordinary Voigt wave, and the linearly polarised wave with the field Ê||î described by equation (2.32) - the ordinary Voigt wave (4). When (2.31) is substituted in (2.33), we obtain for the extraordinary wave
to) c2 =
^
(2.35)
1 ^ ^ — - ω ω + — Ρ; ^ Ρ'
For the collisionless case (1.3)
k
ω 2 (ω 2
P\,P.
ω
72
- p- *>
;
(,
ω2) (2.36)
ω2)
There are two typical non-transmission bands (k2 ζ 0 ) , one in the low frequency region, and the other near the resonance. This is shown on Fig. 2.5.II. As follows from (2.36), the resonance occurs at a frequency ω
Τ/
= ίω
D2r
(2.37)
P'
It is called shifted by the plasma frequency or the hybrid cyclotron resonance, and also Voigt cyclotron resonance. There is no resonance for ω = ω . According to (2.36), the cut-off frequency for non-transmission can be determined from a biquadratic equation
co
ω 2 v(2ω co i
ω 2 ); + ω" = 0. c p
(2.38)
When ω > ω ρ both ordinary and extraordinary waves can propagate with different phase velocities. The result of this is that the projection on the plane yz of the sum wave field has elliptical polarisation, with the slant of the long axis of the sum wave ellipse varying during propagation. This phenomenon is called Voigt effect. Oblique Waves. The geometries of Faraday and Voigt correspond to the so-called direct waves. The waves whose direction of propagation forms a slanting angle with that of the magnetic field are called oblique. Without presenting simple but cumbersome calculations, let us give the general expressions for the waves in a single-component cold plasma for an arbitrary angle Θ between the wave vector k||z and the magnetic field Bo which is in the xz plane (see Fig. 2.3.b)(4, 5 ) . As follows from (2.4), the dispersion equation has" the form 72
ki,2
=
1 -
ω2
£jr Xi
·
2Ωω2 )[2Ω(ω + iv+) L p
- ωω2 s i n 2 θ]J ± ω LΓω2ω2 s i n 4 θ + 4Ω2 cos 2 θ]J " c c c
, (2.39)
66
Plasma and Current Instabilities in Semiconductors
where Ω = ωίω + iv->) - ω 2 ;
v* = — .
The resonance frequencies are given by (v* = 0 ) .
2ω 2
pe2
Fig. 2.8.
= ω 2 + ω 2 ± [(ω2 + ω 2 ) 2 - 4ω 2 ω 2 cos 2 θΐ^.
ρ
c
ικ
ρ
cJ
p c
(2.40)
J
Polarisation of transverse components of waves in a single-component plasma for different values of angles between the direction of the wave propagation (k) and the direction of the magnetising field (B 0 ): Faraday's geometry (a); the intermediate case (b); Voigt's geometry (c) (e - the extraordinary wave, o - the ordinary one). For the extraordinary Voigt wave the polarisation of the longitudinal component of the field is- shown.
67
Waves in Plasma For the polarisation of the transverse components of the wave amplitudes in the reference frame x'y'z' (Fig. 2.3.a) we have ί2ω Ω cos Θ c
E
Xf
E
f\
(2.41)
sin 2 θ ± ω l[ω 2 ω 2 sin 4 θ + 4Ω 2 cos 2 θΐ' J c c.
When Θ = 0, equations (2.39), (2.40), (2.41) correspond to Faraday's geometry, and when θ = π/2 - to Voigt 's one. We would just like to note that formulas (2.31) (2.34) that have been obtained previously for Voigt's configuration are now derived in the other reference frame (k||x) in which coordinates have no prime signs. As the angle Θ varies from 0 to π/2, the waves with circular polarisation are first transformed into the elliptically polarised waves with the normal long axes of the ellipses, and then into two transverse waves of plane polarisation (Fig. 2.8). The rotation of the polarisation plane occurs in the opposite directions for waves with a frequency above ω > ω and for waves with ω < ω (2.41). The longitudinal component of the field amplitude S ,||k , is
x'
p c
ωω 2 v(ω + iv+) c P
sinG - iE
y
,ω2ω (ω + tv+) sin6 P c P
- (ω + -£v->)2 Ω P
(2.42)
3. THE MULTI-COMPONENT PLASMA. MAGNETOHYDRODYNAMIC AND MAGNETOSONIC WAVES The presence of several kinds of mobile carriers affects substantially the dispersion of waves in a solid. Consider a multi-component plasma in the hydrodynamic approximation assuming that the effective masses and relaxation times are isotropic. In this approximation the charge carriers of each kind make their own partial contributions into the dielectric constant or, to put it more precisely, into the plasma electric conductivity. The electric conductivity tensor components represent a sum of the corresponding components of partial electric conductivities (see equation (3.26) of Chapter I) describing each kind (denoted by a) of the charge carriers.
N
^
tja
a=l
where σ.. Èo||î -%iy
B
N
6 . . + u Β0ε . . 7 Q Yl \X
a=l
1
n
B
+
* ^a
KB°î2
B
°iB°j) Bî j
(3.1)
2
is given by (3.26) of Chapter I, and in the reference frame in which (3.28) of Chapter I.
One can say a priori that there will be a qualitative difference between the compensated plasma's response to a perturbation (when the number of mobile carriers with the positive charge is equal to that of the carriers with the negative charge) and the response of the uncompensated plasma. In a compensated plasma, if the collisions are neglected, the resultant drift current produced by the Lorentz force turns out to be equal to zero since the drift velocities u ((3.16) of Chapter I) depend neither on the mass nor on the charge sign. Therefore, exactly those currents which are responsible for the main peculiar features of the singlecomponent plasma in the magnetic field are absent. Besides, in a compensated plasma the concepts of the ordinary and extraordinary waves in Faraday's configuration as they are defined in Section 2 are meaningless since a wave which is "extraordinary" with respect to the electrons is "ordinary" with respect to holes,
68 and vice
Plasma and Current Instabilities in Semiconductors versa.
Consider the peculiar features of the wave dispersion in a multicomponent plasma with the example of a two-component plasma in the following sequence: (a) both kinds of mobile charge carriers have the same sign, ql = q2 = q; (b) the mobile carriers have charges of the opposite signs, q\ - -q2 and ηλ = n2 (compensated plasma). The indices 1 and 2 will denote parameters describing the free charge carriers of the first and second kind, respectively. The Single-Charge Two-Component Plasma. For Faraday's geometry, according to (3.1) and (2.10), the dispersion equation for transverse waves has the form k2
_£JL
1 +
Xi
ω(3ω
cl
ω((3ω c2
- ω) Pi
_£2_ ω)
(3.2) P2
where 3 = +1 for the extraordinary wave and 3 = -1 for the ordinary one. Figure 2.5. Ill presents schematically the quantity N2 as a function of ω for the collisionless case. In the absence of the magnetic field the cut-off frequency ω below which k2 < 0 and no waves propagate in the plasma is
ω2 y
(r2 Pi
CO
(3.3)
P2J
In the magnetised plasma the cut-off frequency for the ordinary wave is shifted towards lower frequencies
-£*-
_£JL Cl
1 +
(3.4)
c2
According to (3.2), for the extraordinary wave the cyclotron resonances for each kind of particles and near the cyclotron frequencies (the non-transmission bands) are observed. For Voigt's configuration, by substituting into the expressions for a^j ((2.21) of Chapter I) the sum of partial electric conductivities for both kinds ot carriers, we obtain a
xx
=
Xl
%
Σ2),
' ^ ω μ °( Σ ι
a = a - k , yy xx a
a
a
=
ωμ
(3.5)
xy
=
Σ
β
" yx
zz
=
" k2 +ÏÏ2-Xl " im°^01
^ ο( ι^ι °
+
Σ
2\ΐ2Βο),
+σ 2
)·
In these formulas θχ = qiniVi,
σ2 =
qinzVi,
(3.6)
69
Waves in Plasma 1 + (uiSo) 2 For the ordinary Voigt wave azz sion equation takes the form ?2
k
=
ω —
Σ2 =
1
+
(3.7)
ÎViBo)2'
Since τ = τ+(1 + ϊωτ+)~ι, P P
= 0 (2.9).
the disper-
P? P2 ωίΐ + ϊωτ+ )
Pi Pi (1 + ϊωτ+ )
1 + i
Xi
,
(3.8)
Dispersion does not depend on magnetic field. In the collisionless case, when ωτ+ » 1, P 2 i ω2 + ω2 (3.9) The square of the cut-off frequency is equal to the sum of squares of plasma frequencies for each kind of carriers. For the extraordinary Voigt wave equations (3.5) and (2.10) yield 7 2
*
=
Bß Σ2υ2#ρ \ Σ, + Σ 2 + JLüLi o0 ^ωε ! + Σι + Σ 2 J j *
ω
— Xi
(3.10)
For the collisionless plasma (ωτ-* » 1 ) , according to (3.7) of Chapter I, u « -^ -r-. m τω
Hence ελω2
ειω
ω
(3.11)
^cT
ω 1
2
ρωο -ω
(3.12)
Substitution of (3.11) into (3.5) results in , ,2
ω
4,
Ί ω2 - ω2 ω' - ω" Ci C2 As follows from (2.33), the resonance will occur if a = 0 . a xx
Xi
1
(3.13)
XX
There will be separate resonances at cyclotron frequencies if ω 2 « ω and ω 2 _« ω, in other words, in very pure samples at large frequencies. In^this case expression (3.13) turns into zero when u)2 « ω 2 - ω 2 = ω 2 , a ω 2 - ω 2 » ω 2 , Pi ci c2 ρ2 or 2
?
ω « ω , a ω C2
9
9
9
-ω » ω . Ci Ρι
For low frequencies, when ω » ω and ω » ω, the resonance will occur if Pi P2
70
Plasma and Current Instabilities in Semiconductors
n2 ω
2x
+
ω^
ω ci c 2
n2
2
2
Pi
Pi
·*.]
(3.14)
This resonance which depends on the ratio of the carrier numbers instead of their absolute concentrations is called the hybrid resonance. For nl - n2 the hybrid resonance will take place at the frequency ω
(3.15)
ω . ci c 2
The cut-off frequency corresponding to non-transmission of waves in Voigt's configuration can be determined, according to (2.33), from the condition a
+ a
xx
= 0
xy
or
a
xx
= τα
xy .
(3.16)
In the collisionless case the substitution of (3.12) and (3.11) in (3.5) and (3.16) results in the following equation
_£l_
ω
_£
- ω
CO
CO
Ci
0.
(3.17)
c2
Compensated Plasma. For a compensated plasma n\ - n2, Qi = -Qi, and, as has been noted already, the plasma's response to a perturbation differs essentially from that of the single-component plasma. It will be assumed, as before, that B01| z and the vector k is in the plane xz (see Fig. 2.3.a). In sufficiently strong magnetic fields ω 0 τχ » 1, ω 0 2 τ 2 » 1> and because of compensation of charges the Hall current as well as αχι. given by (2.2) tend to zero:
a
xy
= a
yx
σ
« to)u0
1
+
μι#ο
Q~2
(3.18)
P2#c
The wave equation (2.1) takes on the form a
0
XX
0
a XZ
a
0
(3.19)
= 0,
yy a
0
zx
a
zz
where, as follows from (2.2) and (2.3), k2 cos 2 Θ - ίωμ (Σχ + Σ 2 ) + ^ o-yy = - k2
Χι>
- ^ωμ0(Σι + Σ 2 ) + ^ - Χι, (3.20)
= - k
α
α
ζχ
= α
χζ
2
2
sin θ - ϊω\ι0 (θ\ - k2 cos θ sin θ.
+ σ2) + —
χι,
71
Waves in Plasma In these formulas olf σ2 and Σι, Σ 2 are determined by (3.6) and (3.7). tion (3.19) two dispersion equations
From equa-
a =0, yy a
a
XX
- a
ZZ
2
(3.21) J
= 0,
(3.22)
XZ
can be obtained which correspond to two types of waves. Consider waves in the low frequency region when the displacement currents can be neglected in comparison with the conduction currents and the terms (ω2/ο2)χχ in (3.20) can be discarded. Assume also that ω « ω
,
For the first type of waves (α
ω
, ω « ω
,
ω
(3.23)
= θ) we obtain
k\ = - ΐωμ,ίΣ, ♦ Σ2) Ξ - ί ω μ . ( ^
+
^ ) ^ .
(3.24)
If collisions are neglected, taking into account (3.7) of Chapter I results in
kl
-^L
eoBÎ
J'
(3 25)
*
Note that the phase velocities of waves, described by (3.25), do not depend on frequency (linear dispersion) and are always less than the velocity of light; this can be seen from the formula y
h A
-
· /w(tf?i +
(3.26)
m2)
Waves described by (3.25) and the velocity (3.26) are usually called Alfven, by name of the scientist who predicted such waves in the gaseous plasma. From equation OLyyEy = 0 (the second row of (3.19)) we immediately obtain polarisation E||y, i.e., tne Alfven waves are linearly polarised with the field vector E being normal to the magneticefield vector B0 and the wave vector ki, regardless of the angle between ki and B0. For the second type of waves ( O L ^ O ^ - a£ 2 = 0), by substituting (3.20) into (3.22), we obtain for arbitrary angle Θ between B0 and k: -£ωμ0(οΊ + σ2) (Σι + Σ2) [(σι + σ2) cos2 θ + (Σι + Σ2) sin2 θ]
(3.27)
When θ = 0 (Faraday's geometry), equation (3.27) coincides with (3.24). If θ ^ 0 (oblique waves) in high magnetic fields (Σ « σ) it can be assumed for all angles Θ except those close to π/2 that
H « — ^ .
(3.28)
J cos2 Θ If collisions are neglected, it follows again from (3.27) and (3.28) that the second type of waves is also described by linear dispersion and the phase velocity is independent of frequency. According to (3.28), however, the phase velocity z
72
Plasma and Current Instabilities in Semiconductors
depends on the angle Θ and for Θ / 0 it is less than the phase velocity of the Alfven wave; it turns into zero when Θ « π/2 (Fig. 2.9). Therefore, the wave of the second type is called slow Alfven wave, and that of the first type - fast Alfven wave. It follows from (3.22) that the slow Alfven wave is polarised in the
plane xz,
Fig. 2.9.
The phase velocities of the fast (vA) and slow (v2) Alfven waves as functions of the direction of wave propagation.
By applying (3.24) we obtain for the imaginary part of the wave vector of the fast Alfven wave:
Ά
(3.29)
._ J mlh-i^l.
The comparison of (3.29) with (1.19) yields that the Alfven wave's absorption is described by the ordinary expression for the skin-layer where the change of the electric conductivity in the magnetic field is taken into account. For the slow Alfven wave it follows from (3.28) that k'ï = k'{
fcrj, J
for
ωϋο(σΐ2+ σ 2 )
Θ « 0, for
(3.30) θ
^
(
(3§31)
73
Waves in Plasma
Magnetohydrodynamic Waves. Although in a compensated magnetised plasma the currents that are transverse with respect to the magnetic field are equal to zero, the transverse fluxes of plasma particles are large. The plasma as a whole, as a certain liquid of mobile charges experiences displacement. This displacement is wave-like, i.e., the propagation of Alfven wave in the plasma is accompanied by a hydrodynamic wave of conducting fluid. In the presence of a magnetic field B 0 the magnetic force lines are as if frozen in the plasma and the vibrations of conducting fluid occur at the same time as the magnetic field vibrations. The waves in a conducting fluid placed in a magnetic field are called magnetohydrodynamic or magnetosonic. The fast Alfven waves are transverse from the point of view of electrodynamics (EiJ_ki). If kJlBo, the plasma subjected to the action of fields Ei and B 0 moves also in a direction which is normal to ki and oscillates together with the magnetic force lines as a string (Fig. 2.1.a). In this case Alfven wave is transverse also from the hydrodynamic point of view. And if kiJ_Bn, the Lorentz forces proportional to Ei x B 0 move the plasma along the vector ki leading to its compression or expansion. A longitudinal hydrodynamic wave appears. In other words, in this case the fast Alfven wave, while remaining transverse in the electrodynamic sense, turns out to be longitudinal in the hydrodynamic one (Fig. 2.1.b). Let us determine the phase velocities of magnetic sounds on the basis of magnetohydrodynamic approach. In the ordinary hydrodynamics the motion of masses of neutral fluid is described by the equation of motion of a "liquid particle" dv
m dt
1 P
(3.32)
In this formula vm is the mass velocity, p is the mass density and VP is the pressure gradient or the force of pressure. In a compensated cold plasma the role of force that acts on the liquid particle belongs to the ponderomotive force.
f = Î x î0.
(3.33)
The electric fi_eld E in a compensated plasma does not act on its neutral element since q\E + q2^ = 0. The hydrodynamic equation of the cold plasma assumes the form ^ -jw
dt
1 -* * = - 0 x 5o).
(3.34)
This equation can be easily obtained from the equations of motion in Drude approximation: dv\ <7i[E + (V! x t0)], (3.35) m ~dt dv2 ÏÏÎ2
"at
q2 [Ê + (v2 x ?o)],
(3.36)
where the subscripts 1 and 2 denote the component number. By multiplying (3.35) by ηλ and (3.36) by n2 and adding them up we obtain (3.34) in which the mass velocity
m
ηλπιλ + n2m2
~
P
'
(3.3/J
74
Plasma and Current Instabilities in Semiconductors
the mass density p = «imi + n2m2
(3.38)
and the current Î = <7i*iVi + q2n2v2.
(3.39)
From the solution of the hydrodynamic equation for neutral fluid (3.32) we obtain for the phase velocity of transverse acoustical wave:
V
ei - V ?
(3.40)
where J is tension along the fluid and p is the mass density. The velocity (3.40) coincides with the ordinary expression for a vibrating string. In the adiabatic approximation, i.e., when PV' = const, where P is pressure and V is the specific volume, the solution of equation (3.32) yields v
s2
V*
( 3 · 41 )
= 1(Ύ7·
The factor γ represents the ratio of the specific heat at a constant pressure to the specific heat at a constant volume. In the magnetohydrodynamic case the action of the force j reduced to isotropic pressure
x S 0 for Jjt0
can be
PH = Ho ~
(3.42)
and tension along the force field lines J H = y 0 #o.
(3.43)
Substituting (3.42) in place of P in (3.41) and taking into account the fact that for the frozen-in field γ = 2, we obtain the following expression for the magnetic sound velocity (3.44) One can see that v
coincides with the Alfven velocity v. given by (3.26).
Substitution of (3.43) in place of J in (3.40) results in (3.45) Again we have vQ - VK, Thus, the expressions for the velocities of fast Alfven waves and the longitudinal and transverse hydrodynamic waves coincide, although they have been obtained on the basis of different initial assumptions. We have considered above the low frequency region for which the inequality (3.23) is valid. When ω « ω 0 the wave dispersion turns out to be strongly dependent on frequency. As has been noted already, such waves are called cyclotron waves. In
75
Waves in Plasma the semiconductor plasma one can distinguish the hole (for ω « u)Cp) and electron (for ω « ω ο η ) cyclotron waves.
In the case of Faraday's geometry in a collisionless plasma we have, according to (3.2) k2 = — 2 *3 Ö
XYl,
1 +
ωρχ ω(3ω
ω
- ω) ci
Ρ2
ω(- 3ω - ω) J c2
J
(3.46)
If 3 = + 1, the cyclotron resonance is observed on carriers of the first kind, and if 3 = -1 - on carriers of the second kind. If it is assumed that ω^ > ω~ , the non-transmission band for the wave with 3 = +1 corresponds only to those ω for which ω > ω ς . The wave with 3 = -1> in addition to the non-transmission region ω > u)c also has, according to (3.46), the non-transmission band in the frequency region'below
rpi
coi
"P2
(3.47)
This is shown on Fig. 2.5.IV. For Voigt's configuration in a compensated plasma hybrid resonances are observed which are similar« to those considered here for a plasma where the mobile particles of two kinds have identical signs of charges (3.14). The waves in the plasma in the vicinity of hybrid resonances are called hybrid waves.
4.
THE EFFECT ON WAVES OF THERMAL MOTION OF PARTICLES. ELECTROSONIC WAVES. LANDAU ATTENUATION.
The dispersion relations given in the preceding sections are based on the hydrodynamic approximation in a cold plasma. We shall determine now the changes produced in the wave propagation in plasma by the thermal motion of particles when the pressure forces associated with this motion turn out to be comparable with electromagnetic ponderomotive forces. A number of non-local effects as well as new types of waves appear in such plasma which is usually called the hot plasma. Note once more that this term is applied only in the above sense and by no means always implies the high temperature of the plasma. Plasma (Electrosonic) Waves. If in a cold plasma in the absence of magnetic field no longitudinal waves emerge (see Section 1)\ in a hot plasma pressure produces longitudinal waves the origin of which is similar to that of ordinary acoustical waves in a gas of neutral particles. The equation of motion for a single-component hot plasma in the absence of magnetic field is =
£E _ VP
(4.1)
In this formula n and m are the concentration and the mass of particles, VP is the pressure force resulting from the thermal motion of the free particles of the plasma. In the ideal gas approximation we have
P = k+T n. B n
(4.2)
In a homogeneous plasma the pressure gradient is determined by the non-equilibrium concentration on. In the adiabatic approximation
76
Plasma and Current Instabilities in Semiconductors —
- Y — ,
(4.3)
where γ is the exponent in the adiabatic formula (Pn
= const).
One has to solve the equation of motion (4.1) for the longitudinal wave together with the Poisson equation
-M-^L
(4.4) £1
and the continuity equation (2.28) of Chapter I 6n =^-(kvV
(4.5)
For the longitudinal wave with k||v it follows from (4.5) that Von = - i ÜÖ- k2v.
(4.6)
The substitution of (4.6) into (4.3) yields VP = - ^γ
- 7
—u
.
(4.7)
The quantity t>
= V γ
= 4/ γ - ^ ^
(4.8)
can be called the velocity of electron (hole) sound. It is totally identical with the expression for the ordinary sound velocity in a neutral medium. By substituting (4.7) into (4.4) and taking into account (4.5) and (4.1) we obtain a dispersion equation for the longitudinal waves in a hot single-component plasma ω 2 - ω 2 + k2v2 = 0. (4.9) J v p sn The term "plasma" or "electrosonic" waves is used to denote these waves which are caused by the electrostatic forces (as in plasma vibrations) or by pressure forces (as in sound). In a cold plasma Tn « 0, vsn « 0, and the plasma waves become electrostatic plasma vibrations with ω = ω ρ . In a magnetised compensated plasma, in addition to the electrosonic waves, magnetohydrodynamic waves (magnetic sounds) appear. We are not going to give here cumbersome expressions for waves in a magnetised hot plasma. Just the main specific features of these waves will be mentioned. In the hot plasma the dispersion of magnetosonic waves undergoes a substantial change. For example, the limitations imposed by the hybrid frequency on the dispersion of the Alfven wave in Voigt*s geometry are removed, and the waves can propagate at the frequencies above the hybrid one. Because of the interaction of magnetic sounds with electron and hole sounds, the accelerated and decelerated magnetosonic waves appear. The velocity of the accelerated magnetosonic wave is greater than the velocity of the Alfven wave and the velocity of sounds in the plasma, whereas the velocity of the decelerated wave is less than each of these quantities.
77
Waves in Plasma
Landau Attenuation. Another very important phenomenon resulting from the thermal motion of the plasma particles, namely, the collisionless Landau attenuation, is not taken into account in the hydrodynamic approximation. The Landau attenuation is non-linear and non-local with respect to the electric field, and for this reason it was not considered when the above approximations were discussed.
Us
v
Fig. 2.10.
u
Landau attenuation
Figure 2.10 shows the physical mechanism that leads to the Landau attenuation. Imagine that a longitudinal wave with the phase velocity Vf propagates in the plasma. The electrons in the plasma the velocity component of which in the direction of the wave propagation is a little greater than the phase velocity of the wave turn out to be in the decelerating field of the wave and transmit their energy to it, while the electrons with the corresponding velocity component which is a little less than the wave phase velocity will be accelerated and consume the wave energy. Since the usual equilibrium velocity distribution of electrons is a decaying function (dn/dv < 0 ) , the number of slow electrons with a velocity less than Vf is greater than the number of fast electrons with a velocity greater than Vf. Tne total effect of the interaction between the electrons and the wave results in the attenuation of the latter. Only the electrons with the close values of the velocity projections on the direction of k and with the direction of projections coinciding with that of the wave phase velocity participate in Landau attenuation. One can say that the latter appears in the case of phase resonance between the particles and the wave. If Vf is much greater than the thermal velocities of electrons, the number of electrons which can participate in the phase resonance is small. For slow waves with the velocities comparable tô the thermal velocities of the plasma particles the Landau attenuation is manifest. The waves the phase velocities of which are close
78
Plasma and Current Instabilities in Semiconductors
to the acoustical velocities experience strong attenuation. In particular, the decelerated magnetosonic waves in the plasma decay rapidly because of the phase resonance. As the wave frequencies approach some resonance ones (for instance, when ω -S ω 0 ) , their phase velocities tend to zero; this phenomenon results in the increase of the number of particles that move in resonance with the wave. In this case taking into account the thermal motion (which can be infinitesimal) leads to collisionless energy absorption through Landau mechanism. For the non-Maxwellian particle velocity distribution, when dn/dv effect, namely, the wave amplification, is possible.
> 0, the reverse
The attenuation or the increase of the amplitude of the plasma waves during phase resonance can be regarded as a form of Cherenkov effect, i.e., as the absorption or emission of waves by particles the velocity of which is close to the wave phase velocity in the medium. The Dispersion Equation for the Unmagnetised Hot Plasma. When pressure and velocity distribution of particles are taken into account, the dispersion relations for a two-component unmagnetised hot plasma can be written as (6): 1 + {I2
k2)~l
[l - φ(3) + νκ%
{l
2
O
2 1
k )'
exp (- z2)]
[l - φ(αζ) + itfiaz
+
exp (- a2z2)}
= 0,
(4.10)
for the longitudinal wave, and k22CΛ 2
2
= 1 -
U)l(]
pn [φ(ζ) - i^z
exp (- 2 2 ) ]
—2~ [Φ(^ 2 ) - ^π2<22 exp (-
a2z2)],
(4.11)
for the transverse wave. In these expressions φ(ζ) = 2z exp (- z2)
I exp Or2)
dx,
(4.11a)
o
"n 2 T
JkJ' The asymptotic expansions of φ(ζ)
into a series are
♦<«■*■- P f l - M · Φ(*) = 1 + 12iT
(4.11b)
h.
IA
15 SzE
for z « 1, for z » 1.
Although in derivation of (4.10)'and (4.11) collisions were neglected, the presence of imaginary terms ih~themv means the existence of attenuation. This attenuation is caused by Landau mechanism. The exponential terms in (4.10) resulted from the averaging of the charge carrier velocities with a distribution function which is
79
Waves in Plasma
assumed to be Boltzmann one (see Section 4 of Chapter I). When the average thermal velocities of particles are close to the wave phase velocity, i.e., when z « 1 and az Ä 1, it follows from (4.10) that the Landau attenuation is so great that it makes no sense to speak of the propagation of waves. If the thermal velocities of particles are small in comparison to the phase velocity of waves (s » 1, \iz » 1), we obtain from (4.10) in the case of long waves (t -+ 0, klD « 1) :
"Pn 1
Γ pn
f
2
«M 1
PP
k2v2n
(4.12)
In the case of a single-component plasma 3 where v^ - S ^ r / V n (ω = 0) the equation (4.12) can be transformed to (4.9).
4 = WV
For high thermal velocities of electrons (2 « 1) and their mass being much less than that of holes (which is typical of many semiconductors) it can be assumed for long waves that « 1 « az
and
kl
« 1
(4.13)
Then from (4.10) we o b t a i n *PP\ 2
7
k
1
. /7 hp 3n exp
(4.14)
Equation (4.14) describes a hole sonic wave, similar to the ion-sonic wave in a gas plasma in which the field of heavy particles (holes) is screened by that of light particles (electrons) and the Landau attenuation is thereby "screened" too. In this case the imaginary term in (4.14) is small owing to the quantities lp and P ω ρ ρ being much less than lD and ω , respectively. The collisionless Landau attenuation is associated with the longitudinal component of the wave electric field. With the helicon and Alfven waves it occurs when θ φ 0, i.e., for oblique waves. The theory of the Landau attenuation for the helicon waves is discussed in (7). The attenuation that appears during interaction of the magnetic moment of a charged particle M = mü2j2Bo with the parallel gradient of the variable magnetic field Bz is in a certain sense similar to the Landau attenuation. In this case the equation of motion of a particle is do dt
z = - M
dB z_ dz
(4.15)
This formula is identical with the equation of motion in an electric field (3.1) of Chapter I if M is replaced by the charge and dBz/dz - by the electric field. This phenomenon is called the magnetic Landau attenuation or the attenuation on transit time (9). The magnetic Landau attenuation increases with the increase of Θ, the angle between k and B0.
80
Plasma and Current Instabilities in Semiconductors 5.
HELICON WAVES IN METALS
Consider another kind of collisionless attenuation of waves in the plasma. From the point of view of an electron that moves counter to the wave with a velocity v the frequency of the field is increased by a quantity kv due to Doppler effect. In metals the Doppler frequency shift is determined by the velocity of electrons on the Fermi surface and can be quite large. It is responsible for the cyclotron resonance in a metal occurring at much smaller frequencies ω = ω
c
- kv„. F
(5.1)
This phenomenon is called Doppler-shifted cyclotron resonance. The condition of the existence of helicon waves in a cold plasma in semiconductors is the inequaliWhen ω = ω , strong absorption of waves is observed. ty ω In connection with the Doppler shift of the cyclotron resonance, in metals the region of existence of helicon waves is determined by inequality (11)
(5.2) ω
ρ^
ω
No helicon wave appears in a metal until inequality (5.2) is satisfied. To put it differently, there is a minimal value of the magnetic field B 0 for a fixed ω below which the wave decays and above which it can propagate. This limiting value of the magnetic field is called the Kjeldaas edge. If inequality (5.2) is satisfied with a large margin, i.e., if ω*α2 c (5.3) ω « 2
ω νΙ'
there is no spatial dispersion of the helicon wave, and the dispersion equation can be written in its ordinary form (2.18). When the margin in inequality (5.2) is small, the helicon wave exhibits spatial dispersion. Indeed, the electron mean free path I «. vFT:n anc* s i - n c e t n e condition of the helicon propagation is ω € τ+ » 1 and ω « ω , inequality (5.2) corresponds to the inequality kZ
» 1,
(5.4)
i.e., to the non-local regime when the wave length is much less than the electron mean free path. For the non-local regime of the helicon wave the dispersion equation assumes the form (10)
(5.5)
Xi where Ω = ω
*ΌΊ)
= 2n2
l +
ω - tv+ P
i - n z In 2n
(i + n) (i - n)
kvr
n=
(5.6)
Formula (5.5) differs from the usual expression for the local case (2.18) by the
Waves in Plasma
81
presence of the factor F(n). If the collisional term is discarded (v+ = 0 ) , the frequency of the Doppler-shifted cyclotron resonance (5.1) corresponds to η = 1. When η > 1, the logarithmic term in the factor F(n) makes it a complex quantity which leads to the wave number k being complex in the frequency range near and above the cyclotron resonance in metals. Therefore, in metals, in the Dopplershifted cyclotron resonance region there is absorption of waves in the absence of collisions. From the physics point of view such absorption of waves in a collisionless non-local case corresponds to transition of electrons from one Landau level to the next. Thus, for helicons in metals one observes, in addition to the collisionless Landau attenuation, the collisionless attenuation due to the Doppler-shifted cyclotron resonance. The Landau attenuation for helicons in metals will manifest itself when ω/k « v$ cos Θ and ω « ω and because of this, as the magnetic field B 0 decreases, the Landau attenuation may occur before cyclotron absorption. On the basis of measurement of Kjeldaas edge it is possible to determine Vp and to use helicons for the study of the electron properties of metals (11). Figure 2.13 depicts an example of dependence of the helicon passage through a metallic sodium plate on the magnitude of magnetic field. The Kjeldaas edge can be clearly seen.
6.
WAVES IN A SEMICONDUCTOR WITH ANISOTROPIC VALLEYS
The complex form of electron dispersion in semiconductors may reveal itself in the dispersion of waves. Thus, in semiconductors characterised by zinc blende structure there are three minima (valleys) in the function SW for the electrons of the conduction band: Γ, X and L (see Section 5 of Chapter I ) . Only in Γ-valley the effective mass is isotropic and only for the electrons of this valley the dispersion equations for waves in the plasma discussed in the preceding sections can be used. These dispersion relations can be also applied in the case of many valleys if the effective mass of the electrons and holes in these valleys is isotropic. Although the wave equation and the dispersion relations written in terms of G£j remain valid also for anisotropic valleys, the expressions for O^J themselves undergo an essential change. It turns out that the components or the electric conductivity tensor depend not only on the mutual orientation of the wave vector and the magnetic field Bo as was the case with the isotropic valley, but on the orientation of these vectors with respect to the anisotropy axes of valleys as well. Let us determine the components of the electric conductivity tensor for anisotropic valleys of the type X and L the minima of which are the absolute extrema in the conduction band of Si and Ge, respectively. In these valleys the surfaces of constant energy are ellipsoids (Fig. 1.11). It will be assumed that the mobility in these valleys is anisotropic the nature of this anisotropy being described also by ellipsoids. The mobility in the direction of the ellipsoid's axis of symmetry will be denoted by u^ and called longitudinal, and the mobility in the direction normal to this axis u^ will be called transverse. Appendix IV contains a calculation of the components of electric conductivity tensor fpr an individual ellipsoidal valley in the Cartesian reference frame. For Bo11 z these components for the valley S are given by (see Appendix IV, (10))
PCI - G
82
Plasma and Current Instabilities in Semiconductors
°xx{S) o
«nS 1 + v\ßV
=
(S) = qnn
, **
?n9,
(6.1)
ij2n
where
v
yy{S)
^
*+ h
- ^ ) cos2 0^(5),
=μ
=
(6.2)
^ ^ z + K " ^)cos 2 e s (S)].
In the above expressions Θ Α (5), θ^(5) and θ 2 (5) are the angles formed by the axis of symmetry of the ellipsoid corresponding to the valley S with the coordinate axes. For transverse waves and Faraday's configuration, when B0||z and Bo is directed along one of the main crystallographic axes so that σ ^ = σ ^ = σ ^ = σ ^ = 0, the yz zy zx dispersion relation coincides with equation (2.10) . xz
α
α
- a a
xx yy Solving (6.3) with respect to k2 obtain
= 0.
(6.3)
xy yx
and taking into account (2.21) of Chapter I, we
k2 = fy Xi - ^ωμ 0 σ β££ ,
(6.4)
where for Faraday's configuration
0" r.c = ^Γ
eff If σ
xx
= σ
yy
.σ
xy
2
= -σ
v
fa
) ±
+0"
yy}
xx
0
0
xy yx
+
Ό - σ xx yy
2
2
(6.5)
J
we have
yx σ ~~ = o ± %o . eff xx xy
(6.6)
According to (2.7), polarisation of transverse waves in the anisotropic medium is determined by E
E y
O rr
a
_x = ^£
a xy
=
eff
- O
yy
o
yx
(6.7)
The components of the tensor of the total electric conductivity are equal to the sums of the corresponding components of the individual valleys' electric
Waves in Plasma
83
conductivity tensor. In the general case the expressions for components 0^η· are quite cumbersome; they can be simplified, however, for certain directions or the magnetic field B 0 . An example of such direction is the symmetry axis of one of the ellipsoids. We shall give the formulas for the components of electric conductivity tensor for Si in which there are six J-valleys (Fig. 2.11), and the magnetic field is parallel to the symmetry axis of one pair of valleys. This example will be used to trace the peculiar features of the wave dispersion that appear in connection with the anisotropy and the presence of many valleys in the electron spectrum.
Fig. 2.11.
The distribution of constant energy ellipsoids in Si. The mobility along the symmetry axis of the ellipsoid is y-, in the transverse direction - y .
The reference frame will be chosen in such a way that the directions of the symmetry axes of the six ellipsoids coincide with the coordinate axes x, y, z (Fig. 2.11). The magnetic field will be assumed to be parallel to the s-axis. Then for valleys on the z-axis θ 2 = 0, θ χ = Qy = 90°; for valleys on the x-axis: ez = 90°, θ^ = 0, Qy = 90° and for valleys on the #--axis 0Z = 90°, θ^ = 90°, Qy = 0. By calculating the sum of the corresponding components (6.1) of the electric conductivity tensor for each of the six valleys we obtain
84
Plasma and Current Instabilities in Semiconductors x = qn{z)
°xx
1 + y2gg
+
l
i«W
<7*Û/)
1 - VtVtB\
a M
t t" (6.8)
qn{z)
yy
Ö
μ+
2S
= q n
^
+
T^W
V
l
+
^W
μ
^ ^
£
+
!+
^ ^
^
= qn(2)
V (6.8a)
y^0 ö
», ♦ <*00 1 + μ α,ΒΙμ * Γ
μ μ
y
Ί" Λι*Β*
+
u
ß
t Z ° 1 + i. i. E2
^ ^
y
+
y
ß
t t ° ^ 0 / ) i + n n ß2«
Here n(#), n{y) and n(s) are the concentrations of particles in pairs of valleys oriented along the axes x, y , z, respectively. In the absence of the deformation of the crystal in the state of equilibrium n{z) = n{x) = n{y) = n 0 /3, where rc0 is the total concentration of electrons in the conduction band. Then we have σ = σ , and the dispersion equation for Faraday's xy geometry (6.4) assumes the form xx Ί 2
ωμο 3ϋχ
ω
Vt{&VtB0
- i)
- i(Mz
1 + viBÎ
+
Vt)
* 2βμί.μιΒ0
(6.9)
V^Bi
1 ♦
By virtue of (6.6) and (6.7) the polarisation of waves remains circular. collisionless case (ωτ-> » 1, y œq/itm) ω) ^
-2-(3ω . 72
ω
*■ - ^ 2 Xi
+ ωυ
qn0
ο ^γ2-
2
ω et, - ω
23ω . -^- + et m,
2
In the
^L + S. (6.10)
et et
or 2 v κk -- rISr Xi
■4
2 2Βωet,ωpi ,
-El.
ω(ω
ρΖ * " p p
.
(6.11)
In these expressions the following notations have been introduced:
ct
=
A. BQ>
Ct
pt
C\m '
^
pi
= 3-2LJL
£irn '
(6.12)
One can see from equation (6.11) that there are two cyclotron resonances: u)C£ = ω and ω0^ω(:^ = ω 2 . The first of them corresponds to the resonance of particles in valleys on the 2-axis the effective mass of which in the plane of the magnetic field is isotropic and equal to m^ (Fig. 2.11). The second one corresponds to the resonance of particles in ellipsoids oriented "sideways" with respect to the magnetic field (valleys on the axes x and y , Fig. 2.11) in which the electron mass is anisotropic in the plane of the magnetic field. It can be said that the particles participate in the cyclotron resonance with the so-called cyclotron mass
85
Waves in Plasma
(6.13) m = Jm m when B0||z. J v c x y " The number of different cyclotron masses for a given direction of the magnetic field determines the number of cyclotron resonances. In the case under consideraίοη with Βο||ΟθΟ> there are two such masses, mQ^ = m^ and mC2 = Jm^m^* If o||, all valleys in Si in the plane of the magnetic field are equivalent, and therefore there is only one cyclotron mass.
έ
In the case of an ellipsoidal valley, assuming that μ ~ 1/m, we obtain from (6.13) and (6.2) for the cyclotron mass for any mutual orientation of the magnetic field and the valley symmetry axis:
m
c
=
4
mm
t l
VmlCos*6
+Wt
sin26'
(6·14)
where Θ is the angle between the magnetic field vector B 0 and the symmetry axis of the valley. As has been shown in Section 2, for the isotropic mass the cyclotron resonance is observed only for the extraordinary wave (3 = +1)· It is obvious that in a valley which is isotropic with respect to the s-axis the cyclotron resonance also takes place only for the extraordinary Wc/e (the first term in the square brackets of (6.11)). And the resonance in anisotropic valleys (oriented along the axes x and y, Fig. 2.11) occurs both for the extraordinary (3 = +1) and for the ordinary (3 = -1) waves (the second term in the square brackets of (6.11)). This is understandable from the physical point of view since the trajectory of the cyclotron motion of particles in anisotropic valleys is elliptic. A circularly polarised wave can be presented as a sum of two waves with the right and left elliptic polarisation. Therefore, both for the ordinary and extraordinary waves there is a component of the drift motion of an electron that coincides with its cyclotron motion. Figure 2.5.V depicts schematically the function N2(ω) for the waves in the above case (6.11). The comparison of Figs. 2.5.V and 2.5.1 enables one to see that the dispersion of waves is changed by the anisotropy of valleys. The cut-off frequency for B 0 = 0 (which is obtained by setting the right-hand side of (6.11) equal to zero) is
w
c o = ί Ht+
w
y
Ξ
"P-
(6 ΐ5)
·
where qZnp and
(6.16)
3m m y
The quantity m^ is called the plasma effective mass or the conductivity mass. Not only u) c0 has been changed but the cyclotron resonance has appeared for the ordinary wave. In spite of the anisotropy of valleys, the polarisation of waves remains circular, which is not surprising since the total electric conductivity is still symmetric
86
Plasma and Current Instabilities in Semiconductors
with respect to the direction of the magnetic field. Naturally, in an anisotropic semiconductor the wave polarisation will be anisotropic. Let us assume that there is an ellipsoidal anisotropy of conductivity (a single valley) and that the magnetic field B0||z is parallel to the short axis of the ellipsoid. Then it can be obtained from (6.1) that
o
xx
=
L
—^r,
i + v^-jBV
σ
-
yy
t
-
l + vtvzBV
(6.18)
qnQ\it\i^B0 σ
xy ~ T + VtVzBl'
For the c o l l i s i o n l e s s case (ωτ+ » 1) P σ
335
= ^ω
Vcl
·
ω
xy where ω
ρίο = ω ρ* £ ι
and ω
«-,
pt
σ
yy
= ^ω
u
è t 'cl -
ω
^, (6.19)
ω^ω^
ρΐο = ω ρΐ ε »·
It follows from (6.18) that σ ^ Φ Oyy and, according to (6.7), instead of the circular polarisation of waves in Faraday's configuration we are dealing with the elliptic one. One can see from (6.19) that there is only one cyclotron resonance with the cyclotron mass mQ = ^πΐητη^-, but it occurs for both directions of polarisation, i.e., for the extraordinary as well as for the ordinary wave. A conclusion can be drawn from the above that the presence of the cyclotron resonance for the ordinary wave gives evidence of the existence of valleys with the anisotropic effective mass, regardless of whether the total electric conductivity of the crystal is isotropic or not. Note that in high magnetic fields (ω , « » ω) there are no more peculiar features of dispersion that are associated witn'the anisotropy of valleys in an isotropic or anisotropic crystal. Indeed, in this case equation (6.11) can be transformed to
ω2
1 +
Βωω
(6.20) c
and the dispersion law is independent of the effective mass.
7.
EXPERIMENTAL OBSERVATION OF MAGNETOPLASMA WAVES IN SEMICONDUCTORS
Consider the main experimental methods of observing magnetoplasma waves and the possibilities of using them in order to study the electron dispersion in semiconductors . Since a semiconductor sample is a bounded medium, it is obvious that the conditions at the boundaries will affect strongly the wave dispersion in it. As is clear a priori, the reflection from the boundaries will lead to the interference
87
Waves in Plasma of waves and to geometric resonances, as in ordinary cavity resonators. In what follows we shall discuss the simplest of such resonators, namely, an infinite semiconductor plate with parallel walls and a thickness d.
This case corresponds to real samples that are used in most experiments and represent plates with the thickness being much less than their other dimensions. It is possible to produce waves in such a plate if one subjects it to the action of a variable magnetic or electric field. Both the variation of the input parameters of the excitation system and the special systems for indication of the magnetic or electric field can be used for wave indication. The methods of observation of magnetoplasma waves in semiconductors discussed below are classified according to the techniques of excitation and indication of these waves.
=>A
={>«.
Fig. 2.12. a) the crossed coils method; b) the method of passing the waves through a plate.
Plasma and Current Instabilities in Semiconductors
88
The Crossed Coils Method. Figure 2.12.a depicts schematically these techniques. A semiconductor plate is placed in a variable magnetic field which is produced by a current J\ that passes through an inductance coil; the latter will be called the excitation winding. Electro-magnetic waves are generated in the plate. If the external constant magnetic field is absent, in an isotropic medium there is only 5^-component of these waves, and neither e.m.f. nor current will be produced in the second winding crossed with the first; the second winding will be called the indication one. In the presence of an external magnetic field B 0 such that BoZ φ 0 there will also be By-component of the electro-magnetic waves in an uncompensated semiconductor (see Section 2). In this case e.m.f. V2 will be induced in the indication winding. £L-component will be the greatest if the conditions of the experiment are such that a helicon wave is excited in the semiconductor plate (Section 2). For this reason the device for the observation of magnetoplasma waves depicted on Fig. 2.12.a is also called a helicon transformer. By analogy with the ordinary transformer, we have V2 = igM^x.
(7.1)
where g is a geometrical factor (the number of turns, the winding configuration, the volume, etc.), and Mj is the "magnetic permeability":
1 MTT = H #.
(7.2)
m
In this expression +d/2 +a/ z
Ti * k [ H1y y
d
)
dz
(7.3)
,
-all
and Hm is the magnetic field amplitude in the excitation winding. Let us determine how Mj is related to the parameters of the wave and the medium (11, 17). In a semiconductor resonator shaped as a plane-parallel plate charac^ terised by a high dielectric constant standing waves produced by waves with k||z appear. Therefore, no matter what the direction of the external magnetic field Bo is, only kz-components of waves form a standing wave, and it can be assumed that Bx = 0. The boundary conditions for the standing waves in a semiconductor plate are determined by the conditions of continuity of the magnetic field tangential components: H ^ e x p UL't
- k+ | ] + Ηχ_ exp
tl't
- k_ |
= Hm exp(tu)'t), (7.4)
H y*
exp Κ[ω'£ - k+ |
+ H _ exp U L ' t - k_ |
= 0.
The subscript "plus" corresponds to the extraordinary, and "minus" - to the ordinary wave. The subscript z near the wave vector is omitted for the sake of simplicity. One can easily see that, if the boundary conditions (7.4) are taken into account, the components of the magnetic field in the semiconductor are
Waves in Plasma B D
exp(iü)'t)
f+ - f,
X
B z
In these formulas /
cos
\
\d
f_
m
f+ - f.
COS
\ V
cos[k_z]
cos(k
z)
cos \— k_d
= 0,
|a|
and /
describe the wave polarisation x+
B ~ f+' y+
y-
Ί
cos f i k_d
e x p ( i a ) ' t ) Γ cos(fc s)
B B y
" / + cos(fe+g;
89
-î
J
(7.5)
f·
(7.6)
It follows from (7.5) and (7.3) that
M
'T
=
cos(kz)
]
cr+ - fjd
-d/2
cos
\k+d
cos(k_z) cos
— k_d
dz
(7.7)
L
or
Μ
τ-ΤΓ
- f )
1
jL M
tg
k+d
k d
tg
fc d
(7.8)
In the problem that we are dealing with we are interested only in the magnetic components of the wave in the semiconductor, i.e., only in transverse waves. For this reason, it is convenient to determine the dispersion and polarisation of waves on the basis of the wave equation for the magnetic field (equation (2.23) of Chapter I): k x (pk x H) = τωμ 0 Η.
(7.9)
In this expression p is the resistance tensor which appears in the equation connecting the total current and the field:
pJ = Î.
(7.10)
Assuming for the transverse wave with k||z that Hz - 0 and p(H0 + H) « p(II0), we obtain from (7.9) p k2H + p k2H = ίω\ι0Η , xx y ^xy x y
(7.11) p k2H + p k2H = - ·ζ;ωμο# . yx y yy χ χ Note that only four components of the resistance tensor determine the wave dispersion. The dispersion equation expressed through these four components of the resistance tensor results from setting the determinant of the system (7.11) equal to zero:
90
Plasma and Current Instabilities in Semiconductors
k2 =
j. i Ü2ÜL15UL ± [.
ωμ0 P
P
xx yy
- P
P
(p.XX
P
xy yx
)
2
w
P
P
f\
(7.12)
xy yx
The polarisation is given by
f
= -Jü =
2/
2P
ay
— (P. 2/2/
(7.13)
- P )2
) ± [(P
4p
Q
xy xyi
Now we can find the relation between Mj and the components of the resistance tensor and thereby to establish connection between e.m.f. produced in the indication winding and the semiconductor plasma parameters. By substituting (7.13) and (7.12) into (7.7) and making use of the series expansion for tg (i k d]
** [i k-d) - - Σ [[ikA+ ηπ - hπ] ·
(7.14)
we obtain xy
π2 p
yy
xx
Σ
(7.15)
γ=ι
l + H
Here ω = — V2 p p r \iQd ^xx^yy o =
(p
χχ
p
-
p
xx
p
xy yxJ
yy 0
- p p I , xy yx'
+
)^
(7.16) (7.17)
P
yy
and γ is an odd number. This property of γ reflects the fact that when the number of half-waves on the sample thickness d is even, the average magnetic field By = 0. The meaning of Q is the quality of the semiconductor resonator, and ω ρ is the resonance frequency which corresponds to the formation of a standing wave in the plate. Standing waves can be produced for relatively small values of frequency. This is associated with the possibility of the phase velocities of the extraordinary waves being much less than the velocity of light (Section 2) and, therefore, of their wave length being much smaller than the thickness of the semiconductor plate. The method of crossed coils is convenient for observing the low frequency helicon waves. The spatial resonances of such waves in semiconductors have been observed by many authors (11, 12, 14, 15, 16). Figure 2.18 presents the voltage on the indication coil as a function of the constant magnetic field which is normal to the plane of the semiconductor plate. Below, in Section 8, we shall discuss the information concerning the plasma properties which can be obtained from such experiments. The Method of Transmitting Waves Through a Plate. The schematic outline of the method is shown on Fig. 2.12.b. The excitation and indication coils are on the different sides of the semiconductor plate and normal to each other. It can be
91
Waves in Plasma
shown that the magnetic field H2 of the wave that has passed through a plate with a refractive index N and a thickness d is (12, 17, 27)
», = [1
t (N + /I/"1) sin
(*j
ML· o
HIX'
(7.18)
Here Hlr is the magnetic field near the plate on the side of the excitation coil. For helicon waves the refractive index can be as high as 10 2 . Therefore, in (7.18) the term N'1 can be neglected. After some transformations we obtain for the amplitude of the field that has passed through the plate (11, 17, 21):
** ■'-Ψ[ψ * [Ί M) TTd ^ To find the quantity M
[l hd
which in this case is given by
Mt
= ^
(7.20)
we again make use of the series expansion for the tangent.
4
-ίωά a
,,t T
(7.19)
Then
2P,xy
^xy , 4 V π2
2
^ μ0ω
£_j γ=1
P
C-Φ 1 +
^ + P^ ^ MS- - —
(7.21)
If the axis of the indication coil is parallel to that of the excitation one, the "magnetic permeability of the plate"
(7.22)
is equal to 2p
MTl -
-iud G
.
4^P
,/yy
d μοω
x x
oo
+
_i_
π2
Σ γ=1
(-l)Y pxx
+ p
yy
1 + ÎQ
·
i °^C
' ~"*
c
(7.23)
—
Note that in this expression γ may be even, too. The fact that in the absence of resonance in the plate M^ and M^ are not equal to zero (because of the term before the sum) corresponds to the penetration of the field through the plate due to skineffect. It is obvious that the geometrical factor g and, therefore, the signal at the output is much less in the case of the method of wave transmission through the plate than in the case of crossed coils method. The measurement of a small signal at the output is not, however, a major obstacle for the experimentalist since there is a high reference voltage on the excitation winding, which is coherent with the measured signal; this makes it possible to employ phase-sensitive rectification. In the method of wave transmission the interference between the input and the output signals is used for the indication of the wave phase. If the reference signal
92
Plasma and Current Instabilities in Semiconductors V = | 7 0 | sin ω£,
(7.23a)
which does not depend on the magnetic field is fed to the indication winding from the excitation one, it will be mixed at the output with the field-dependent signal (7.23b)
V{H) = 17^| sin [ω£ + α(#0)]>
leading to a typical oscillation pattern. Since the phase of the signal transmitted through the sample varies with the variation of the magnetic field, it at one moment is added to and at another is subtracted from, the direct transmission signal. The total voltage at the output is given by 27* = \V0\2
+ | ^ | 2 + 2\V
\VI
cos a.
(7.24)
The displacement of signals at the output is shown on Fig. 2.13 which presents the experimental results on transmission of helicons through a sodium plate (18).
0
J
Fig. 2.13.
L
20
UO
60
80
100 Bo, kG
The amplitude (in arbitrary units) of the helicon wave with a frequency 50 MHz that has passed through a sodium plate of thickness 0.48 mm. In magnetic fields below 50 kG no signal is observed because of strong absorption near the Doppler-shifted cyclotron resonance (18).
93
Waves in Plasma
Obviously, the method of interference of the direct signal and the signal transmitted through the semiconductor is a general one. It can be used for the measurement of the magnetoplasma waves' phase velocity in semiconductors as a function of the magnetic field also with other techniques of producing waves. There is an analogue of this method in optics, which is called the Rayleigh interferometry method. The phenomenon similar to the spatial resonance in optics is the Fabry-Perot resonance. Superhigh Frequency Methods. In this case the semiconductor plate is placed on the propagation line of an SHF signal and either the transmitted through or the reflected from the plate signal is observed. For wave transmission through the plate the amplitude of the transmitted signal is connected through equations (7.20) - (7.23) with the amplitude of the incident one, and geometric resonances take place under the conditions of weak attenuation. By means of these techniques the authors of (14) discovered for the first time the helicons in semiconductors. Figure 2.14 depicts the amplitude of a signal with a frequency of 10 GHz, transmitted through an InSb plate, as a function of the magnetic field strength.
7A
/.* Fig. 2.14.
B0
kG
The SHF-power P(So) that has passed through a sample rc-InSb of thickness 2 mm covering the aperture of the three centimetre range waveguide whose axis is parallel to the magnetic field, as a function of B0, in arbitrary units (14): 1 - the transmitted power P under the conditions when the direction of B 0 corresponds to the excitation of a helicon wave in the sample; 2 - the transmitted power for the reverse direction
of So; 3 - the initial section of curve 1 amplified by 12 db. When the attenuation of waves in a semiconductor is high and no geometric resonances are observed, the Rayleigh interferometry method is employed (16, 19).
94
Plasma and Current Instabilities in Semiconductors
A number of authors used the geometric SHF-resonance and interferometry for detection of Alfven waves in semiconductors (20). Such waves were observed for the first time in (21). The polarisation of Alfven waves is linear and their phase velocity does not depend on frequency (Section 3 ) . These properties are used in experiments for identification of Alfven waves. In the SHF-range the method of reflection is convenient. In the signal reflected from a thin sample the same geometric resonances are revealed as in the transmitted signal. When the method of reflection is employed, the requirements to the sample shape are simplified. As far as this method is concerned, the exact correspondence between the sample shape and the wave guide window is not necessary as it is in the study of wave transmission. It is sufficient that the sample is larger than the wave guide open end window. Besides, thick (semi-infinite)samples can be analysed by the reflection method. Figure 2.15 presents the dependence of
0 Fig. 2.15.
0.2
0A
0.6
0.8 1.0 B0J0* G
The reflection coefficient R and the amplitude U of the transmitted signal (in relative units) as functions of the magnetic field for a plate of n-InSb thickness d = 2.64 mm for / = 37.7 GHz, T = 77 K, n = 4.4*1011+cm"3, y = 4. l*105cm2/V-sec; 1 - the experimental curve i?(S0)i 2 - the experimental curve #(B 0 ) obtained when the impedances of the waveguide and the sample are matched by means of a λ/4 dielectric plate; 3 - the experimental curve i/(B0); the dashed curve is the theoretical one. The numbers 5-10 denote Fabry-Perot resonances (22).
95
Waves in Plasma
the signal reflected from an n-InSb plate on B0. Geometric resonances manifest themselves more clearly in the reflected signal than in the transmitted one. The method of wave reflection is the only available techniques for thick samples. In this case the reflection method amounts to varying the input impedance of the sample, which depends essentially on the type of waves in the plasma as well as on its parameters. Figure 2.16 depicts the reflection coefficient for a thick sample, in which a helicon wave was produced, as a function of B0.
R
t.OV
I
0 Fig. 2.16.
_
i
02
i
OU
""
i
i
0.6 0.8 B0,10* G
The power reflection coefficient iî as a function of magnetic induction for a "semi-infinite" sample with the following parameters: thickness d ~ 1.5 cm (the back wall of the sample is not parallel to the front one), n - 1.15*10 cm" 3 , p = 6.5-103cm2/V*sec, χλ = 17. Curve 1 corresponds to the extraordinary, and curve 2 - to the ordinary wave; the dashed lines are theoretical, the solid ones are experimental (23).
In addition to geometric resonances, Faraday- and Voigt-types of rotation of the polarisation plane, and also the cyclotron and other kinds of resonances considered in Sections 2 and 3 can be observed in the optical and the SHF-range. The cyclotron resonance forms the basis of the main method for measuring the effective masses of the charge carriers in semiconductors and it has been studied by many authors. Figure 2.17 shows an example of the amplitude of the signal transmitted through the semiconductor as a function of the magnetic field strength; one can see the cyclotron resonances in Faraday's and Voigt's geometry. Many authors have also observed the Faraday rotation of the polarisation plane, which has been used for obtaining information on the non-parabolicity of the effective mass and on heating of current carriers in semiconductors. The experiments on the cyclotron resonance and the Faraday rotation are mainly performed in the optical frequency
96
Plasma and Current Instabilities in Semiconductors
range, and we shall not dwell on them here.
Fig. 2.17.
The cyclotron resonance in n-InSb in Faraday's and Voigt's geometries for ω = 295 cm"1 (24). Along the ordinate axis the power P of the signal that has passed through the semiconductor is plorteçj. in normalised units: l 1 - k_|_B0_[E; min P corresponds to (ω£ + ω£)^> 2 - k||B0J_E, min P corresponds to ω = ω .
8.
HELICON SPECTROSCOPY
On the basis of studying the electromagnetic wave propagation in the plasma it is possible to obtain considerable information on parameters and the energy spectrum of free charge carriers. Various resonances in the optical range that are associated with quantisation of the motion of electrons (holes) by the magnetic field provide especially rich material. Different types of magnetic resonances enable one to establish the effective masses of carriers and the details of the band structure in semiconductors, the shape of Fermi surfaces in metals. All these studies initiated a new trend in solid state physics - the magneto-optics (25). It is difficult to observe magnetic and plasma resonances in the microwave range. For example, the necessary condition of the cyclotron resonance is ω = o)c and U) C T± » 1, and even under the conditions of very low temperatures the cyclotron frequencies turn out to be in the optical range or in the one close to it. Still, as has been demonstrated in (26), substantial information on the intraband structure of a semiconductor can be obtained from the studies of propagation of microwaves in a magnetised plasma. This can be done by means of techniques that are simple in comparison with magneto-optical ones. In this section we shall consider several examples of determining the charge carrier parameters in semiconductors by means of the helicon waves in the microwave range. Since these waves penetrate
97
Waves in Plasma deep into the conducting medium, they are most convenient for such studies.
The dispersion relations for the microwaves in the solid state plasma are specified by the electric conductivity tensor. By means of the experimental measurement of waves' absorption, reflection, phase velocities, resonances, polarisation and of their dependence on the geometry of the experiment and the external fields Eo and Bo it is possible to find the components of the electric conductivity tensor, and then the concentration distribution over valleys inside a band, the magnitude and the anisotropy of mobility inside valleys, the signs of the charge carriers. First, consider a semiconductor with one spherical valley. The main parameters that describe the properties of such a semiconductor, the mobility and the concentration of charge carriers in it are usually determined from the measurements of electric conductivity and Hall effect. We shall demonstrate how these parameters can be found from the measurements^of helicons' geometric resonances. In the case of isotropic mobility, when B0||z||k (Appendix IV, 12) we obtain p
xx
= p
yy
=
y30
1
p
qn]i'
= - p
^xy
=
^yx
.
an\x
(8.1)
K
J
The geometric resonance frequencies (7.16) in a plate made of such semiconductor are i
(1 ♦ V2Bl)h.
(8.2)
For \i2B20 »
1
(8.3)
we have
Ay2 4r>
(8.4)
where A
rad =-ll, = 4.91-1015 q\i0d2 ~ H x i U G-sec*cm3
As follows from (8.4), if one determines the resonance frequency (or the resonance field S 0 for a given frequency) one can find the current carrier concentration. In the crossed coils method the resonance values ω^ or BQr are obtained from the curve representing the dependence of the voltage V2 on the output coil (Section 7) on the signal frequency on the primary winding for a constant B 0 or on the magnetic field strength B 0 for a constant frequency. From the point of view of the experiment the second variant is simpler and more convenient since it requires no coordination of the circuits in a broad frequency range. In the general case, The maxima of Mj are determined from the maxima of V2. according to (7.15), the values of the frequency ω^ or of magnetic induction BQm corresponding to the maxima of V2 do not coincide with the resonance values of the It can be easily shown, however, by frequency u)r or of magnetic induction Bor. differentiating (7.1) with respect to S 0 or with respect to ω and by taking into account (7.15) and (8.1) that
98
Plasma and Current Instabilities in Semiconductors
MV)
dV2 dB0
dV2 du
and
V2(ur)
(8.5)
where V2 (Sop) anc^ ^2 (ωρ) are the values of voltage on the output coil that correspond to the resonance values of magnetic induction or frequency (12^. Equations (8.5) make it possible to find from the experimental dependence ^2(^0) for a fixed frequency (Fig. 2.18) or Κ 2 (ω) for fixed B 0 the quantities BQr or ω^.
B
oi
Fig. 2.18.
B
or Bom
B
0,W
B
o2
The dependence of the output signal Vz (in arbitrary units) on the helicon transformer made of alloy Bi0.8 9 S b 0 . n on the constant magnetic field S 0 for a fixed frequency f = 10 7 Hz of the current through the primary winding and T = 77 K; S o r is the magnetic field corresponding to the dimensional resonance, BQm is the magnetic field characteristic of ^ 2 max'
By transforming (7.17) we arrive at op
0)2
- U>i
(8.6)
(8.7)
Here S 0 2 and S 0 i are the values of magnetic fields corresponding to the points of intersection of a straight line that passes through the origin and through the point 0.707 V2{Bm) with the curve that depicts the function y2(#o)(Fig. 2.18). Figure 2.18 presents an experimental curve obtained for a fixed frequency. The values of ω„ and ω 2 - ωι can be found from a similar curve for a fixed magnetic field. Note that when y2So » 1
99
Waves in Plasma
and from (8.6) we obtain
(Bo
2 VBo
(8.8)
-*»w·
(8.9)
Thus, on the basis of measurement of curves representing the function ^ 2 ( B o) for U 2 £ Q >> 1 i1: i s possible to determine the concentration and the mobility of charge carriers in a semiconductor. In comparison with the method of measurement of Hall voltage and electric conductivity, the advantage of this method is that it is not necessary to prepare a sample of a special shape and, what is more important, the ohmic measurement contacts. To conduct the experiment, only a plane parallel plate is necessary, even if it is of an irregular shape. The shortcoming of the method is the limited number of materials for which condition (8.3) can be realised. It should be noted that when inequality (8.3) is valid, it can be assumed with a high degree of accuracy that ω^ « ω^ and Bor Ä 5 0 ^, where ω^ and Bm are the values of the frequency and the magnetic field which correspond to the maxima of V2 (Fig· 2.18). As the quantity \iB0 decreases, the difference between ω^ and ω^, and also between £ n y ) and Bnrn increases, and when \iB0 « 1, there is no resemblance Thus, for μ*£$ > 1 the maximum ot ΜΊ at all between the curves ω/Β0Γ and m/Bm. (and, therefore, of 7 2 ) corresponds to a condition (26)
2\s0d2 qn\x]
2
VKm- 1)
(8.10)
As the quantity \iB0 approaches unity, the curve oi/Bom experiences a decrease and for yBo = Ι,ω/B m = 0. At the same time, according to (8.2), the function u)/Bor exhibits an increase as \iB0 decreases, and for μ5 0 « 1 it is greater than its value at P 2 B Q >:> 1 by a factor of /2. Figure 2.19 presents examples of these
f/Bn r -Or
f/B om
MHZ kG
8.0 6.0\ 4.0\
J_ n
2.0
0 Fig. 2.19.
1
2
3
4
5
The quantities / / B „ r (1) and f/Bm tions of μΒ 0 for rc-InSb.
6
μΒ0
(2) as func-
100
Plasma and Current Instabilities in Semiconductors
dependences. Equation (8.10) makes it possible, after having determined the value of B m for several values of ω under the condition of relatively small \iB0 >, 1, to 2 2 5m until the curve intersects with the axis extrapolate the dependence of ω 2 on μ K ^ \iBQ and to find the mobility (Fig 2.20). On the basis of the slope of the straight line in the plot of B2Qm as a function of ω2ζ (Fig. 2.20) one can, according to (8.10), also determine concentration. Thus, the experimental measurement of the quantity Bm as a function of ω for \iB0 ^ 1 is another method for obtaining current carrier concentration and mobility.
fztn MHz
ZOOMr
W00Y-
0
2 0.2 8L,10*G om*
at
Fig. 2.20. The dependence of B2m on fz Hgo.82 Te for T = 180 K.
for Cd0.i8
In thin or sufficiently pure samples the value of ω ρ determined from (8.2) may turn out to be quite large, such that (8.11)
ω « ω In this case (7.15) yields T
,'
8u
0^2(jd
π1*
P
O P y
xx yy
3#
(8.12)
- O p
xy yx
For a spherical valley .. Τ
. 8μ 0 α 2ω "% ΉΗ
8\ι0ά2ωςη\ι xy
The function (8.13) has a maximum ,at \iB0 = 1·
B
\ιΒ0
(8.13)
Y measuring the dependence of V2
101
Waves in Plasma on Bo under the condition (8.11) it is possible to find the mobility. Thus, the measurement of helicon concentrations and mobilities of tial factor is that although the valid, the frequencies for which
geometric resonances enables one to determine current carriers by various methods. An esseninequality \ιΒ0 ϊ 1, i.e., ω τ » 1, should be the measurements are conducted may be quite small.
Many-Valley Semiconductors. The charge carrier concentration and mobility in each valley, as well as the asymmetry of mobility in valleys and their mutual orientation determine the gaivanomagnetic properties of many-valley semiconductors. For spherical valleys, if the condition (8.11) is satisfied, equation (8.13) in the many-valley case assumes the form Μ
Ί
τ
8y0d2uy?ny f μ ^ ρ π* [l + ν\Β\
μ2£ρ 1 + ν\Β\
·'·]'
C8
·143
and if there is a noticeable difference between the values of mobility in valleys, in the experimental dependence of Vz on Bo each sort of carriers will correspond to its own maximum. Although formulas (7.15), (7.21), (7.23) and (8.12) remain valid for anisotropic valleys, the expression for the components of the resistance tensor becomes more complex; this leads not only to quantitative changes of the dependences of Mj and ω^ on B0 but to qualitative ones as well. One such qualitative change has already been discussed in Section 6; it amounts to the fact that for anisotropic valleys the cyclotron resonance occurs not just for the extraordinary wave but also for the ordinary one. Equation (8.10) can be demonstrated to remain valid for the anisotropic ellipsoidal valley as well, if μ is replaced by a certain parameter μ that can be called the cyclotron mobility y
c=
y
*iyy
for
B||Z
(8.15)
by analogy with the cyclotron mass mc = Jm^my. The cyclotron mobility depends on the orientation of the magnetic field with respect to the axes of the ellipsoid of mobility. It describes a certain mean value of mobility in the plane normal to the magnetic field B 0 . If the direction of the latter is parallel to the ellipsoid's axis of symmetry, the electron mobility in the plane normal to the magnetic field is isotropic and equal to μ^. When B 0 is normal to the symmetry axis of the ellipsoid, we have
yc = ^ψΐ-
(8.16)
The measurement of y c based on geometric resonances for \iQB0 > 1 (8.10) and various orientations of B 0 with respect to the crystallographic axes makes it possible to determine the mobility components μ^ and μ^ (Appendix IV, 3 ) . Figure 2.21 shows examples of the experimental determination of μ . One can also find the structure of mobility in a valley by measuring the dependence of ur/B0 on B0 in the transition region of magnetic fields. As an example, consider n-Ge semiconductor (26, 29, 30). In the presence of only a helicon wave the real part of the helicon frequency ω^ in n-Ge, corresponding to the wave length λ = 2d, i.e., to the geometric resonance ω ρ , is
102
Plasma and Current Instabilities in Semiconductors
BorJ0*G 2.0 2.5
r
2.0
δ,ont Fig. 2.21.
2.5
The ratio f/BQr as a function of B with only the helicon wave taken into account (dashed curves) and f/B as a function of Bnm (solid curves) for n-Ge (n = 1.49-10 15 cm" 5 , y 0 = 2.101* cm2/V sec, K = 10, T - 80 K) (28). The curves are obtained by calculation, the points - from £he experiment: k|| B01|^110>; 1 - P||<100> and |P| = 1.1-103 bar; 2 - P = 0; <1Î0> and = 2.03-103 bar.
- ?
103
Waves in Plasma
B*n.\A AS
V2.Bl + 1
.
^ 2'
B^n.vi y 2 .* 2 + 1
+
A^vffl
+
(8.17)
i
where Ύ2π2 c7Po^2 γ . = y Ku J i T 1 s i n 2 6 ^
2 t
(8.17a)
cos20. + ΐ ) ( μ . ) " \
+
(8.17b) 2
y. = μ μ
+ y (y
2
- y-) cos Θ .,
X = —,
^ and θ^ is the angle between the symmetry axis of the i-th
ellipsoid and the s-axis.
It is possible to assume, up to the accuracy of 5%, that the parameter 2
S = 1-4
σ
- σ
χχ
yy
σ
k 2
(8.18)
xy
in n-Ge for ïc||<100>, <111> and <110> in the field ran^e 5 0 «0-2.10 ,+ is equal to unity (29). The changes in the dependence of BQr on ω^ = ω^ that are associated with the valley anisotropy are determined by a parameter γ. Figure 2.22 presents the quantity BQr as a function of ω^ = ω ρ for various values £f the mobility anisotropy and for various orientations of valleys with respect to z||Bo » the frequency ω^ being calculated from (8.17). Curve 3 corresponds to the equivalence of all valleys in n-Ge (i.e., they all have the same cyclotron mass) and to Bo||. One can see that the quantity ω/Β Ο Ρ does not remain constant as was the case with the isotropic mass (curve 1 ) . In n-Ge there are two groups of valleys in the directions B0||<110> and Β0||<111> with different cyclotron mass. This corresponds to a minimum (curve 4) in the dependence of BQr on ω. The position of the minimum is specified by parameter γ. One can determine K = U-^/Ui or y* = (2y^ + y^)/3 from the position of the minimum (26). The fact that the quantity BQr as a function of ω is very sensitive to the shape and the orientation of valleys makes it possible to find the main features of a semiconductor's intraband structure even from the qualitative form of this function. The intraband structure of a semiconductor changes during the deformation of the crystal (Section 5 of Chapter I ) . By combining the deformation of the crystal with the simultaneous measurements of geometric and other resonances in the semiconductor plasma one can obtain all the basic parameters of the intraband structure and its pressure-produced change. Figure 2.21 presents examples of the dependence of u/BQr on BQ for deformed n-Ge (28). The change of dispersion is associated with the redistribution of electrons between valleys. In a many-valley structure the quantity Mj may also depend on the orientation of the excitation field with respect to crystallographic directions (26).
104
Plasma and Current Instabilities in Semiconductors
1.5 2.0 B0 JO* G Fig. 2.22.
The dependence of ω on B$v for various struetures of valleys in the semiconductor (26): -3. 1 - a single spherical group, n0 - 1 0 1 5 cm" 2 - two spherical groups, μ χ = 101* cm2/V«sec, y 2 = 2.5-101* cm2/V*sec, nx = nz = 0.5; 3 - a single group consisting of four ellipsoids, y c = 2Ί0 1 * cm 2 /V # sec, n^ = 0.25 n0; 4 - two groups containing two ellipsoids each, y c = 2Ί0 1 * cm2/V*sec, η; = 0.25 n0.
For example, in n-Ge for B0||k||<110> the components of the electric conductivity tensor turn out to be dependent on rotation of the reference frame around the axis
XX
yy
xy
σ
cos φ + σ Ύ
xx σ
sin2 φ + σ Ύ
xx σ
xy
sin
yy
+ — Kίσ
2
cos2
(8.19)
yy yy
- σ
) sin2
xxJ
In these formulas φ is the angle between the a;-axis and the crystallographic direction <001>. In the direction B0||k|| or t0 ||£||<100> we have σ,ocx °yy>
105
Waves in Plasma
= σ;yx> and the angular dependence disappears. Under the condition ω « ω ' ï ~ ^xy> a n d t n e v ° l t a g e o n t n e indication winding 7 2 (see Fig. 2.12.a) varies as the semiconductor plate rotates around the direction of B 0 (Fig. 2.23)(26, 33)
J L xy r
60° \
Ό0 v V V
vx/
^
\
» ™ ^
«
/
"f^2 - k||B||<ÎÎ0>; 3 - k||B||.
£70'
It is possible to determine on the basis of the angular dependence of Vm the ratio of the difference between the diagonal components of the electric conductivity tensor to its non-diagonal component: V (φ
45°) - V (φ m V (φ (Fj m
0°) 1
σ =
yy
- σ 2σ xy
χχ
(8.20)
Equation (8.20) allows one to find the anisotropy parameter K from the known y c and vice versa, u c from the known K. The angular dependence of V2 can be quite pronounced for large K. Figure 2.24 shows the function V2(Φ) for Bi (31). We have considered several examples of applying the crossed coils method for determining the intraband structure. Obviously, other techniques, such as the method of transmission, reflection, etc., (see Section 7 ) , can be also used for these studies. For instance, Baynham (32) used helicons in p-Ge to distinguish between heavy and light holes. All this confirms the high potential of the helicon spectroscopy in the studies of the band structure in many-valley semiconductors .
106
Plasma and Current Instabilities in Semiconductors
Fig. 2.24.
The functions ν^(φ) in polar coordinates obtained in the experiment for pure Bi0.9 2 Sbo.os (dots) and the sinusoidal functions V = \V0 + A sin 2φ| (solid lines), superposed on the experimental points for / = 0.8 MHz; 1 - So = 0-3 kG, B0 > 0; 2 - Bo = 0.3 kG, Bo < 0; 3 - Bo = 0 (the scale of curve 3 is fifty times greater than that of curves 1 and 2 (31).
Waves in Plasma
In a conducting medium, such as plasma, the propagating electric field waves may produce the waves of particle density, which, in their turn, may again induce the electric field waves. By making use of the models of particle motion in electric and magnetic fields, considered in Section 3 of Chapter I, one can construct elementary models of all possible kinds of waves in the semiconductor plasma. When the magnetic field B 0 is absent, only one type of waves, the acoustical ones, can exist in the plasma. The cause of these compression-expansion waves is the thermal motion of the plasma particles; this is similar to the way of formation of sound waves in a gas consisting of neutral particles. In the plasma, however, these waves of particle density are accompanied by a variable electric field. For instance, in the single-component electron plasma the acoustical vibrations of the electron density produce waves of space charge which is negative in the region of the excessive electron concentration and positive where the uncompensated donor charge exists. In an unmagnetised plasma the sound waves are called the electron or hole sounds. In the case of a plasma placed in a magnetic field the wave spectrum is much broader. Consider the perturbation of the particle density produced in the intrinsic plasma by a plane variable transverse wave of the electric field E. Assume that the direction of the wave propagation is parallel to that of the external constant magnetic field: k||B0 (Fig. 2.1.a). According to (3.16) of Chapter I, the plasma particles drift independently of their mass and charge sign the drift direction being normal to both acting fields, the electric E and the magnetic B 0 . When the sign of the wave electric field changes, the drift direction changes its sign too. As a result of this, the propagation of the transverse electric field wave is accompanied by transverse hydrodynamic vibrations of the plasma particles on the whole (Fig. 2.1.a). In their own turn, these displacements of the plasma induce an additional electric field which is caused by the cyclotron rotation of particles with different masses (Fig. 1.3). By the same kind of reasoning we arrive at a conclusion that if kJ_B0, the transverse wave of the electric field is accompanied by the longitudinal wave of the plasma particle density compressionexpansion (see Fig. 2.1.b). In the intermediate case corresponding to an arbitrary angle between the vectors k and B 0 the hybrid waves of the plasma particle density will appear. The above hydrodynamic waves in the plasma are called magnetic sounds. The electric field waves which are accompanied by magnetic sounds, in the intrinsic plasma, are called Alfven waves.
46
Waves in Plasma
Fig. 2.1.
47
The formation of hydrodynamic waves in magnetised intrinsic plasma during propagation of the electric field wave: B 0 corresponds to the transverse hydrodynamic wave (a); o - to the longitudinal hydrodynamic wave (b).
As in the unmagnetised plasma, the longitudinal electric field wave for k || Bo produces electron sound. And if k/jfBo, the longitudinal fields E will excite also the transverse waves of the plasma density, in addition to the longitudinal oscillations. In the single-component plasma the particles with a charge of one sign are motionless, and the plasma does not drift on the whole. The electric and magnetic fields cause only the mobile particles to drift. Therefore, the flux of free charged particles (e.g., electrons) with a drift velocity u (see (3.16), Chapter I) creates an electric current j = qn0u which is uncompensated by particles with the charge of the other sign. This current whose direction is normal to that of the vector E produces, according to (1.2) of Chapter I, an electric field Which is normal to the initial one. The resultant field vector proves to be turned by a certain angle with respect to the initial direction of the vector E. Because of the fact that the direction of the electron drift u always remains normal to that of the acting fields, the rotation of the polarization of the field vector results in the rotation of the drift direction by the same angle. This leads to twisting into a kink of the transverse wave for k||B0 (Fig. 2.2). The kink direction depends on the sign of free charges, namely, it coincides, as will be shown below, with the direction of the cyclotron rotation of particles. This is natural as the direction of the drift u of particles is independent of the charge sign and the
48
Plasma and Current Instabilities in Semiconductors
currents caused by the drift have opposite directions for particles with charges of different signs. In the plasma, owing to kinking of the transverse wave, the waves with circular polarization emerge which are called helicon. The helicon waves experience weak absorption. This property of helicon waves made it possible to use them for the study of the plasma properties in semiconductors and metals, as well as for constructing a number of devices.
J=9«QU
In the present chapter we deduce the dispersion relations for plane monochromatic waves in a plasma in the absence of a Fig. 2.2. Polarisation of the transverse wave constant field E 0 and the drift in a magnetised single-component plasma. of the charge carriers associated with it. From the dispersion relation we obtain the wave propagation characteristics: the amplitude attenuation in space, the wave polarization, the phase velocity and, finally, the ranges of frequencies with which waves can propagate in a plasma. Let us present the main formulas that characterize the propagation of a weakly decaying monochromatic plane wave of the form E = E for the case k||r.
exp[t((jo£ - kr)]
(0.1)
The wave length λ =
2π
(0.2)
The phase velocity of the wave ω'
(0.3)
The attenuation factor for the wave amplitude 1
(0.4)
The refractive index N =
kl2~2"| a
(0.5)
The radiation intensity reflection coefficient for the case of normal incidence 2+ (fe'-M ( fc ") 2 , where ka 2
(k' - ko)
* (k"Y
°
^ .
One can see that all these quantities as well as their dependence on the wave frequency ω' can be found if one solves the dispersion equations for the real frequencies with respect to the complex wave number ?c(u)').
(0.6)
49
Waves in Plasma 1.
WAVES IN A COLD PLASMA IN THE ABSENCE OF MAGNETIC FIELD
In the linear approximation, in the absence of magnetic field B 0 the determinant of the wave equation (2.20) of Chapter I contains only the diagonal components a , a , a . The non-diagonal components are equal to zero. The only direction in^the Isotropie unmagnetized plasma which is singled out is the direction of the wave propagation. We shall assume that this direction is parallel to the 2-axis. The transverse and longitudinal waves in an unbounded isotropic plasma are not related to each other and can be considered separately. The Longitudinal Waves. It is possible to reduce the equation (2.20) of Chapter I for the longitudinal wave which propagates, for instance, along the 2-axis to an equation OL E = 0 (since Εχ - Ey = 0 ) , and the dispersion relation (2.22) of Chapter I reduces to azz = 0. On the basis of (2.21) of Chapter I and taking into account the fact that κχ - ky = 0, we obtain immediately
ω = i ^
(1.1)
As was noted in Section 2 of Chapter I, in a cold plasma σ z z does not depend on k. Therefore, equation (1.1) cannot be called a dispersion relation since it does not contain k and, because of this, does not yield the dependence of k on ω. From the physical point of view this corresponds to the fact that in this case waves do not propagate as such. Two extreme modes can be distinguished: the collisional one when ωτ+ « 1, (1.2) J P and the collisionless for which ωτ+ » 1. P
(1.3)
In the first case the electric conductivity σ 2 2 , according to (3.8) of Chapter I, is a real quantity. Then one obtains from (1.1) that ω' = 0 which corresponds to the absence of periodic oscillations. If σ > 0, the perturbation of charges in the plasma decays exponentially, with the decay constant
zz
(1.4)
The time interval τ ^ is called the dielectric relaxation time or the Maxwell M relaxation time. There is an electrical engineering analog of the Maxwell relaxation time, namely, the time of discharge of a charged capacitor through a resistance τ = RC. Indeed, assume that a certain perturbation results in that the negative charge in a semiconductor is displaced with respect to the positive by a distance 1. The corresponding situation can be imagined as that of a charged capacitor with a capacitance C = Z\{S/l), where S is the semiconductor cross-section, the capacitor being shunted by a resistance R = 1/(So). If one neglects diffusion, the discharge time τ = RC coincides with (1.4). Thus, the charges in the plasma that are separated in space move to each other, but friction (collisions with the lattice) prevent the appearance of plasma oscillations: ω' = 0. In the collisionless plasma (1.3) there is no such friction, and because of
50
Plasma and Current Instabilities in Semiconductors
inertia the motion of charges towards each other results in their new separation. The oscillations arise. Indeed, when the inequality (1.3) is valid, according to (3.8) of Chapter I the electric conductivity σ is an imaginary quantity which is given for the single-component plasma by an equation: *2 3- n0 -J-. m ^ω
σ zz
(1.5)
After substitution of (1.5) into (1.1) we obtain ω' - ω , P
(1.6)
where ω =4 V \y me ι
(1.7)
The frequency ωρ, already familiar to us from the first chapter, is called the plasma or Langmuir frequency. It is equal to
where n0
is expressed in cm" 3 , m* -
m/m0.
The plasma frequency is related to the Maxwell relaxation time through a simple equation + =
£
i 23
=
1 7
(1.8) p
The plasma frequency represents the most important parameter of the plasma. As we have seen, the frequency ω ρ corresponds to the oscillations of the negative charges with respect to the positive. These oscillations are sometimes called the eigen modes of plasma oscillations. The energy quantum of plasma oscillations ?2ωρ is called plasmon. The plasma oscillations in a partially ionized gas which were observed in Tonks' and Langmuir's experiments were characterized by a frequency with the order of magnitude of 100 MHz. The energy of plasmons, corresponding to this frequency, is rather smallftcup~ 10" 5 eV, and the oscillations could be excited by the thermal motion of the gas ions. The frequency of plasma oscillations in metals is of the order of magnitude of 1 0 1 5 Hz which corresponds to the order of magnitude of plasmon energy 10 eV. Therefore, they are practically not excited by the thermal motion of the atoms of the solid. Ruthemann (1) and Lang (2) were the first who excited and observed plasmons in a solid in 1948. In their experiments the plasmons were produced in thin metal films by an electron beam with the energy of several KeV. The plasmon energy was determined on the basis of the analysis of the losses1 spectrum for the electrons that passed through the metal film. The following values were obtained: for Al ?ζω « 14.7 eV, for Sb Τίω^ = 22.6 eV, for Be Τζωρ = 19.0 eV. In semiconductors ?2ü)p ~ 10" 3 eV, and the excitation of plasmons can occur as a result of thermal electron oscillations. The Transverse Wave. In the absence of magnetic field the dispersion equation for a transverse wave propagating in the direction of s-axis [E = 0 , fc = k = 0 ) , y according to (2.20) and (2.22) of Chapter I, has the form:
51
Waves in Plasma (χ
= ot
yy
= 0.
(1.9)
xx
= σ = σ, and from (1.9) and (2.21) of Chapter For an isotropic semiconductor σ xx yy I we obtain k2 - ^ S - + ^ωμ0ο* = 0.
(1.10)
For a collisionless plasma, when the inequality (1.3) is valid, the substitution of (1.5) into (1.10) results in
*».ίομ
1
ω2
(1.11)
If ω < ü)p, we have kz < 0 or kf = 0. This means that the transverse waves whose frequency is smaller than the plasma frequency cannot be excited in the plasma. The waves with frequencies smaller than uu are totally reflected from the plasma since the reflection coefficient R (0.6) equals 1 for kr = 0. In this case the so-called non-transmission of the wave is observed (see Section 1 of Chapter III). The physical meaning of non-transmission is that the electrons in a plasma have enough time to become redistributed in such a way that they screen the penetration of waves into the plasma. For ωί/ω 2 » 1, according to (1.11), the depth of the transition layer a = l/ktf inside which the amplitude of the wave incident on the plasma surface decreases by a factor of e equals to a =
(1.12)
UpSxï
Note that the cause of the wave attenuation is not the energy losses but the formation of the reverse wave. This constitutes the difference between nontransmission and absorption of waves. When ω > ω ρ we have k2 > 0, and undamped waves can propagate through a collisionless plasma (with k' Φ 0, k" = 0). As follows from (1.11), the phase velocity of the penetrating wave is v
f
=
ω' k'
Xi 1 -
(o>'r
(1.13)
The free carriers contribute the negative term into the dielectric constant, and the phase velocity turns out to exceed the velocity of light in a medium with a lattice dielectric constant χχ. When ω = ω , we have v ~ -+ °°, and the group velocity
rp
3ω'
dk'
Xit>
(1.14)
/
tends to zero. The equation ω = au corresponds to the resonance increase of the amplitude of the plasma eigen modes. For ω = ΐύρ the displacement current equals the drift current of free charges. Since the direction of these currents is opposite, the total current in the plasma
52
Plasma and Current Instabilities in Semiconductors jf = qriv - ίωελΪ
= i ^-
(ω2 - ω2)ί
(1.15)
proves to be zero when ω = ω . P In order to compute the attenuation depth a = 1/k" of the penetrating wave, it is necessary to take into account the collisions of electrons with the scattering centres which represent the physical factor responsible for the energy losses in the wave and for the attenuation of its amplitude. In solids τ-»· ~ 10" 13 s, and the inequality (1.2) corresponding to a collisional plasma is sStisfied over the whole microwave range. In this case, according to (3.8) of Chapter I, σ is a real quantity. On the basis of (1.10) we obtain for the imaginary and the real components of the wave vector:
W
^f */*/l + S +
cW
2 W
1.
(1-16)
(1.17) The quantity σ / ^ ω ) Ξ1/(ωτ+) equals the ratio of conduction currents oE to displacement currents ε\ωΕ. For a medium with high conductivity °
£lU)
» 1
(1.18)
and k"
= t/SUS, ■V
k' = k".
(1.19)
Since in this case there are no physical phenomena that can supply the energy to the electromagnetic wave, we choose the negative root [k" < 0) among those of (1.19), which corresponds to attenuation. The depth inside which the wave amplitude decreases by a factor of e is equal to α0 Ξ Jfr
= 1.26·10'*(σω)~'ί cm,
(1.20)
where σ is expressed in ohm"1•cm"1. The power flux S ~ (E x H) attenuates by a factor of e inside a depth δ = 2αο. This depth is called the skin-layer, and the limitation of the electromagnetic wave penetrating into a conducting medium - the skin-effect. When the displacement currents are greater than the conduction currents, 7~-«
1,
(1.21)
and from (1.16) and (1.17) we obtain
" '- i a Vl"' k'= * ^'
k
(1 22)
·
53
Waves in Plasma If σ is expressed in ohm"1·cm"1; 1 = 1 .88σχϊ^ cm" .
k"
(1.23)
For a medium with anisotropic conductivity ( σ ^ ί σ ) the dispersion of waves will also depend on the orientation of the wave polarisation plane with respect In analogy with birefringence in optics, which is to the reference frame x and y. caused by the anisotropy of χχ, in this case we observe birefringence on free carriers (3). 2.
WAVES IN A SINGLE-COMPONENT PLASMA PLACED IN MAGNETIC FIELD
The Dispersion Equation. When a magnetic field is applied, the medium becomes anisotropic and, in the general case, all components of the matrix of equations (2.20) of Chapter I are not equal to zero. In a plasma placed in a magnetic field the directions of vectors k and Bo are singled out. Let us choose the reference frame in such a way that the vectors k and B 0 are in the plane xz, with B0||z and B 0 > 0 (Fig. 2.3.a). Then the wave equation will have the form α xx
α xy
a yx
a
a
'xz xz
X
0
yy 0
1E1
α
a
= 0.
E y
(2.1)
E
zz '
1
z ■ In this expression, for a single-component cold plasma with the isotropic electric conductivity (in the absence of magnetic field) σ = qn0\i the components α.λ IJ' according to (2.21) and (3.28) of Chapter I, are 12
= - K
+
ω
k2 - k2
yy
Z
X
- K + '?T Xi α
= α
(uSo):
'^2 Xl " ^ U C -
X1 -
ϊων
"
1 + (μΒ 0 ) 2 '
^ωμ0σ,
(2.2)
= k k , \iB0
yx
xy
1 + (y#o) 2
Note that k2 = k2 cos29, z
k2 = k2 sin26, x
k k ' = k2 cos Θ sin9, x z
where Θ is the angle between the vectors k and Bo (Fig. 2.3.a).
(2.3)
54
Plasma and Current Instabilities in Semiconductors
y'fu
Fig. 2.3. Reference frames for the magnetized plasma. Sometimes it is convenient to consider waves in the plasma with k||z (Fig. 2.3.b), Then in equation (2.20) of Chapter I components a. ., according to (3.26) of Chapter I, are given by 1 + (yBotf)2 Α ω α = - k22 + 7 Xi - ™μ„σ χ + (μΒο)2
xx
o.„. = - k2 ♦ ^ 2/î/
Xl
- «ομ„σ
1
, ^
a
,
USo„ a#
ιωμοσ VBo„
V
= tü)p ö
°
i ♦ (yBo) 2 '
(2.4)
Waves in Plasma 1
. «zz
α
a
χζ
21/
=
S*
= α
Xl
" ^
°
σ
* (l*o a ) 1 + (μ^ο) 2
tü)p0ö
ζχ
= - α
μ
yz
=
- ^
55 2
i + (ys0)2
0 0
1 + (ρΒο) 2
where Β0
Β 0 sin θ,
= ß 0 cos θ.
(2.5)
The dispersion equation formed by the components otVj that are determined by (2.2) or (2.4) is a quadratic equation with respect to k = (ω2/ο2)Ν2 (Ν is the refractive index): Akk + Bk2 + C = 0,
(2.6)
which means the existence of two types of waves with opposite polarisation. The latter can be found from (2.1). From the equation of the middle row a E + a E = 0 we obtain yx x yy y a JÜL (2.7a) a * yx E
and from the third row equation a XX
E
+a ZZ
X
= 0 it follows that Z
(2.7b) We are going to give now general expressions for cases that occur most frequently in the experiment, namely, when the direction of the wave propagation is parallel to the magnetic field ïc||S0 (Θ = 0) and normal to it tjß0 (θ = π/2); in the first case we are dealing with configuration of Faraday, in the second case - with Voigt's configuration. Assume that B0||k||z for configuration of Faraday and B0||z and k||x for Voigt's configuration. Then for both configurations from (2.1) and (2.2) it can be obtained that a xx a yx 0
1E 1
0
a
X
xy
0
a 0yy
= 0.
y
a
zz '
i
z
(2.8)
it
When the determinant of the system of equations (2.8) is set equal to zero, two dispersion equations result:
o,
Vyy
vw = °-
(2.9) (2.10)
56
Plasma and Current Instabilities in Semiconductors
which correspond to different types of waves, depending on configuration. Equation (2.9) describes the waves the field vector E of which is polarised along the magnetic field B 0 . For Faraday's configuration this is a longitudinal wave, and for the configuration of Voigt it is transverse. Equation (2.10) corresponds to waves that are polarised in the plane xy; in the case of Faraday's configuration it is transverse waves, and for Voigt's configuration - the hybrid ones. Consider the propagation of waves in Faraday's and Voigt's configurations separately. Faraday's Configuration. a
xx
= a
xy
yy
= - a
7,2
=-
ω2 a
For S0||k||z equation (2.2) yields
Helicon Waves.
K
+
ω2 β2
.
Χι
ιωμ„σ
1
1 +
(yflo)2.
- ·£ωμ0σ.
(2.11)
μδο σ yx - " ^ » i V ( p B 0 ) 2 ·
The substitution of (2.11) into equation (2.9) for the longitudinal component of the field E results for ωτ+ » 1 in the solution which is known to us already from Section 1: ω = ω ρ . In other words, for Faraday's geometry no longitudinal waves propagate in the magnetised cold plasma. By substituting (2.11) into (2.10) we obtain for the transverse waves k
l
2
- "TXi
+
^ y 0 a e f f ± = 0,
(2.12)
where °eff±
=
1 + ivB0)2
C1 ± ^ 0 ) ·
<2·13)
When (2.11) is substituted into (2.7a) and (2.12) is taken into account, an equation for the polarisation of the transverse waves results: E χ E y
(2.14)
The factor i means the phase shift by π/2. Therefore, formula (2.14) describes transverse waves with circular polarisation. Thus, in the case under consideration the wave in the plasma can be decomposed into two waves with right and left circular polarisation, which are characterised by different velocities and attenuation. For these waves we have E + = E — (x ± iy)
exp(io)t - k + l ) .
(2.15)
v2
The law of the wave dispersion (2.12) depends both on the direction of the wave polarisation (the sign before the imaginary unit) and on the sign of the product u£ 0 = ω 0 τ , where ooc is the cyclotron frequency. For an observer whose direction of observation is parallel to that of the vector B 0 the cyclotron rotation of electrons occurs in the clockwise direction, and that of holes - in the counterclockwise direction (Fig. 1.2). Accordingly, when So > 0, in the electron plasma we have UBQ < 0, and in the hole plasma - μ5 0 > 0. Finally, the term ±i\iB0 in
57
Waves in Plasma
(2.13) will have a positive sign if the direction of rotation of the wave field vector coincides with the direction of the cyclotron rotation of the plasma charge mobile carrier. Such wave is called extraordinary. The second wave the direction of polarisation of which does not coincide with that of the cyclotron rotation of charges is called ordinary. For an electron plasma the wave with the right polarisation for an observer who looks along the vector B 0 is extraordinary. For a hole plasma, on the contrary, the wave with the left polarisation is extraordinary. This is illustrated by Fig. 2.4.
/?=♦/
7=-/
a Fig. 2.4.
a) The direction of rotation of the field vector in a wave with circular polarisation; b) The directions of the cyclotron rotation of electrons and holes; c) The directions of circular polarisation of the extraordinary wave for the electron (1) and hole (2) plasma.
In order to avoid confusion with the signs, we shall write equation (2.13) as
Vfß
=
i + ok)2
i_ i + β | μ β ο | )
(2.16)
where 3 = η
qBp
(2.17)
η = -1 for the right polarisation of the field vector in the case of observation in the direction of the wave propagation and η = +1 for the left polarisation. The quantity 3 = + 1 for the extraordinary wave and 3 = -1 for the ordinary wave. If one takes into account the fact that |μ£ 0 | = |ω0|ΐ and σ ~ μ ~ τ, where τ = τ+/(1 + ίωτ·+), equation (2.12) can be transformed to
?2
ω
(2.18) ω(3ω
- ω) + ίω
In order to simplify the notations, it is assumed here and in what follows that
58
Plasma and Current Instabilities in Semiconductors
ω 0 Ξ |u)c| and that its sign is determined by the coefficient ß. If (3ω v
c
(2.19)
- ω)τ->- » 1 'p
the collisional term in (2.18) can be neglected and we obtain that k2
- ÜL. v
1 +
ω(Βω
Ά
ο
(2.20)
- ω)
h\\8n
^^^..•^Τ^Κ I
■rf—τ j
{
1
ω
l/l
- ^ /?=*/
Βη*0
w
E
ω F i g . 2 . 5 . 1 F, I I .
59
Waves in Plasma
ω
Fig. 2 . 5 . I l l ç IV
60
Plasma and Current Instabilities in Semiconductors
Fig. 2.5.
Schematic drawings of dependences of N2 on ω (a); the displacement of the regions of non-transmission of waves (shaded areas) by magnetic field (b). I - the single-component plasma in Faraday's configuration (2.20): 1 - β = +1; 2 - 3 = -1; 3 = î0 = 0. II - the single-component plasma in Voigt's configuration: 1 - the extraordinary wave; 2 - B0 = 0.
Ill - the singly charged two-component plasma in Faraday's configuration: 1 - 3 = +l; 2 - 3 = -1; 3 - 5 0 = 0. IV - compensated plasma in Faraday's configuration: 1 - 3 = +1; 2 - 3 = -1 (3.46); 3 - B0 = 0. 2
2
ω ω Pi _ _£2_ ω7 ω_ '1 Pi P2 ω ω c\ V - the plasma in a semiconductor with anisotropic valleys in Faraday's configuration: 1 - 3 = +1; 2 - 3 = -1; 3 - B0 = 0. Here ω,
P
2
In t h i s case ω? = -=■ν (2ω* + ω 2 7 ) , ω =K ίω , ωΓ 7 j κ 3 pt pV o\ et cV o2 The cyclotron resonance occurs also for the ordinary wave.
ot
Figure 2.5.1 presents schematically the dependence of the refractive indices squared N
A c,2 on the wave frequency for a collisionless case (2.20).
As the
magnetic field increases, for the extraordinary wave the transmission band becomes more broad from the side of the lower frequencies up to ω , and the non-transmission band becomes more narrow near wc. For the ordinary wave the transmission band increases because of the decrease of the cut-off frequency ω ς 0 from the value equal to ω ρ down to lower frequencies and to values determined from the condition
61
Waves in Plasma 2
of k
turning into zero: ω
No propagation of
ίω + ω ) 1 ω 2 . co^ c co ; p
(2.21)
the ordinary wave (B = -1) is possible in the plasma if ω < ω
,
(2.22) J
co since in this case k2 nary wave (3 = +1) k2
< 0 and kr = 0, R = 1 (0.6) and vr~ > 0 for all ω < ω
= 0.
For the extraordi-
(2.23)
and the reflection coefficient R < 1. Inequality (2.23) is certainly valid for the whole microwave range since in the fields of the order of magnitude 101* G the cyclotron frequency ω 0 « 1 0 1 2 Hz. Thus, when inequalities (2.22) and (2.23) hold, out of two waves with the left and right circular polarisation the wave with only one polarisation can propagate in the plasma, namely, the extraordinary wave. As the wave with the plane polarisation penetrates into a magnetised plasma, it twists into a helix. This wave is called the helicon; it has been already mentioned in the introduction to this Chapter. When ω = ω 0 , the so-called cyclotron resonance occurs. In the vicinity of this point the dependence of the refractive index on ω is very pronounced. Waves in the plasma whose frequencies are close to o)c are called cyclotron waves. In the cyclotron resonance region the drift current produced by the crossed fields E and B 0 turns out to be co-phasal with the particle cyclotron rotation currents. Thus, when the magnetic field is switched on, the response of the plasma to the electromagnetic wave changes substantially. First, the condition of collision neglect is changed. In the absence of magnetic field the role of such condition was played by the inequality ωτ-> » 1 (1.3) which is practically not valid for the whole microwave range. When a plasma is subject to the action of the magnetic field, inequality (2.19) becomes the condition of collision neglect; this inequality holds for all frequencies ω « ω if
V p Ξ V° *
lf
(2 24)
*
which is quite possible in materials characterised by a high mobility. The plasma for which condition (2.24) is valid is called magnetised. This term, though, is applied also in general to describe a plasma placed in a magnetic field. Second, there is no total reflection of the extraordinary wave which penetrates into the plasma at frequencies below ω , which cannot happen in the absence of the magnetic P field È 0 . As has been already noted, the main physical cause responsible for such striking changes in the plasma response are the drift currents of particles that are normal to the electric and magnetic fields. When there are no collisions, the electric and magnetic fields produced by these currents are normal to the corresponding fields of the incident signal and can change only polarisation but not the amplitude of the latter. Although because of collisions the drift currents are deflected from the normal direction by a certain angle, the value of this angle is smaller for stronger magnetic fields, tg Θ = (yBo)" 1 ; therefore, the energy losses in the wave that are associated with the current component parallel to the electric field decrease as the magnetic field increases. To put it differently, the increase of the magnetic field is accompanied by the increase of magnetic resistance and the decrease of losses.
62
Plasma and Current Instabilities in Semiconductors
Let us estimate the depth of penetration of waves from the microwave range. Assume that ωτ* « 1,
ω « ω ,
ω 2 » ωω .
(2.25)
In t h i s c a s e , from ( 2 . 1 8 ) , we o b t a i n (ω2τ+)Η \ [l + (ω τ-*) 2 ]^ - βωc τ+Ι I JLZ p p| J c P _ PI 1 + (ω ο τ-)
(2.26)
and the penetration depth for the extraordinary helicon wave is
a
H
=
W
a(0)
1 + (ωc τ+) 2 P /l + (ω "τ->)ζ - 'ω~τ^ c p c p
(2.27)
where a(0) is the depth of the skin-layer in the unmagnetised plasma (1.20). The penetration depth increases rapidly with the increase of the magnetic field (Fig. 2.6). For the ordinary wave ($ = -1) the initial decrease of the penetration depth is followed by its slow increase accompanying the growth of the magnetic field.
äffi
μΒ0 Fig. 2.6. The penetration depth for the helicon wave a H and the ordinary wave a 0 as a function of the quantity uB0 = ω τ+ in skin-layer depth units for B 0 = 0. c p Because of the large depth of penetration of the helicon wave into the conducting media it is possible to use it for studying the properties of solid state plasma.
63
Waves in Plasma We shall discuss this in Sections 6 and 8. For very high magnetic fields and sufficiently high frequencies when ($ω -ω)τ-* » 1 and ω(ω + 3ω ) » ω* , c ρ c ρ
(2.28)
instead of (2.18) we have 2 k*3 = Hl v C 2 Χι>
(2.29)
in other words, there is no interaction between the wave and the free carriers, and the wave propagates in the conducting medium as in a dielectric with a dielectric constant χ 1β Since in this case the dispersion law is the same for both waves, no twisting of the wave with plane polarisation into a helix occurs. There is also an essential difference between the phase velocity of the helicon wave and the phase velocity of waves in the plasma for B 0 = 0. For the collisionless plasma _ ω'
i
βΧι
1 +
ω(3ω
(2.30) - ω)
For the ordinary wave when 3 = -1 tl^e phase velocity is always greater than the velocity of light in the medium ol\{2 . In the case of the extraordinary or helicon wave 3 = +1 and if ω < ω 0 the phase velocity is always less than the velocity of light in the medium; it approaches zero when ω = ω (Fig. 2.7). 7f/2
c
Fig. 2.7.
The 1 -
dependence corresponding to (2.30): ω /ω = 1, 3 = +1; 2 - ω /ω = 2 , ω_/ω„ c' ρ c' ρ
= +1;
3 - ω ο /ω ρ = 2, 3
64
Plasma and Current Instabilities in Semiconductors
The physical explanation of the small value of the phase velocity for the helicon wave is that the contribution of free carriers in the dielectric constant in the magnetised plasma turns out to be positive (Fig. 1.3). In the low frequency range and in the vicinity of the cyclotron resonance the small phase velocities of the extraordinary waves correspond to short wave lengths. For instance, in vacuum the frequency ω « 10 7 Hz corresponds to the wave length À « 3 0 m, while in the semiconductor for ω ρ « 1 0 1 2 Hz, ω 0 « 1 0 1 2 Hz and χχ « 10 the wave length λ « 3 cm, i.e., is less by a factor of 1000. One can see from Fig. 2.5.I.b that there are frequency ranges for which both the ordinary and the extraordinary waves can propagate in the plasma at the same time. The difference between the waves' phase velocities will lead to rotation of the long axis of the resultant wave polarisation ellipse as the wave propagates in the medium. This phenomenon is called the Faraday effect or the magnetic Kerr effect. Vo ig t ' s Configurât ion. For this geometry tjßQ, only k φ 0. The coefficients a., are given by X
t\\x,
?ο||Ϊ*
θ = π/2 and in (2.2)
tj
«xx = ^ Xl " ^ y ° G 1 + (U) 2 ' k2,
a = a XX yy a
a
zz
xy
= -
(2.31)
2
k
■+
ω2 ^2 Xi * *-ωρ0σ,
- -v
tU)Po°"
y£o i +"(y£0)2·
Again, as in the case of Faraday geometry, there are two types described by dispersion equations (2.9) and (2.10).
of waves which are
Equation (2.9) in Voigt's configuration describes a transverse wave with plane polarisation E||z and dispersion which does not depend on magnetic field: k2
2
= -J2- Xi - ^ων0σ.
(2.32)
Equation (2.32) coincides with the dispersion equation (1.10) that corresponds to transverse waves in the plasma in the absence of magnetic field. From equation (2.10) and by taking into account that α ^ = αχχ - k2 (see (2.31)) we can obtain for the second type of waves the electric field vectors of which are in the plane xy a dispersion equation of the form -2 _
a2
xx
xy
a
(2.33)
XX
According to (2.31) and (2.7), the field components Εχ and Ey form a wave with elliptical polarisation in the plane xy. The component Εχ is longitudinal; it results from the drift of the plasma charges produced by the Lorentz force which is normal to y and B 0 . If the wave is observed along the vector k, it appears to have a linear polarisation along the y axis. The field of the wave with elliptical polarisation can be written as
65
Waves in Plasma
xy
î-ΐο
x + y
exp[t(u)t - kx)] .
(2.34)
xx Such wave is called the extraordinary Voigt wave, and the linearly polarised wave with the field Ê||î described by equation (2.32) - the ordinary Voigt wave (4). When (2.31) is substituted in (2.33), we obtain for the extraordinary wave
to) c2 =
^
(2.35)
1 ^ ^ — - ω ω + — Ρ; ^ Ρ'
For the collisionless case (1.3)
k
ω 2 (ω 2
P\,P.
ω
72
- p- *>
;
(,
ω2) (2.36)
ω2)
There are two typical non-transmission bands (k2 ζ 0 ) , one in the low frequency region, and the other near the resonance. This is shown on Fig. 2.5.II. As follows from (2.36), the resonance occurs at a frequency ω
Τ/
= ίω
D2r
(2.37)
P'
It is called shifted by the plasma frequency or the hybrid cyclotron resonance, and also Voigt cyclotron resonance. There is no resonance for ω = ω . According to (2.36), the cut-off frequency for non-transmission can be determined from a biquadratic equation
co
ω 2 v(2ω co i
ω 2 ); + ω" = 0. c p
(2.38)
When ω > ω ρ both ordinary and extraordinary waves can propagate with different phase velocities. The result of this is that the projection on the plane yz of the sum wave field has elliptical polarisation, with the slant of the long axis of the sum wave ellipse varying during propagation. This phenomenon is called Voigt effect. Oblique Waves. The geometries of Faraday and Voigt correspond to the so-called direct waves. The waves whose direction of propagation forms a slanting angle with that of the magnetic field are called oblique. Without presenting simple but cumbersome calculations, let us give the general expressions for the waves in a single-component cold plasma for an arbitrary angle Θ between the wave vector k||z and the magnetic field Bo which is in the xz plane (see Fig. 2.3.b)(4, 5 ) . As follows from (2.4), the dispersion equation has" the form 72
ki,2
=
1 -
ω2
£jr Xi
·
2Ωω2 )[2Ω(ω + iv+) L p
- ωω2 s i n 2 θ]J ± ω LΓω2ω2 s i n 4 θ + 4Ω2 cos 2 θ]J " c c c
, (2.39)
66
Plasma and Current Instabilities in Semiconductors
where Ω = ωίω + iv->) - ω 2 ;
v* = — .
The resonance frequencies are given by (v* = 0 ) .
2ω 2
pe2
Fig. 2.8.
= ω 2 + ω 2 ± [(ω2 + ω 2 ) 2 - 4ω 2 ω 2 cos 2 θΐ^.
ρ
c
ικ
ρ
cJ
p c
(2.40)
J
Polarisation of transverse components of waves in a single-component plasma for different values of angles between the direction of the wave propagation (k) and the direction of the magnetising field (B 0 ): Faraday's geometry (a); the intermediate case (b); Voigt's geometry (c) (e - the extraordinary wave, o - the ordinary one). For the extraordinary Voigt wave the polarisation of the longitudinal component of the field is- shown.
67
Waves in Plasma For the polarisation of the transverse components of the wave amplitudes in the reference frame x'y'z' (Fig. 2.3.a) we have ί2ω Ω cos Θ c
E
Xf
E
f\
(2.41)
sin 2 θ ± ω l[ω 2 ω 2 sin 4 θ + 4Ω 2 cos 2 θΐ' J c c.
When Θ = 0, equations (2.39), (2.40), (2.41) correspond to Faraday's geometry, and when θ = π/2 - to Voigt 's one. We would just like to note that formulas (2.31) (2.34) that have been obtained previously for Voigt's configuration are now derived in the other reference frame (k||x) in which coordinates have no prime signs. As the angle Θ varies from 0 to π/2, the waves with circular polarisation are first transformed into the elliptically polarised waves with the normal long axes of the ellipses, and then into two transverse waves of plane polarisation (Fig. 2.8). The rotation of the polarisation plane occurs in the opposite directions for waves with a frequency above ω > ω and for waves with ω < ω (2.41). The longitudinal component of the field amplitude S ,||k , is
x'
p c
ωω 2 v(ω + iv+) c P
sinG - iE
y
,ω2ω (ω + tv+) sin6 P c P
- (ω + -£v->)2 Ω P
(2.42)
3. THE MULTI-COMPONENT PLASMA. MAGNETOHYDRODYNAMIC AND MAGNETOSONIC WAVES The presence of several kinds of mobile carriers affects substantially the dispersion of waves in a solid. Consider a multi-component plasma in the hydrodynamic approximation assuming that the effective masses and relaxation times are isotropic. In this approximation the charge carriers of each kind make their own partial contributions into the dielectric constant or, to put it more precisely, into the plasma electric conductivity. The electric conductivity tensor components represent a sum of the corresponding components of partial electric conductivities (see equation (3.26) of Chapter I) describing each kind (denoted by a) of the charge carriers.
N
^
tja
a=l
where σ.. Èo||î -%iy
B
N
6 . . + u Β0ε . . 7 Q Yl \X
a=l
1
n
B
+
* ^a
KB°î2
B
°iB°j) Bî j
(3.1)
2
is given by (3.26) of Chapter I, and in the reference frame in which (3.28) of Chapter I.
One can say a priori that there will be a qualitative difference between the compensated plasma's response to a perturbation (when the number of mobile carriers with the positive charge is equal to that of the carriers with the negative charge) and the response of the uncompensated plasma. In a compensated plasma, if the collisions are neglected, the resultant drift current produced by the Lorentz force turns out to be equal to zero since the drift velocities u ((3.16) of Chapter I) depend neither on the mass nor on the charge sign. Therefore, exactly those currents which are responsible for the main peculiar features of the singlecomponent plasma in the magnetic field are absent. Besides, in a compensated plasma the concepts of the ordinary and extraordinary waves in Faraday's configuration as they are defined in Section 2 are meaningless since a wave which is "extraordinary" with respect to the electrons is "ordinary" with respect to holes,
68 and vice
Plasma and Current Instabilities in Semiconductors versa.
Consider the peculiar features of the wave dispersion in a multicomponent plasma with the example of a two-component plasma in the following sequence: (a) both kinds of mobile charge carriers have the same sign, ql = q2 = q; (b) the mobile carriers have charges of the opposite signs, q\ - -q2 and ηλ = n2 (compensated plasma). The indices 1 and 2 will denote parameters describing the free charge carriers of the first and second kind, respectively. The Single-Charge Two-Component Plasma. For Faraday's geometry, according to (3.1) and (2.10), the dispersion equation for transverse waves has the form k2
_£JL
1 +
Xi
ω(3ω
cl
ω((3ω c2
- ω) Pi
_£2_ ω)
(3.2) P2
where 3 = +1 for the extraordinary wave and 3 = -1 for the ordinary one. Figure 2.5. Ill presents schematically the quantity N2 as a function of ω for the collisionless case. In the absence of the magnetic field the cut-off frequency ω below which k2 < 0 and no waves propagate in the plasma is
ω2 y
(r2 Pi
CO
(3.3)
P2J
In the magnetised plasma the cut-off frequency for the ordinary wave is shifted towards lower frequencies
-£*-
_£JL Cl
1 +
(3.4)
c2
According to (3.2), for the extraordinary wave the cyclotron resonances for each kind of particles and near the cyclotron frequencies (the non-transmission bands) are observed. For Voigt's configuration, by substituting into the expressions for a^j ((2.21) of Chapter I) the sum of partial electric conductivities for both kinds ot carriers, we obtain a
xx
=
Xl
%
Σ2),
' ^ ω μ °( Σ ι
a = a - k , yy xx a
a
a
=
ωμ
(3.5)
xy
=
Σ
β
" yx
zz
=
" k2 +ÏÏ2-Xl " im°^01
^ ο( ι^ι °
+
Σ
2\ΐ2Βο),
+σ 2
)·
In these formulas θχ = qiniVi,
σ2 =
qinzVi,
(3.6)
69
Waves in Plasma 1 + (uiSo) 2 For the ordinary Voigt wave azz sion equation takes the form ?2
k
=
ω —
Σ2 =
1
+
(3.7)
ÎViBo)2'
Since τ = τ+(1 + ϊωτ+)~ι, P P
= 0 (2.9).
the disper-
P? P2 ωίΐ + ϊωτ+ )
Pi Pi (1 + ϊωτ+ )
1 + i
Xi
,
(3.8)
Dispersion does not depend on magnetic field. In the collisionless case, when ωτ+ » 1, P 2 i ω2 + ω2 (3.9) The square of the cut-off frequency is equal to the sum of squares of plasma frequencies for each kind of carriers. For the extraordinary Voigt wave equations (3.5) and (2.10) yield 7 2
*
=
Bß Σ2υ2#ρ \ Σ, + Σ 2 + JLüLi o0 ^ωε ! + Σι + Σ 2 J j *
ω
— Xi
(3.10)
For the collisionless plasma (ωτ-* » 1 ) , according to (3.7) of Chapter I, u « -^ -r-. m τω
Hence ελω2
ειω
ω
(3.11)
^cT
ω 1
2
ρωο -ω
(3.12)
Substitution of (3.11) into (3.5) results in , ,2
ω
4,
Ί ω2 - ω2 ω' - ω" Ci C2 As follows from (2.33), the resonance will occur if a = 0 . a xx
Xi
1
(3.13)
XX
There will be separate resonances at cyclotron frequencies if ω 2 « ω and ω 2 _« ω, in other words, in very pure samples at large frequencies. In^this case expression (3.13) turns into zero when u)2 « ω 2 - ω 2 = ω 2 , a ω 2 - ω 2 » ω 2 , Pi ci c2 ρ2 or 2
?
ω « ω , a ω C2
9
9
9
-ω » ω . Ci Ρι
For low frequencies, when ω » ω and ω » ω, the resonance will occur if Pi P2
70
Plasma and Current Instabilities in Semiconductors
n2 ω
2x
+
ω^
ω ci c 2
n2
2
2
Pi
Pi
·*.]
(3.14)
This resonance which depends on the ratio of the carrier numbers instead of their absolute concentrations is called the hybrid resonance. For nl - n2 the hybrid resonance will take place at the frequency ω
(3.15)
ω . ci c 2
The cut-off frequency corresponding to non-transmission of waves in Voigt's configuration can be determined, according to (2.33), from the condition a
+ a
xx
= 0
xy
or
a
xx
= τα
xy .
(3.16)
In the collisionless case the substitution of (3.12) and (3.11) in (3.5) and (3.16) results in the following equation
_£l_
ω
_£
- ω
CO
CO
Ci
0.
(3.17)
c2
Compensated Plasma. For a compensated plasma n\ - n2, Qi = -Qi, and, as has been noted already, the plasma's response to a perturbation differs essentially from that of the single-component plasma. It will be assumed, as before, that B01| z and the vector k is in the plane xz (see Fig. 2.3.a). In sufficiently strong magnetic fields ω 0 τχ » 1, ω 0 2 τ 2 » 1> and because of compensation of charges the Hall current as well as αχι. given by (2.2) tend to zero:
a
xy
= a
yx
σ
« to)u0
1
+
μι#ο
Q~2
(3.18)
P2#c
The wave equation (2.1) takes on the form a
0
XX
0
a XZ
a
0
(3.19)
= 0,
yy a
0
zx
a
zz
where, as follows from (2.2) and (2.3), k2 cos 2 Θ - ίωμ (Σχ + Σ 2 ) + ^ o-yy = - k2
Χι>
- ^ωμ0(Σι + Σ 2 ) + ^ - Χι, (3.20)
= - k
α
α
ζχ
= α
χζ
2
2
sin θ - ϊω\ι0 (θ\ - k2 cos θ sin θ.
+ σ2) + —
χι,
71
Waves in Plasma In these formulas olf σ2 and Σι, Σ 2 are determined by (3.6) and (3.7). tion (3.19) two dispersion equations
From equa-
a =0, yy a
a
XX
- a
ZZ
2
(3.21) J
= 0,
(3.22)
XZ
can be obtained which correspond to two types of waves. Consider waves in the low frequency region when the displacement currents can be neglected in comparison with the conduction currents and the terms (ω2/ο2)χχ in (3.20) can be discarded. Assume also that ω « ω
,
For the first type of waves (α
ω
, ω « ω
,
ω
(3.23)
= θ) we obtain
k\ = - ΐωμ,ίΣ, ♦ Σ2) Ξ - ί ω μ . ( ^
+
^ ) ^ .
(3.24)
If collisions are neglected, taking into account (3.7) of Chapter I results in
kl
-^L
eoBÎ
J'
(3 25)
*
Note that the phase velocities of waves, described by (3.25), do not depend on frequency (linear dispersion) and are always less than the velocity of light; this can be seen from the formula y
h A
-
· /w(tf?i +
(3.26)
m2)
Waves described by (3.25) and the velocity (3.26) are usually called Alfven, by name of the scientist who predicted such waves in the gaseous plasma. From equation OLyyEy = 0 (the second row of (3.19)) we immediately obtain polarisation E||y, i.e., tne Alfven waves are linearly polarised with the field vector E being normal to the magneticefield vector B0 and the wave vector ki, regardless of the angle between ki and B0. For the second type of waves ( O L ^ O ^ - a£ 2 = 0), by substituting (3.20) into (3.22), we obtain for arbitrary angle Θ between B0 and k: -£ωμ0(οΊ + σ2) (Σι + Σ2) [(σι + σ2) cos2 θ + (Σι + Σ2) sin2 θ]
(3.27)
When θ = 0 (Faraday's geometry), equation (3.27) coincides with (3.24). If θ ^ 0 (oblique waves) in high magnetic fields (Σ « σ) it can be assumed for all angles Θ except those close to π/2 that
H « — ^ .
(3.28)
J cos2 Θ If collisions are neglected, it follows again from (3.27) and (3.28) that the second type of waves is also described by linear dispersion and the phase velocity is independent of frequency. According to (3.28), however, the phase velocity z
72
Plasma and Current Instabilities in Semiconductors
depends on the angle Θ and for Θ / 0 it is less than the phase velocity of the Alfven wave; it turns into zero when Θ « π/2 (Fig. 2.9). Therefore, the wave of the second type is called slow Alfven wave, and that of the first type - fast Alfven wave. It follows from (3.22) that the slow Alfven wave is polarised in the
plane xz,
Fig. 2.9.
The phase velocities of the fast (vA) and slow (v2) Alfven waves as functions of the direction of wave propagation.
By applying (3.24) we obtain for the imaginary part of the wave vector of the fast Alfven wave:
Ά
(3.29)
._ J mlh-i^l.
The comparison of (3.29) with (1.19) yields that the Alfven wave's absorption is described by the ordinary expression for the skin-layer where the change of the electric conductivity in the magnetic field is taken into account. For the slow Alfven wave it follows from (3.28) that k'ï = k'{
fcrj, J
for
ωϋο(σΐ2+ σ 2 )
Θ « 0, for
(3.30) θ
^
(
(3§31)
73
Waves in Plasma
Magnetohydrodynamic Waves. Although in a compensated magnetised plasma the currents that are transverse with respect to the magnetic field are equal to zero, the transverse fluxes of plasma particles are large. The plasma as a whole, as a certain liquid of mobile charges experiences displacement. This displacement is wave-like, i.e., the propagation of Alfven wave in the plasma is accompanied by a hydrodynamic wave of conducting fluid. In the presence of a magnetic field B 0 the magnetic force lines are as if frozen in the plasma and the vibrations of conducting fluid occur at the same time as the magnetic field vibrations. The waves in a conducting fluid placed in a magnetic field are called magnetohydrodynamic or magnetosonic. The fast Alfven waves are transverse from the point of view of electrodynamics (EiJ_ki). If kJlBo, the plasma subjected to the action of fields Ei and B 0 moves also in a direction which is normal to ki and oscillates together with the magnetic force lines as a string (Fig. 2.1.a). In this case Alfven wave is transverse also from the hydrodynamic point of view. And if kiJ_Bn, the Lorentz forces proportional to Ei x B 0 move the plasma along the vector ki leading to its compression or expansion. A longitudinal hydrodynamic wave appears. In other words, in this case the fast Alfven wave, while remaining transverse in the electrodynamic sense, turns out to be longitudinal in the hydrodynamic one (Fig. 2.1.b). Let us determine the phase velocities of magnetic sounds on the basis of magnetohydrodynamic approach. In the ordinary hydrodynamics the motion of masses of neutral fluid is described by the equation of motion of a "liquid particle" dv
m dt
1 P
(3.32)
In this formula vm is the mass velocity, p is the mass density and VP is the pressure gradient or the force of pressure. In a compensated cold plasma the role of force that acts on the liquid particle belongs to the ponderomotive force.
f = Î x î0.
(3.33)
The electric fi_eld E in a compensated plasma does not act on its neutral element since q\E + q2^ = 0. The hydrodynamic equation of the cold plasma assumes the form ^ -jw
dt
1 -* * = - 0 x 5o).
(3.34)
This equation can be easily obtained from the equations of motion in Drude approximation: dv\ <7i[E + (V! x t0)], (3.35) m ~dt dv2 ÏÏÎ2
"at
q2 [Ê + (v2 x ?o)],
(3.36)
where the subscripts 1 and 2 denote the component number. By multiplying (3.35) by ηλ and (3.36) by n2 and adding them up we obtain (3.34) in which the mass velocity
m
ηλπιλ + n2m2
~
P
'
(3.3/J
74
Plasma and Current Instabilities in Semiconductors
the mass density p = «imi + n2m2
(3.38)
and the current Î = <7i*iVi + q2n2v2.
(3.39)
From the solution of the hydrodynamic equation for neutral fluid (3.32) we obtain for the phase velocity of transverse acoustical wave:
V
ei - V ?
(3.40)
where J is tension along the fluid and p is the mass density. The velocity (3.40) coincides with the ordinary expression for a vibrating string. In the adiabatic approximation, i.e., when PV' = const, where P is pressure and V is the specific volume, the solution of equation (3.32) yields v
s2
V*
( 3 · 41 )
= 1(Ύ7·
The factor γ represents the ratio of the specific heat at a constant pressure to the specific heat at a constant volume. In the magnetohydrodynamic case the action of the force j reduced to isotropic pressure
x S 0 for Jjt0
can be
PH = Ho ~
(3.42)
and tension along the force field lines J H = y 0 #o.
(3.43)
Substituting (3.42) in place of P in (3.41) and taking into account the fact that for the frozen-in field γ = 2, we obtain the following expression for the magnetic sound velocity (3.44) One can see that v
coincides with the Alfven velocity v. given by (3.26).
Substitution of (3.43) in place of J in (3.40) results in (3.45) Again we have vQ - VK, Thus, the expressions for the velocities of fast Alfven waves and the longitudinal and transverse hydrodynamic waves coincide, although they have been obtained on the basis of different initial assumptions. We have considered above the low frequency region for which the inequality (3.23) is valid. When ω « ω 0 the wave dispersion turns out to be strongly dependent on frequency. As has been noted already, such waves are called cyclotron waves. In
75
Waves in Plasma the semiconductor plasma one can distinguish the hole (for ω « u)Cp) and electron (for ω « ω ο η ) cyclotron waves.
In the case of Faraday's geometry in a collisionless plasma we have, according to (3.2) k2 = — 2 *3 Ö
XYl,
1 +
ωρχ ω(3ω
ω
- ω) ci
Ρ2
ω(- 3ω - ω) J c2
J
(3.46)
If 3 = + 1, the cyclotron resonance is observed on carriers of the first kind, and if 3 = -1 - on carriers of the second kind. If it is assumed that ω^ > ω~ , the non-transmission band for the wave with 3 = +1 corresponds only to those ω for which ω > ω ς . The wave with 3 = -1> in addition to the non-transmission region ω > u)c also has, according to (3.46), the non-transmission band in the frequency region'below
rpi
coi
"P2
(3.47)
This is shown on Fig. 2.5.IV. For Voigt's configuration in a compensated plasma hybrid resonances are observed which are similar« to those considered here for a plasma where the mobile particles of two kinds have identical signs of charges (3.14). The waves in the plasma in the vicinity of hybrid resonances are called hybrid waves.
4.
THE EFFECT ON WAVES OF THERMAL MOTION OF PARTICLES. ELECTROSONIC WAVES. LANDAU ATTENUATION.
The dispersion relations given in the preceding sections are based on the hydrodynamic approximation in a cold plasma. We shall determine now the changes produced in the wave propagation in plasma by the thermal motion of particles when the pressure forces associated with this motion turn out to be comparable with electromagnetic ponderomotive forces. A number of non-local effects as well as new types of waves appear in such plasma which is usually called the hot plasma. Note once more that this term is applied only in the above sense and by no means always implies the high temperature of the plasma. Plasma (Electrosonic) Waves. If in a cold plasma in the absence of magnetic field no longitudinal waves emerge (see Section 1)\ in a hot plasma pressure produces longitudinal waves the origin of which is similar to that of ordinary acoustical waves in a gas of neutral particles. The equation of motion for a single-component hot plasma in the absence of magnetic field is =
£E _ VP
(4.1)
In this formula n and m are the concentration and the mass of particles, VP is the pressure force resulting from the thermal motion of the free particles of the plasma. In the ideal gas approximation we have
P = k+T n. B n
(4.2)
In a homogeneous plasma the pressure gradient is determined by the non-equilibrium concentration on. In the adiabatic approximation
76
Plasma and Current Instabilities in Semiconductors —
- Y — ,
(4.3)
where γ is the exponent in the adiabatic formula (Pn
= const).
One has to solve the equation of motion (4.1) for the longitudinal wave together with the Poisson equation
-M-^L
(4.4) £1
and the continuity equation (2.28) of Chapter I 6n =^-(kvV
(4.5)
For the longitudinal wave with k||v it follows from (4.5) that Von = - i ÜÖ- k2v.
(4.6)
The substitution of (4.6) into (4.3) yields VP = - ^γ
- 7
—u
.
(4.7)
The quantity t>
= V γ
= 4/ γ - ^ ^
(4.8)
can be called the velocity of electron (hole) sound. It is totally identical with the expression for the ordinary sound velocity in a neutral medium. By substituting (4.7) into (4.4) and taking into account (4.5) and (4.1) we obtain a dispersion equation for the longitudinal waves in a hot single-component plasma ω 2 - ω 2 + k2v2 = 0. (4.9) J v p sn The term "plasma" or "electrosonic" waves is used to denote these waves which are caused by the electrostatic forces (as in plasma vibrations) or by pressure forces (as in sound). In a cold plasma Tn « 0, vsn « 0, and the plasma waves become electrostatic plasma vibrations with ω = ω ρ . In a magnetised compensated plasma, in addition to the electrosonic waves, magnetohydrodynamic waves (magnetic sounds) appear. We are not going to give here cumbersome expressions for waves in a magnetised hot plasma. Just the main specific features of these waves will be mentioned. In the hot plasma the dispersion of magnetosonic waves undergoes a substantial change. For example, the limitations imposed by the hybrid frequency on the dispersion of the Alfven wave in Voigt*s geometry are removed, and the waves can propagate at the frequencies above the hybrid one. Because of the interaction of magnetic sounds with electron and hole sounds, the accelerated and decelerated magnetosonic waves appear. The velocity of the accelerated magnetosonic wave is greater than the velocity of the Alfven wave and the velocity of sounds in the plasma, whereas the velocity of the decelerated wave is less than each of these quantities.
77
Waves in Plasma
Landau Attenuation. Another very important phenomenon resulting from the thermal motion of the plasma particles, namely, the collisionless Landau attenuation, is not taken into account in the hydrodynamic approximation. The Landau attenuation is non-linear and non-local with respect to the electric field, and for this reason it was not considered when the above approximations were discussed.
Us
v
Fig. 2.10.
u
Landau attenuation
Figure 2.10 shows the physical mechanism that leads to the Landau attenuation. Imagine that a longitudinal wave with the phase velocity Vf propagates in the plasma. The electrons in the plasma the velocity component of which in the direction of the wave propagation is a little greater than the phase velocity of the wave turn out to be in the decelerating field of the wave and transmit their energy to it, while the electrons with the corresponding velocity component which is a little less than the wave phase velocity will be accelerated and consume the wave energy. Since the usual equilibrium velocity distribution of electrons is a decaying function (dn/dv < 0 ) , the number of slow electrons with a velocity less than Vf is greater than the number of fast electrons with a velocity greater than Vf. Tne total effect of the interaction between the electrons and the wave results in the attenuation of the latter. Only the electrons with the close values of the velocity projections on the direction of k and with the direction of projections coinciding with that of the wave phase velocity participate in Landau attenuation. One can say that the latter appears in the case of phase resonance between the particles and the wave. If Vf is much greater than the thermal velocities of electrons, the number of electrons which can participate in the phase resonance is small. For slow waves with the velocities comparable tô the thermal velocities of the plasma particles the Landau attenuation is manifest. The waves the phase velocities of which are close
78
Plasma and Current Instabilities in Semiconductors
to the acoustical velocities experience strong attenuation. In particular, the decelerated magnetosonic waves in the plasma decay rapidly because of the phase resonance. As the wave frequencies approach some resonance ones (for instance, when ω -S ω 0 ) , their phase velocities tend to zero; this phenomenon results in the increase of the number of particles that move in resonance with the wave. In this case taking into account the thermal motion (which can be infinitesimal) leads to collisionless energy absorption through Landau mechanism. For the non-Maxwellian particle velocity distribution, when dn/dv effect, namely, the wave amplification, is possible.
> 0, the reverse
The attenuation or the increase of the amplitude of the plasma waves during phase resonance can be regarded as a form of Cherenkov effect, i.e., as the absorption or emission of waves by particles the velocity of which is close to the wave phase velocity in the medium. The Dispersion Equation for the Unmagnetised Hot Plasma. When pressure and velocity distribution of particles are taken into account, the dispersion relations for a two-component unmagnetised hot plasma can be written as (6): 1 + {I2
k2)~l
[l - φ(3) + νκ%
{l
2
O
2 1
k )'
exp (- z2)]
[l - φ(αζ) + itfiaz
+
exp (- a2z2)}
= 0,
(4.10)
for the longitudinal wave, and k22CΛ 2
2
= 1 -
U)l(]
pn [φ(ζ) - i^z
exp (- 2 2 ) ]
—2~ [Φ(^ 2 ) - ^π2<22 exp (-
a2z2)],
(4.11)
for the transverse wave. In these expressions φ(ζ) = 2z exp (- z2)
I exp Or2)
dx,
(4.11a)
o
"n 2 T
JkJ' The asymptotic expansions of φ(ζ)
into a series are
♦<«■*■- P f l - M · Φ(*) = 1 + 12iT
(4.11b)
h.
IA
15 SzE
for z « 1, for z » 1.
Although in derivation of (4.10)'and (4.11) collisions were neglected, the presence of imaginary terms ih~themv means the existence of attenuation. This attenuation is caused by Landau mechanism. The exponential terms in (4.10) resulted from the averaging of the charge carrier velocities with a distribution function which is
79
Waves in Plasma
assumed to be Boltzmann one (see Section 4 of Chapter I). When the average thermal velocities of particles are close to the wave phase velocity, i.e., when z « 1 and az Ä 1, it follows from (4.10) that the Landau attenuation is so great that it makes no sense to speak of the propagation of waves. If the thermal velocities of particles are small in comparison to the phase velocity of waves (s » 1, \iz » 1), we obtain from (4.10) in the case of long waves (t -+ 0, klD « 1) :
"Pn 1
Γ pn
f
2
«M 1
PP
k2v2n
(4.12)
In the case of a single-component plasma 3 where v^ - S ^ r / V n (ω = 0) the equation (4.12) can be transformed to (4.9).
4 = WV
For high thermal velocities of electrons (2 « 1) and their mass being much less than that of holes (which is typical of many semiconductors) it can be assumed for long waves that « 1 « az
and
kl
« 1
(4.13)
Then from (4.10) we o b t a i n *PP\ 2
7
k
1
. /7 hp 3n exp
(4.14)
Equation (4.14) describes a hole sonic wave, similar to the ion-sonic wave in a gas plasma in which the field of heavy particles (holes) is screened by that of light particles (electrons) and the Landau attenuation is thereby "screened" too. In this case the imaginary term in (4.14) is small owing to the quantities lp and P ω ρ ρ being much less than lD and ω , respectively. The collisionless Landau attenuation is associated with the longitudinal component of the wave electric field. With the helicon and Alfven waves it occurs when θ φ 0, i.e., for oblique waves. The theory of the Landau attenuation for the helicon waves is discussed in (7). The attenuation that appears during interaction of the magnetic moment of a charged particle M = mü2j2Bo with the parallel gradient of the variable magnetic field Bz is in a certain sense similar to the Landau attenuation. In this case the equation of motion of a particle is do dt
z = - M
dB z_ dz
(4.15)
This formula is identical with the equation of motion in an electric field (3.1) of Chapter I if M is replaced by the charge and dBz/dz - by the electric field. This phenomenon is called the magnetic Landau attenuation or the attenuation on transit time (9). The magnetic Landau attenuation increases with the increase of Θ, the angle between k and B0.
80
Plasma and Current Instabilities in Semiconductors 5.
HELICON WAVES IN METALS
Consider another kind of collisionless attenuation of waves in the plasma. From the point of view of an electron that moves counter to the wave with a velocity v the frequency of the field is increased by a quantity kv due to Doppler effect. In metals the Doppler frequency shift is determined by the velocity of electrons on the Fermi surface and can be quite large. It is responsible for the cyclotron resonance in a metal occurring at much smaller frequencies ω = ω
c
- kv„. F
(5.1)
This phenomenon is called Doppler-shifted cyclotron resonance. The condition of the existence of helicon waves in a cold plasma in semiconductors is the inequaliWhen ω = ω , strong absorption of waves is observed. ty ω In connection with the Doppler shift of the cyclotron resonance, in metals the region of existence of helicon waves is determined by inequality (11)
(5.2) ω
ρ^
ω
No helicon wave appears in a metal until inequality (5.2) is satisfied. To put it differently, there is a minimal value of the magnetic field B 0 for a fixed ω below which the wave decays and above which it can propagate. This limiting value of the magnetic field is called the Kjeldaas edge. If inequality (5.2) is satisfied with a large margin, i.e., if ω*α2 c (5.3) ω « 2
ω νΙ'
there is no spatial dispersion of the helicon wave, and the dispersion equation can be written in its ordinary form (2.18). When the margin in inequality (5.2) is small, the helicon wave exhibits spatial dispersion. Indeed, the electron mean free path I «. vFT:n anc* s i - n c e t n e condition of the helicon propagation is ω € τ+ » 1 and ω « ω , inequality (5.2) corresponds to the inequality kZ
» 1,
(5.4)
i.e., to the non-local regime when the wave length is much less than the electron mean free path. For the non-local regime of the helicon wave the dispersion equation assumes the form (10)
(5.5)
Xi where Ω = ω
*ΌΊ)
= 2n2
l +
ω - tv+ P
i - n z In 2n
(i + n) (i - n)
kvr
n=
(5.6)
Formula (5.5) differs from the usual expression for the local case (2.18) by the
Waves in Plasma
81
presence of the factor F(n). If the collisional term is discarded (v+ = 0 ) , the frequency of the Doppler-shifted cyclotron resonance (5.1) corresponds to η = 1. When η > 1, the logarithmic term in the factor F(n) makes it a complex quantity which leads to the wave number k being complex in the frequency range near and above the cyclotron resonance in metals. Therefore, in metals, in the Dopplershifted cyclotron resonance region there is absorption of waves in the absence of collisions. From the physics point of view such absorption of waves in a collisionless non-local case corresponds to transition of electrons from one Landau level to the next. Thus, for helicons in metals one observes, in addition to the collisionless Landau attenuation, the collisionless attenuation due to the Doppler-shifted cyclotron resonance. The Landau attenuation for helicons in metals will manifest itself when ω/k « v$ cos Θ and ω « ω and because of this, as the magnetic field B 0 decreases, the Landau attenuation may occur before cyclotron absorption. On the basis of measurement of Kjeldaas edge it is possible to determine Vp and to use helicons for the study of the electron properties of metals (11). Figure 2.13 depicts an example of dependence of the helicon passage through a metallic sodium plate on the magnitude of magnetic field. The Kjeldaas edge can be clearly seen.
6.
WAVES IN A SEMICONDUCTOR WITH ANISOTROPIC VALLEYS
The complex form of electron dispersion in semiconductors may reveal itself in the dispersion of waves. Thus, in semiconductors characterised by zinc blende structure there are three minima (valleys) in the function SW for the electrons of the conduction band: Γ, X and L (see Section 5 of Chapter I ) . Only in Γ-valley the effective mass is isotropic and only for the electrons of this valley the dispersion equations for waves in the plasma discussed in the preceding sections can be used. These dispersion relations can be also applied in the case of many valleys if the effective mass of the electrons and holes in these valleys is isotropic. Although the wave equation and the dispersion relations written in terms of G£j remain valid also for anisotropic valleys, the expressions for O^J themselves undergo an essential change. It turns out that the components or the electric conductivity tensor depend not only on the mutual orientation of the wave vector and the magnetic field Bo as was the case with the isotropic valley, but on the orientation of these vectors with respect to the anisotropy axes of valleys as well. Let us determine the components of the electric conductivity tensor for anisotropic valleys of the type X and L the minima of which are the absolute extrema in the conduction band of Si and Ge, respectively. In these valleys the surfaces of constant energy are ellipsoids (Fig. 1.11). It will be assumed that the mobility in these valleys is anisotropic the nature of this anisotropy being described also by ellipsoids. The mobility in the direction of the ellipsoid's axis of symmetry will be denoted by u^ and called longitudinal, and the mobility in the direction normal to this axis u^ will be called transverse. Appendix IV contains a calculation of the components of electric conductivity tensor fpr an individual ellipsoidal valley in the Cartesian reference frame. For Bo11 z these components for the valley S are given by (see Appendix IV, (10))
PCI - G
82
Plasma and Current Instabilities in Semiconductors
°xx{S) o
«nS 1 + v\ßV
=
(S) = qnn
, **
?n9,
(6.1)
ij2n
where
v
yy{S)
^
*+ h
- ^ ) cos2 0^(5),
=μ
=
(6.2)
^ ^ z + K " ^)cos 2 e s (S)].
In the above expressions Θ Α (5), θ^(5) and θ 2 (5) are the angles formed by the axis of symmetry of the ellipsoid corresponding to the valley S with the coordinate axes. For transverse waves and Faraday's configuration, when B0||z and Bo is directed along one of the main crystallographic axes so that σ ^ = σ ^ = σ ^ = σ ^ = 0, the yz zy zx dispersion relation coincides with equation (2.10) . xz
α
α
- a a
xx yy Solving (6.3) with respect to k2 obtain
= 0.
(6.3)
xy yx
and taking into account (2.21) of Chapter I, we
k2 = fy Xi - ^ωμ 0 σ β££ ,
(6.4)
where for Faraday's configuration
0" r.c = ^Γ
eff If σ
xx
= σ
yy
.σ
xy
2
= -σ
v
fa
) ±
+0"
yy}
xx
0
0
xy yx
+
Ό - σ xx yy
2
2
(6.5)
J
we have
yx σ ~~ = o ± %o . eff xx xy
(6.6)
According to (2.7), polarisation of transverse waves in the anisotropic medium is determined by E
E y
O rr
a
_x = ^£
a xy
=
eff
- O
yy
o
yx
(6.7)
The components of the tensor of the total electric conductivity are equal to the sums of the corresponding components of the individual valleys' electric
Waves in Plasma
83
conductivity tensor. In the general case the expressions for components 0^η· are quite cumbersome; they can be simplified, however, for certain directions or the magnetic field B 0 . An example of such direction is the symmetry axis of one of the ellipsoids. We shall give the formulas for the components of electric conductivity tensor for Si in which there are six J-valleys (Fig. 2.11), and the magnetic field is parallel to the symmetry axis of one pair of valleys. This example will be used to trace the peculiar features of the wave dispersion that appear in connection with the anisotropy and the presence of many valleys in the electron spectrum.
Fig. 2.11.
The distribution of constant energy ellipsoids in Si. The mobility along the symmetry axis of the ellipsoid is y-, in the transverse direction - y .
The reference frame will be chosen in such a way that the directions of the symmetry axes of the six ellipsoids coincide with the coordinate axes x, y, z (Fig. 2.11). The magnetic field will be assumed to be parallel to the s-axis. Then for valleys on the z-axis θ 2 = 0, θ χ = Qy = 90°; for valleys on the x-axis: ez = 90°, θ^ = 0, Qy = 90° and for valleys on the #--axis 0Z = 90°, θ^ = 90°, Qy = 0. By calculating the sum of the corresponding components (6.1) of the electric conductivity tensor for each of the six valleys we obtain
84
Plasma and Current Instabilities in Semiconductors x = qn{z)
°xx
1 + y2gg
+
l
i«W
<7*Û/)
1 - VtVtB\
a M
t t" (6.8)
qn{z)
yy
Ö
μ+
2S
= q n
^
+
T^W
V
l
+
^W
μ
^ ^
£
+
!+
^ ^
^
= qn(2)
V (6.8a)
y^0 ö
», ♦ <*00 1 + μ α,ΒΙμ * Γ
μ μ
y
Ί" Λι*Β*
+
u
ß
t Z ° 1 + i. i. E2
^ ^
y
+
y
ß
t t ° ^ 0 / ) i + n n ß2«
Here n(#), n{y) and n(s) are the concentrations of particles in pairs of valleys oriented along the axes x, y , z, respectively. In the absence of the deformation of the crystal in the state of equilibrium n{z) = n{x) = n{y) = n 0 /3, where rc0 is the total concentration of electrons in the conduction band. Then we have σ = σ , and the dispersion equation for Faraday's xy geometry (6.4) assumes the form xx Ί 2
ωμο 3ϋχ
ω
Vt{&VtB0
- i)
- i(Mz
1 + viBÎ
+
Vt)
* 2βμί.μιΒ0
(6.9)
V^Bi
1 ♦
By virtue of (6.6) and (6.7) the polarisation of waves remains circular. collisionless case (ωτ-> » 1, y œq/itm) ω) ^
-2-(3ω . 72
ω
*■ - ^ 2 Xi
+ ωυ
qn0
ο ^γ2-
2
ω et, - ω
23ω . -^- + et m,
2
In the
^L + S. (6.10)
et et
or 2 v κk -- rISr Xi
■4
2 2Βωet,ωpi ,
-El.
ω(ω
ρΖ * " p p
.
(6.11)
In these expressions the following notations have been introduced:
ct
=
A. BQ>
Ct
pt
C\m '
^
pi
= 3-2LJL
£irn '
(6.12)
One can see from equation (6.11) that there are two cyclotron resonances: u)C£ = ω and ω0^ω(:^ = ω 2 . The first of them corresponds to the resonance of particles in valleys on the 2-axis the effective mass of which in the plane of the magnetic field is isotropic and equal to m^ (Fig. 2.11). The second one corresponds to the resonance of particles in ellipsoids oriented "sideways" with respect to the magnetic field (valleys on the axes x and y , Fig. 2.11) in which the electron mass is anisotropic in the plane of the magnetic field. It can be said that the particles participate in the cyclotron resonance with the so-called cyclotron mass
85
Waves in Plasma
(6.13) m = Jm m when B0||z. J v c x y " The number of different cyclotron masses for a given direction of the magnetic field determines the number of cyclotron resonances. In the case under consideraίοη with Βο||ΟθΟ> there are two such masses, mQ^ = m^ and mC2 = Jm^m^* If o||
έ
In the case of an ellipsoidal valley, assuming that μ ~ 1/m, we obtain from (6.13) and (6.2) for the cyclotron mass for any mutual orientation of the magnetic field and the valley symmetry axis:
m
c
=
4
mm
t l
VmlCos*6
+Wt
sin26'
(6·14)
where Θ is the angle between the magnetic field vector B 0 and the symmetry axis of the valley. As has been shown in Section 2, for the isotropic mass the cyclotron resonance is observed only for the extraordinary wave (3 = +1)· It is obvious that in a valley which is isotropic with respect to the s-axis the cyclotron resonance also takes place only for the extraordinary Wc/e (the first term in the square brackets of (6.11)). And the resonance in anisotropic valleys (oriented along the axes x and y, Fig. 2.11) occurs both for the extraordinary (3 = +1) and for the ordinary (3 = -1) waves (the second term in the square brackets of (6.11)). This is understandable from the physical point of view since the trajectory of the cyclotron motion of particles in anisotropic valleys is elliptic. A circularly polarised wave can be presented as a sum of two waves with the right and left elliptic polarisation. Therefore, both for the ordinary and extraordinary waves there is a component of the drift motion of an electron that coincides with its cyclotron motion. Figure 2.5.V depicts schematically the function N2(ω) for the waves in the above case (6.11). The comparison of Figs. 2.5.V and 2.5.1 enables one to see that the dispersion of waves is changed by the anisotropy of valleys. The cut-off frequency for B 0 = 0 (which is obtained by setting the right-hand side of (6.11) equal to zero) is
w
c o = ί Ht+
w
y
Ξ
"P-
(6 ΐ5)
·
where qZnp and
(6.16)
3m m y
The quantity m^ is called the plasma effective mass or the conductivity mass. Not only u) c0 has been changed but the cyclotron resonance has appeared for the ordinary wave. In spite of the anisotropy of valleys, the polarisation of waves remains circular, which is not surprising since the total electric conductivity is still symmetric
86
Plasma and Current Instabilities in Semiconductors
with respect to the direction of the magnetic field. Naturally, in an anisotropic semiconductor the wave polarisation will be anisotropic. Let us assume that there is an ellipsoidal anisotropy of conductivity (a single valley) and that the magnetic field B0||z is parallel to the short axis of the ellipsoid. Then it can be obtained from (6.1) that
o
xx
=
L
—^r,
i + v^-jBV
σ
-
yy
t
-
l + vtvzBV
(6.18)
qnQ\it\i^B0 σ
xy ~ T + VtVzBl'
For the c o l l i s i o n l e s s case (ωτ+ » 1) P σ
335
= ^ω
Vcl
·
ω
xy where ω
ρίο = ω ρ* £ ι
and ω
«-,
pt
σ
yy
= ^ω
u
è t 'cl -
ω
^, (6.19)
ω^ω^
ρΐο = ω ρΐ ε »·
It follows from (6.18) that σ ^ Φ Oyy and, according to (6.7), instead of the circular polarisation of waves in Faraday's configuration we are dealing with the elliptic one. One can see from (6.19) that there is only one cyclotron resonance with the cyclotron mass mQ = ^πΐητη^-, but it occurs for both directions of polarisation, i.e., for the extraordinary as well as for the ordinary wave. A conclusion can be drawn from the above that the presence of the cyclotron resonance for the ordinary wave gives evidence of the existence of valleys with the anisotropic effective mass, regardless of whether the total electric conductivity of the crystal is isotropic or not. Note that in high magnetic fields (ω , « » ω) there are no more peculiar features of dispersion that are associated witn'the anisotropy of valleys in an isotropic or anisotropic crystal. Indeed, in this case equation (6.11) can be transformed to
ω2
1 +
Βωω
(6.20) c
and the dispersion law is independent of the effective mass.
7.
EXPERIMENTAL OBSERVATION OF MAGNETOPLASMA WAVES IN SEMICONDUCTORS
Consider the main experimental methods of observing magnetoplasma waves and the possibilities of using them in order to study the electron dispersion in semiconductors . Since a semiconductor sample is a bounded medium, it is obvious that the conditions at the boundaries will affect strongly the wave dispersion in it. As is clear a priori, the reflection from the boundaries will lead to the interference
87
Waves in Plasma of waves and to geometric resonances, as in ordinary cavity resonators. In what follows we shall discuss the simplest of such resonators, namely, an infinite semiconductor plate with parallel walls and a thickness d.
This case corresponds to real samples that are used in most experiments and represent plates with the thickness being much less than their other dimensions. It is possible to produce waves in such a plate if one subjects it to the action of a variable magnetic or electric field. Both the variation of the input parameters of the excitation system and the special systems for indication of the magnetic or electric field can be used for wave indication. The methods of observation of magnetoplasma waves in semiconductors discussed below are classified according to the techniques of excitation and indication of these waves.
=>A
={>«.
Fig. 2.12. a) the crossed coils method; b) the method of passing the waves through a plate.
Plasma and Current Instabilities in Semiconductors
88
The Crossed Coils Method. Figure 2.12.a depicts schematically these techniques. A semiconductor plate is placed in a variable magnetic field which is produced by a current J\ that passes through an inductance coil; the latter will be called the excitation winding. Electro-magnetic waves are generated in the plate. If the external constant magnetic field is absent, in an isotropic medium there is only 5^-component of these waves, and neither e.m.f. nor current will be produced in the second winding crossed with the first; the second winding will be called the indication one. In the presence of an external magnetic field B 0 such that BoZ φ 0 there will also be By-component of the electro-magnetic waves in an uncompensated semiconductor (see Section 2). In this case e.m.f. V2 will be induced in the indication winding. £L-component will be the greatest if the conditions of the experiment are such that a helicon wave is excited in the semiconductor plate (Section 2). For this reason the device for the observation of magnetoplasma waves depicted on Fig. 2.12.a is also called a helicon transformer. By analogy with the ordinary transformer, we have V2 = igM^x.
(7.1)
where g is a geometrical factor (the number of turns, the winding configuration, the volume, etc.), and Mj is the "magnetic permeability":
1 MTT = H #.
(7.2)
m
In this expression +d/2 +a/ z
Ti * k [ H1y y
d
)
dz
(7.3)
,
-all
and Hm is the magnetic field amplitude in the excitation winding. Let us determine how Mj is related to the parameters of the wave and the medium (11, 17). In a semiconductor resonator shaped as a plane-parallel plate charac^ terised by a high dielectric constant standing waves produced by waves with k||z appear. Therefore, no matter what the direction of the external magnetic field Bo is, only kz-components of waves form a standing wave, and it can be assumed that Bx = 0. The boundary conditions for the standing waves in a semiconductor plate are determined by the conditions of continuity of the magnetic field tangential components: H ^ e x p UL't
- k+ | ] + Ηχ_ exp
tl't
- k_ |
= Hm exp(tu)'t), (7.4)
H y*
exp Κ[ω'£ - k+ |
+ H _ exp U L ' t - k_ |
= 0.
The subscript "plus" corresponds to the extraordinary, and "minus" - to the ordinary wave. The subscript z near the wave vector is omitted for the sake of simplicity. One can easily see that, if the boundary conditions (7.4) are taken into account, the components of the magnetic field in the semiconductor are
Waves in Plasma B D
exp(iü)'t)
f+ - f,
X
B z
In these formulas /
cos
\
\d
f_
m
f+ - f.
COS
\ V
cos[k_z]
cos(k
z)
cos \— k_d
= 0,
|a|
and /
describe the wave polarisation x+
B ~ f+' y+
y-
Ί
cos f i k_d
e x p ( i a ) ' t ) Γ cos(fc s)
B B y
" / + cos(fe+g;
89
-î
J
(7.5)
f·
(7.6)
It follows from (7.5) and (7.3) that
M
'T
=
cos(kz)
]
cr+ - fjd
-d/2
cos
\k+d
cos(k_z) cos
— k_d
dz
(7.7)
L
or
Μ
τ-ΤΓ
- f )
1
jL M
tg
k+d
k d
tg
fc d
(7.8)
In the problem that we are dealing with we are interested only in the magnetic components of the wave in the semiconductor, i.e., only in transverse waves. For this reason, it is convenient to determine the dispersion and polarisation of waves on the basis of the wave equation for the magnetic field (equation (2.23) of Chapter I): k x (pk x H) = τωμ 0 Η.
(7.9)
In this expression p is the resistance tensor which appears in the equation connecting the total current and the field:
pJ = Î.
(7.10)
Assuming for the transverse wave with k||z that Hz - 0 and p(H0 + H) « p(II0), we obtain from (7.9) p k2H + p k2H = ίω\ι0Η , xx y ^xy x y
(7.11) p k2H + p k2H = - ·ζ;ωμο# . yx y yy χ χ Note that only four components of the resistance tensor determine the wave dispersion. The dispersion equation expressed through these four components of the resistance tensor results from setting the determinant of the system (7.11) equal to zero:
90
Plasma and Current Instabilities in Semiconductors
k2 =
j. i Ü2ÜL15UL ± [.
ωμ0 P
P
xx yy
- P
P
(p.XX
P
xy yx
)
2
w
P
P
f\
(7.12)
xy yx
The polarisation is given by
f
= -Jü =
2/
2P
ay
— (P. 2/2/
(7.13)
- P )2
) ± [(P
4p
Q
xy xyi
Now we can find the relation between Mj and the components of the resistance tensor and thereby to establish connection between e.m.f. produced in the indication winding and the semiconductor plasma parameters. By substituting (7.13) and (7.12) into (7.7) and making use of the series expansion for tg (i k d]
** [i k-d) - - Σ [[ikA+ ηπ - hπ] ·
(7.14)
we obtain xy
π2 p
yy
xx
Σ
(7.15)
γ=ι
l + H
Here ω = — V2 p p r \iQd ^xx^yy o =
(p
χχ
p
-
p
xx
p
xy yxJ
yy 0
- p p I , xy yx'
+
)^
(7.16) (7.17)
P
yy
and γ is an odd number. This property of γ reflects the fact that when the number of half-waves on the sample thickness d is even, the average magnetic field By = 0. The meaning of Q is the quality of the semiconductor resonator, and ω ρ is the resonance frequency which corresponds to the formation of a standing wave in the plate. Standing waves can be produced for relatively small values of frequency. This is associated with the possibility of the phase velocities of the extraordinary waves being much less than the velocity of light (Section 2) and, therefore, of their wave length being much smaller than the thickness of the semiconductor plate. The method of crossed coils is convenient for observing the low frequency helicon waves. The spatial resonances of such waves in semiconductors have been observed by many authors (11, 12, 14, 15, 16). Figure 2.18 presents the voltage on the indication coil as a function of the constant magnetic field which is normal to the plane of the semiconductor plate. Below, in Section 8, we shall discuss the information concerning the plasma properties which can be obtained from such experiments. The Method of Transmitting Waves Through a Plate. The schematic outline of the method is shown on Fig. 2.12.b. The excitation and indication coils are on the different sides of the semiconductor plate and normal to each other. It can be
91
Waves in Plasma
shown that the magnetic field H2 of the wave that has passed through a plate with a refractive index N and a thickness d is (12, 17, 27)
», = [1
t (N + /I/"1) sin
(*j
ML· o
HIX'
(7.18)
Here Hlr is the magnetic field near the plate on the side of the excitation coil. For helicon waves the refractive index can be as high as 10 2 . Therefore, in (7.18) the term N'1 can be neglected. After some transformations we obtain for the amplitude of the field that has passed through the plate (11, 17, 21):
** ■'-Ψ[ψ * [Ί M) TTd ^ To find the quantity M
[l hd
which in this case is given by
Mt
= ^
(7.20)
we again make use of the series expansion for the tangent.
4
-ίωά a
,,t T
(7.19)
Then
2P,xy
^xy , 4 V π2
2
^ μ0ω
£_j γ=1
P
C-Φ 1 +
^ + P^ ^ MS- - —
(7.21)
If the axis of the indication coil is parallel to that of the excitation one, the "magnetic permeability of the plate"
(7.22)
is equal to 2p
MTl -
-iud G
.
4^P
,/yy
d μοω
x x
oo
+
_i_
π2
Σ γ=1
(-l)Y pxx
+ p
yy
1 + ÎQ
·
i °^C
' ~"*
c
(7.23)
—
Note that in this expression γ may be even, too. The fact that in the absence of resonance in the plate M^ and M^ are not equal to zero (because of the term before the sum) corresponds to the penetration of the field through the plate due to skineffect. It is obvious that the geometrical factor g and, therefore, the signal at the output is much less in the case of the method of wave transmission through the plate than in the case of crossed coils method. The measurement of a small signal at the output is not, however, a major obstacle for the experimentalist since there is a high reference voltage on the excitation winding, which is coherent with the measured signal; this makes it possible to employ phase-sensitive rectification. In the method of wave transmission the interference between the input and the output signals is used for the indication of the wave phase. If the reference signal
92
Plasma and Current Instabilities in Semiconductors V = | 7 0 | sin ω£,
(7.23a)
which does not depend on the magnetic field is fed to the indication winding from the excitation one, it will be mixed at the output with the field-dependent signal (7.23b)
V{H) = 17^| sin [ω£ + α(#0)]>
leading to a typical oscillation pattern. Since the phase of the signal transmitted through the sample varies with the variation of the magnetic field, it at one moment is added to and at another is subtracted from, the direct transmission signal. The total voltage at the output is given by 27* = \V0\2
+ | ^ | 2 + 2\V
\VI
cos a.
(7.24)
The displacement of signals at the output is shown on Fig. 2.13 which presents the experimental results on transmission of helicons through a sodium plate (18).
0
J
Fig. 2.13.
L
20
UO
60
80
100 Bo, kG
The amplitude (in arbitrary units) of the helicon wave with a frequency 50 MHz that has passed through a sodium plate of thickness 0.48 mm. In magnetic fields below 50 kG no signal is observed because of strong absorption near the Doppler-shifted cyclotron resonance (18).
93
Waves in Plasma
Obviously, the method of interference of the direct signal and the signal transmitted through the semiconductor is a general one. It can be used for the measurement of the magnetoplasma waves' phase velocity in semiconductors as a function of the magnetic field also with other techniques of producing waves. There is an analogue of this method in optics, which is called the Rayleigh interferometry method. The phenomenon similar to the spatial resonance in optics is the Fabry-Perot resonance. Superhigh Frequency Methods. In this case the semiconductor plate is placed on the propagation line of an SHF signal and either the transmitted through or the reflected from the plate signal is observed. For wave transmission through the plate the amplitude of the transmitted signal is connected through equations (7.20) - (7.23) with the amplitude of the incident one, and geometric resonances take place under the conditions of weak attenuation. By means of these techniques the authors of (14) discovered for the first time the helicons in semiconductors. Figure 2.14 depicts the amplitude of a signal with a frequency of 10 GHz, transmitted through an InSb plate, as a function of the magnetic field strength.
7A
/.* Fig. 2.14.
B0
kG
The SHF-power P(So) that has passed through a sample rc-InSb of thickness 2 mm covering the aperture of the three centimetre range waveguide whose axis is parallel to the magnetic field, as a function of B0, in arbitrary units (14): 1 - the transmitted power P under the conditions when the direction of B 0 corresponds to the excitation of a helicon wave in the sample; 2 - the transmitted power for the reverse direction
of So; 3 - the initial section of curve 1 amplified by 12 db. When the attenuation of waves in a semiconductor is high and no geometric resonances are observed, the Rayleigh interferometry method is employed (16, 19).
94
Plasma and Current Instabilities in Semiconductors
A number of authors used the geometric SHF-resonance and interferometry for detection of Alfven waves in semiconductors (20). Such waves were observed for the first time in (21). The polarisation of Alfven waves is linear and their phase velocity does not depend on frequency (Section 3 ) . These properties are used in experiments for identification of Alfven waves. In the SHF-range the method of reflection is convenient. In the signal reflected from a thin sample the same geometric resonances are revealed as in the transmitted signal. When the method of reflection is employed, the requirements to the sample shape are simplified. As far as this method is concerned, the exact correspondence between the sample shape and the wave guide window is not necessary as it is in the study of wave transmission. It is sufficient that the sample is larger than the wave guide open end window. Besides, thick (semi-infinite)samples can be analysed by the reflection method. Figure 2.15 presents the dependence of
0 Fig. 2.15.
0.2
0A
0.6
0.8 1.0 B0J0* G
The reflection coefficient R and the amplitude U of the transmitted signal (in relative units) as functions of the magnetic field for a plate of n-InSb thickness d = 2.64 mm for / = 37.7 GHz, T = 77 K, n = 4.4*1011+cm"3, y = 4. l*105cm2/V-sec; 1 - the experimental curve i?(S0)i 2 - the experimental curve #(B 0 ) obtained when the impedances of the waveguide and the sample are matched by means of a λ/4 dielectric plate; 3 - the experimental curve i/(B0); the dashed curve is the theoretical one. The numbers 5-10 denote Fabry-Perot resonances (22).
95
Waves in Plasma
the signal reflected from an n-InSb plate on B0. Geometric resonances manifest themselves more clearly in the reflected signal than in the transmitted one. The method of wave reflection is the only available techniques for thick samples. In this case the reflection method amounts to varying the input impedance of the sample, which depends essentially on the type of waves in the plasma as well as on its parameters. Figure 2.16 depicts the reflection coefficient for a thick sample, in which a helicon wave was produced, as a function of B0.
R
t.OV
I
0 Fig. 2.16.
_
i
02
i
OU
""
i
i
0.6 0.8 B0,10* G
The power reflection coefficient iî as a function of magnetic induction for a "semi-infinite" sample with the following parameters: thickness d ~ 1.5 cm (the back wall of the sample is not parallel to the front one), n - 1.15*10 cm" 3 , p = 6.5-103cm2/V*sec, χλ = 17. Curve 1 corresponds to the extraordinary, and curve 2 - to the ordinary wave; the dashed lines are theoretical, the solid ones are experimental (23).
In addition to geometric resonances, Faraday- and Voigt-types of rotation of the polarisation plane, and also the cyclotron and other kinds of resonances considered in Sections 2 and 3 can be observed in the optical and the SHF-range. The cyclotron resonance forms the basis of the main method for measuring the effective masses of the charge carriers in semiconductors and it has been studied by many authors. Figure 2.17 shows an example of the amplitude of the signal transmitted through the semiconductor as a function of the magnetic field strength; one can see the cyclotron resonances in Faraday's and Voigt's geometry. Many authors have also observed the Faraday rotation of the polarisation plane, which has been used for obtaining information on the non-parabolicity of the effective mass and on heating of current carriers in semiconductors. The experiments on the cyclotron resonance and the Faraday rotation are mainly performed in the optical frequency
96
Plasma and Current Instabilities in Semiconductors
range, and we shall not dwell on them here.
Fig. 2.17.
The cyclotron resonance in n-InSb in Faraday's and Voigt's geometries for ω = 295 cm"1 (24). Along the ordinate axis the power P of the signal that has passed through the semiconductor is plorteçj. in normalised units: l 1 - k_|_B0_[E; min P corresponds to (ω£ + ω£)^> 2 - k||B0J_E, min P corresponds to ω = ω .
8.
HELICON SPECTROSCOPY
On the basis of studying the electromagnetic wave propagation in the plasma it is possible to obtain considerable information on parameters and the energy spectrum of free charge carriers. Various resonances in the optical range that are associated with quantisation of the motion of electrons (holes) by the magnetic field provide especially rich material. Different types of magnetic resonances enable one to establish the effective masses of carriers and the details of the band structure in semiconductors, the shape of Fermi surfaces in metals. All these studies initiated a new trend in solid state physics - the magneto-optics (25). It is difficult to observe magnetic and plasma resonances in the microwave range. For example, the necessary condition of the cyclotron resonance is ω = o)c and U) C T± » 1, and even under the conditions of very low temperatures the cyclotron frequencies turn out to be in the optical range or in the one close to it. Still, as has been demonstrated in (26), substantial information on the intraband structure of a semiconductor can be obtained from the studies of propagation of microwaves in a magnetised plasma. This can be done by means of techniques that are simple in comparison with magneto-optical ones. In this section we shall consider several examples of determining the charge carrier parameters in semiconductors by means of the helicon waves in the microwave range. Since these waves penetrate
97
Waves in Plasma deep into the conducting medium, they are most convenient for such studies.
The dispersion relations for the microwaves in the solid state plasma are specified by the electric conductivity tensor. By means of the experimental measurement of waves' absorption, reflection, phase velocities, resonances, polarisation and of their dependence on the geometry of the experiment and the external fields Eo and Bo it is possible to find the components of the electric conductivity tensor, and then the concentration distribution over valleys inside a band, the magnitude and the anisotropy of mobility inside valleys, the signs of the charge carriers. First, consider a semiconductor with one spherical valley. The main parameters that describe the properties of such a semiconductor, the mobility and the concentration of charge carriers in it are usually determined from the measurements of electric conductivity and Hall effect. We shall demonstrate how these parameters can be found from the measurements^of helicons' geometric resonances. In the case of isotropic mobility, when B0||z||k (Appendix IV, 12) we obtain p
xx
= p
yy
=
y30
1
p
qn]i'
= - p
^xy
=
^yx
.
an\x
(8.1)
K
J
The geometric resonance frequencies (7.16) in a plate made of such semiconductor are i
(1 ♦ V2Bl)h.
(8.2)
For \i2B20 »
1
(8.3)
we have
Ay2 4r>
(8.4)
where A
rad =-ll, = 4.91-1015 q\i0d2 ~ H x i U G-sec*cm3
As follows from (8.4), if one determines the resonance frequency (or the resonance field S 0 for a given frequency) one can find the current carrier concentration. In the crossed coils method the resonance values ω^ or BQr are obtained from the curve representing the dependence of the voltage V2 on the output coil (Section 7) on the signal frequency on the primary winding for a constant B 0 or on the magnetic field strength B 0 for a constant frequency. From the point of view of the experiment the second variant is simpler and more convenient since it requires no coordination of the circuits in a broad frequency range. In the general case, The maxima of Mj are determined from the maxima of V2. according to (7.15), the values of the frequency ω^ or of magnetic induction BQm corresponding to the maxima of V2 do not coincide with the resonance values of the It can be easily shown, however, by frequency u)r or of magnetic induction Bor. differentiating (7.1) with respect to S 0 or with respect to ω and by taking into account (7.15) and (8.1) that
98
Plasma and Current Instabilities in Semiconductors
MV)
dV2 dB0
dV2 du
and
V2(ur)
(8.5)
where V2 (Sop) anc^ ^2 (ωρ) are the values of voltage on the output coil that correspond to the resonance values of magnetic induction or frequency (12^. Equations (8.5) make it possible to find from the experimental dependence ^2(^0) for a fixed frequency (Fig. 2.18) or Κ 2 (ω) for fixed B 0 the quantities BQr or ω^.
B
oi
Fig. 2.18.
B
or Bom
B
0,W
B
o2
The dependence of the output signal Vz (in arbitrary units) on the helicon transformer made of alloy Bi0.8 9 S b 0 . n on the constant magnetic field S 0 for a fixed frequency f = 10 7 Hz of the current through the primary winding and T = 77 K; S o r is the magnetic field corresponding to the dimensional resonance, BQm is the magnetic field characteristic of ^ 2 max'
By transforming (7.17) we arrive at op
0)2
- U>i
(8.6)
(8.7)
Here S 0 2 and S 0 i are the values of magnetic fields corresponding to the points of intersection of a straight line that passes through the origin and through the point 0.707 V2{Bm) with the curve that depicts the function y2(#o)(Fig. 2.18). Figure 2.18 presents an experimental curve obtained for a fixed frequency. The values of ω„ and ω 2 - ωι can be found from a similar curve for a fixed magnetic field. Note that when y2So » 1
99
Waves in Plasma
and from (8.6) we obtain
(Bo
2 VBo
(8.8)
-*»w·
(8.9)
Thus, on the basis of measurement of curves representing the function ^ 2 ( B o) for U 2 £ Q >> 1 i1: i s possible to determine the concentration and the mobility of charge carriers in a semiconductor. In comparison with the method of measurement of Hall voltage and electric conductivity, the advantage of this method is that it is not necessary to prepare a sample of a special shape and, what is more important, the ohmic measurement contacts. To conduct the experiment, only a plane parallel plate is necessary, even if it is of an irregular shape. The shortcoming of the method is the limited number of materials for which condition (8.3) can be realised. It should be noted that when inequality (8.3) is valid, it can be assumed with a high degree of accuracy that ω^ « ω^ and Bor Ä 5 0 ^, where ω^ and Bm are the values of the frequency and the magnetic field which correspond to the maxima of V2 (Fig· 2.18). As the quantity \iB0 decreases, the difference between ω^ and ω^, and also between £ n y ) and Bnrn increases, and when \iB0 « 1, there is no resemblance Thus, for μ*£$ > 1 the maximum ot ΜΊ at all between the curves ω/Β0Γ and m/Bm. (and, therefore, of 7 2 ) corresponds to a condition (26)
2\s0d2 qn\x]
2
VKm- 1)
(8.10)
As the quantity \iB0 approaches unity, the curve oi/Bom experiences a decrease and for yBo = Ι,ω/B m = 0. At the same time, according to (8.2), the function u)/Bor exhibits an increase as \iB0 decreases, and for μ5 0 « 1 it is greater than its value at P 2 B Q >:> 1 by a factor of /2. Figure 2.19 presents examples of these
f/Bn r -Or
f/B om
MHZ kG
8.0 6.0\ 4.0\
J_ n
2.0
0 Fig. 2.19.
1
2
3
4
5
The quantities / / B „ r (1) and f/Bm tions of μΒ 0 for rc-InSb.
6
μΒ0
(2) as func-
100
Plasma and Current Instabilities in Semiconductors
dependences. Equation (8.10) makes it possible, after having determined the value of B m for several values of ω under the condition of relatively small \iB0 >, 1, to 2 2 5m until the curve intersects with the axis extrapolate the dependence of ω 2 on μ K ^ \iBQ and to find the mobility (Fig 2.20). On the basis of the slope of the straight line in the plot of B2Qm as a function of ω2ζ (Fig. 2.20) one can, according to (8.10), also determine concentration. Thus, the experimental measurement of the quantity Bm as a function of ω for \iB0 ^ 1 is another method for obtaining current carrier concentration and mobility.
fztn MHz
ZOOMr
W00Y-
0
2 0.2 8L,10*G om*
at
Fig. 2.20. The dependence of B2m on fz Hgo.82 Te for T = 180 K.
for Cd0.i8
In thin or sufficiently pure samples the value of ω ρ determined from (8.2) may turn out to be quite large, such that (8.11)
ω « ω In this case (7.15) yields T
,'
8u
0^2(jd
π1*
P
O P y
xx yy
3#
(8.12)
- O p
xy yx
For a spherical valley .. Τ
. 8μ 0 α 2ω "% ΉΗ
8\ι0ά2ωςη\ι xy
The function (8.13) has a maximum ,at \iB0 = 1·
B
\ιΒ0
(8.13)
Y measuring the dependence of V2
101
Waves in Plasma on Bo under the condition (8.11) it is possible to find the mobility. Thus, the measurement of helicon concentrations and mobilities of tial factor is that although the valid, the frequencies for which
geometric resonances enables one to determine current carriers by various methods. An esseninequality \ιΒ0 ϊ 1, i.e., ω τ » 1, should be the measurements are conducted may be quite small.
Many-Valley Semiconductors. The charge carrier concentration and mobility in each valley, as well as the asymmetry of mobility in valleys and their mutual orientation determine the gaivanomagnetic properties of many-valley semiconductors. For spherical valleys, if the condition (8.11) is satisfied, equation (8.13) in the many-valley case assumes the form Μ
Ί
τ
8y0d2uy?ny f μ ^ ρ π* [l + ν\Β\
μ2£ρ 1 + ν\Β\
·'·]'
C8
·143
and if there is a noticeable difference between the values of mobility in valleys, in the experimental dependence of Vz on Bo each sort of carriers will correspond to its own maximum. Although formulas (7.15), (7.21), (7.23) and (8.12) remain valid for anisotropic valleys, the expression for the components of the resistance tensor becomes more complex; this leads not only to quantitative changes of the dependences of Mj and ω^ on B0 but to qualitative ones as well. One such qualitative change has already been discussed in Section 6; it amounts to the fact that for anisotropic valleys the cyclotron resonance occurs not just for the extraordinary wave but also for the ordinary one. Equation (8.10) can be demonstrated to remain valid for the anisotropic ellipsoidal valley as well, if μ is replaced by a certain parameter μ that can be called the cyclotron mobility y
c=
y
*iyy
for
B||Z
(8.15)
by analogy with the cyclotron mass mc = Jm^my. The cyclotron mobility depends on the orientation of the magnetic field with respect to the axes of the ellipsoid of mobility. It describes a certain mean value of mobility in the plane normal to the magnetic field B 0 . If the direction of the latter is parallel to the ellipsoid's axis of symmetry, the electron mobility in the plane normal to the magnetic field is isotropic and equal to μ^. When B 0 is normal to the symmetry axis of the ellipsoid, we have
yc = ^ψΐ-
(8.16)
The measurement of y c based on geometric resonances for \iQB0 > 1 (8.10) and various orientations of B 0 with respect to the crystallographic axes makes it possible to determine the mobility components μ^ and μ^ (Appendix IV, 3 ) . Figure 2.21 shows examples of the experimental determination of μ . One can also find the structure of mobility in a valley by measuring the dependence of ur/B0 on B0 in the transition region of magnetic fields. As an example, consider n-Ge semiconductor (26, 29, 30). In the presence of only a helicon wave the real part of the helicon frequency ω^ in n-Ge, corresponding to the wave length λ = 2d, i.e., to the geometric resonance ω ρ , is
102
Plasma and Current Instabilities in Semiconductors
BorJ0*G 2.0 2.5
r
2.0
δ,ont Fig. 2.21.
2.5
The ratio f/BQr as a function of B with only the helicon wave taken into account (dashed curves) and f/B as a function of Bnm (solid curves) for n-Ge (n = 1.49-10 15 cm" 5 , y 0 = 2.101* cm2/V sec, K = 10, T - 80 K) (28). The curves are obtained by calculation, the points - from £he experiment: k|| B01|^110>; 1 - P||<100> and |P| = 1.1-103 bar; 2 - P = 0; <1Î0> and = 2.03-103 bar.
- ?
103
Waves in Plasma
B*n.\A AS
V2.Bl + 1
.
^ 2'
B^n.vi y 2 .* 2 + 1
+
A^vffl
+
(8.17)
i
where Ύ2π2 c7Po^2 γ . = y Ku J i T 1 s i n 2 6 ^
2 t
(8.17a)
cos20. + ΐ ) ( μ . ) " \
+
(8.17b) 2
y. = μ μ
+ y (y
2
- y-) cos Θ .,
X = —,
^ and θ^ is the angle between the symmetry axis of the i-th
ellipsoid and the s-axis.
It is possible to assume, up to the accuracy of 5%, that the parameter 2
S = 1-4
σ
- σ
χχ
yy
σ
k 2
(8.18)
xy
in n-Ge for ïc||<100>, <111> and <110> in the field ran^e 5 0 «0-2.10 ,+ is equal to unity (29). The changes in the dependence of BQr on ω^ = ω^ that are associated with the valley anisotropy are determined by a parameter γ. Figure 2.22 presents the quantity BQr as a function of ω^ = ω ρ for various values £f the mobility anisotropy and for various orientations of valleys with respect to z||Bo » the frequency ω^ being calculated from (8.17). Curve 3 corresponds to the equivalence of all valleys in n-Ge (i.e., they all have the same cyclotron mass) and to Bo||
104
Plasma and Current Instabilities in Semiconductors
1.5 2.0 B0 JO* G Fig. 2.22.
The dependence of ω on B$v for various struetures of valleys in the semiconductor (26): -3. 1 - a single spherical group, n0 - 1 0 1 5 cm" 2 - two spherical groups, μ χ = 101* cm2/V«sec, y 2 = 2.5-101* cm2/V*sec, nx = nz = 0.5; 3 - a single group consisting of four ellipsoids, y c = 2Ί0 1 * cm 2 /V # sec, n^ = 0.25 n0; 4 - two groups containing two ellipsoids each, y c = 2Ί0 1 * cm2/V*sec, η; = 0.25 n0.
For example, in n-Ge for B0||k||<110> the components of the electric conductivity tensor turn out to be dependent on rotation of the reference frame around the axis
XX
yy
xy
σ
cos φ + σ Ύ
xx σ
sin2 φ + σ Ύ
xx σ
xy
sin
yy
+ — Kίσ
2
cos2
(8.19)
yy yy
- σ
) sin2
xxJ
In these formulas φ is the angle between the a;-axis and the crystallographic direction <001>. In the direction B0||k||
105
Waves in Plasma
= σ;yx> and the angular dependence disappears. Under the condition ω « ω ' ï ~ ^xy> a n d t n e v ° l t a g e o n t n e indication winding 7 2 (see Fig. 2.12.a) varies as the semiconductor plate rotates around the direction of B 0 (Fig. 2.23)(26, 33)
J L xy r
60° \
Ό0 v V V
vx/
^
\
» ™ ^
«
/
"f^2 - k||B||<ÎÎ0>; 3 - k||B||
£70'
It is possible to determine on the basis of the angular dependence of Vm the ratio of the difference between the diagonal components of the electric conductivity tensor to its non-diagonal component: V (φ
45°) - V (φ m V (φ (Fj m
0°) 1
σ =
yy
- σ 2σ xy
χχ
(8.20)
Equation (8.20) allows one to find the anisotropy parameter K from the known y c and vice versa, u c from the known K. The angular dependence of V2 can be quite pronounced for large K. Figure 2.24 shows the function V2(Φ) for Bi (31). We have considered several examples of applying the crossed coils method for determining the intraband structure. Obviously, other techniques, such as the method of transmission, reflection, etc., (see Section 7 ) , can be also used for these studies. For instance, Baynham (32) used helicons in p-Ge to distinguish between heavy and light holes. All this confirms the high potential of the helicon spectroscopy in the studies of the band structure in many-valley semiconductors .
106
Plasma and Current Instabilities in Semiconductors
Fig. 2.24.
The functions ν^(φ) in polar coordinates obtained in the experiment for pure Bi0.9 2 Sbo.os (dots) and the sinusoidal functions V = \V0 + A sin 2φ| (solid lines), superposed on the experimental points for / = 0.8 MHz; 1 - So = 0-3 kG, B0 > 0; 2 - Bo = 0.3 kG, Bo < 0; 3 - Bo = 0 (the scale of curve 3 is fifty times greater than that of curves 1 and 2 (31).