The object of this thesis has been to show the effect o f the small scale structure o f a superlattice on the transport propreties. In each chapter, results were presented for both acoustic phonon limited transport and polar optic phonon limited transport. The acoustic phonon calculation is the simplest, where the relaxation time approximation holds. This was used to gain an insight into the behaviour of the electrons in the superlattice under various conditions. The polar optic calculation was more involved, and two methods were used. The simple relaxation time approximation, which is not valid for
this case, and the iterative approach. The iterative is the exact calculation and the relaxation time method was used to demonstrate its own failings due to the inelasticity of the scattering and the anisotropy of the miniband structure. This was shown particularly clearly in the Hall factor calculation. In this calculation, regimes could be found where either the scattering or the miniband structure dominated the results, and the effect of each could be looked at individually.
Finally, the method used for the Hall factor calculations is as important as the results themselves. This is a novel method and could be used for many other systems. It is particularly useful if the relaxation time approximation is not valid and the Jones-Zener expansion cannot be used. Using this method, little further work is needed if the zero magnetic field Boltzmann equation has been solved for the system in question.
A P P E N D I X A
Symmetry of the Transport Tensors <7,y and Cjyj,
In the effective mass approximation, a superlattice has all the symmetry of the crystal point group 4/mmm, ie. a fourfold rotation axis and three mirror
( °
1
!
U l i)
Th e transport tensors and C.y* transform in the standard way when considering a symmetry operator /*/.
° \ j = U k l j l ° k l
/ , l i ll j m l k n s-,
Cijk = |~ lm*
I f the conductivity tensor <7,y is transformed under mirror plane m „ it follows that,
<7i2 = ( — l)(+ l)< 7 ij = —012 = 0 (^2)
Similarly, it can be shown that the only non-zero components are those with repeated indices, and these can be represented by two parameters.
<7u = <722 = <7|| (i43a)
(Ala)
(A16)
the operation o f of the
(¿ 3 6 )
For C ijk, the transformation under the operation of the the mirror plane m s produces the result
c „ , = ( 1){_ | j (+1>c ' » =
~c“‘
= 0 (.44) Similarly, any component with repeated indices will be zero. For the remaining six terms, operating with the rotation axis 4, gives,Cuj = — Cjia
Cjsi = — C |3J
C321 = — C312
Finally, the conductivity tensor has Onsager symmetry.
< M B ) = (A4o) (.448) (A4c) (X5) where = ct,j(0) + C ljkB k + dijkiB kBi + . . . (¿ 6 )
Therefore it is also true that,
B ) = O ji(0) - CjutBk + djikiB kB i + . . . (¿7 )
Comparing equations (A 5), (A 6 ) and (A 7 ) gives
c i it
=
- c i it(.48)
So, using this result along with the relations in equation (A 4 ) the Cijk can be expressed with just two independent parameters.
C m = —C m — c Cj3i = Cj u — — C321 — - C13J — i 76 tensor (A9o) (A9b)
V
A P P E N D I X B
F o rm o f g ,(k ) w h en the a p p lie d electric fie ld is p a ra lle l t o th e layers For polar optic phonon scattering, the iterative equation for g (k ) is
a ; 8,; ( ! ■ ■ ) _ { E . V t /„
»■+i(k ) - --- J « . J J '; ~ rh - A > t o lw A / » * l/ y W IW + t / « T » / » l)
(fll)
where 7 and N are constants, and the limits r * are given in table 3.1. The Fermi-Dirac distribution function /o(k) is independent of $, ie. /o(k ) = fo{k^,kt ) where kn = (fc* + For E = (£ ,0 ,0 ), E • V h/0 = Edfo dk, E d td k , = - E0fo(\ - h2 = - E P f0[ l - f o )— fc|| cos0 (B 2)
Therefore, the first iteration is proportional to cosfl, and can be written in the form,
*i( 1 0 = 3i(*ii, * . )cob*
Hence, the upper integral for the second iteration is,
(B 3)
Jo J r * •
\ *j> + *} + (*i
-it,)* - 2k(|t|| coa(fl - »<) J
where
F ( * „ * , ) = (1 - /o)[W + 1/2 =F 1/2] + /„[AT + 1/2 ± 1/2] (B 5) The order of integration can now be reversed as the k'M limits are independent o f O', and the integral can be written in the more transparent form,
■ - L « C
*■ 1 - t Z h i)
(B 6 ) whereA =S,(*J, *=-)
b
=*j*+ *} + (*; - *.)J
c=2*1*11
A simple change o f variables transforms the integral into,
/ = / d t ' . r J r* Jo 1 6 — c cos 0 I = / d k ' j ” ^ { A c . » " c ° . » - A . u , . " . i n » | J r* Jo y b — ccos ff ) (B7o) (B ib ) (B io ) (B 8 )
Since *■ antisymmetrical about 0 = k, that part of the integral vanishes, and the integral reduces to,
I ^ cob0 x f
Jr*
(B9)
Therefore, I a cos 0, and thus ¡72(h) oc cos 0. So, by induction, it must also be true that 93, 94, 95 etc are also proportional to cos 6. And so, the iterative equation for 9(h) with the applied electric field in the plane of the superlattice layers can be written in the form,
2 * 7 /r± dkft { * £ [ (>2_ *2)772 - l ] } + \ E P f Q{ l - /o)£j*|| 2,4+1 = 2*7/ ,* dkfM and 9.P0 = * « ) « * * (BIO) ( i l l ) 79
A P P E N D I X C
Symmetry of G o
It has been shown previously by simple arguements (Butcher, 1973) that the collision operator C is symmetrical in an integral, ie.,
/ i(k )C M k ) dk = f M k )C j(k ) Jk (C l)
where g (k ) and h (k) are arbitrary functions of k. Now, let us define some new functions such that,
j ( k ) = C - 'l ( k ) (C2a)
h (k ) = C ~ , m (k ) (C26)
Inserting these definitions into equation (C l ) gives
f ( C -1l)k ))C C -1m (k ) dk = f ( C - 'm ( k ) ) C C - ‘ i(k ) dk
Thus, putting Go = C -1,
f m (k )G 0l(k ) dk = f l(k )G0m (k) <ik
(C 3)
(C 4)
and Go is also symmetric.
A P P E N D I X D
Analytic Calculation of the H all Factors for a Special Case
The calculation here gives an analytical expression for the Hall factors when the following approximations are used.
(i) The relaxation time is a function o f energy only, r = r (t). (ii) The statistics are degenerate.
(iii) The miniband structure is h2k?l
i(k ) = - — Jf- + A (1 - cos kzd) 2mJ
W ith a relaxation time r(e) defined, the solution to the Boltzmann equation is,
*=-«E-v^V(t)
(Dl)
Inserting this solution into equations (5.36), (5.37) and (5.40) gives us the expressions for f
, tj, ox, oy
andaM.
For the degenerate case, with the Fermi level at the energy t f , the integral for f reduces to,
r = ^ / * , i « I * * * . { ( £ - » • ) ^ cos kt d r2( e ) i ( t f ) J (D6)
* / '( ^ f ) dk' {
co*k-i i(t/)}
where U = + A (1 — cos kMd) — ep.
The 0-integral is simple, and the fc|| integral follows from the properties of the ¿-function to give
{ = — 2)r,^ 4 ‘ J _ kt d k - {(< / - A (1 - co« k , d ) ) co. k .J } (D7)
where fci is the maximum value of fc, on the energy surface e = ep. The kt integral is straightforward, and the result is,
r = ( 2 (‘V A ) « ° M + £ . i ~ a M + * . A ) (0 8 )
with
A?i<f = cos-1 ^1 — ; «/r < 2A
=tt ; «#• > 2 A (¿79)
Inserting the value of ki into equation (D8) gives
e3A 2dr2(tp )
2tr*fl4 ep < 2A
e3A2dra(<jr) ; tp > 2A (¿710)
A similar calculation gives
t aA r » (t y ) f r i£ f
* ~ r ’ h 'm id V2
csA ra(cy) _ A
irh2m^d V A /
and for er* and a„, the results are
and
\ ( F\ (
■
‘r
!
s * '
r
' 1
\
1 A .eaA a<fm*r(cy) .
™ 2tt/14
The only other parameter needed before the Hall factors is the electron density. This is given by,
' - x ? i h d k For degenerate statistics, this is
„ = - i / <Ck 4jtsJ t < t r ; t F < 2A eF > 2 A (Z ? ll) ; eF < 2A > 2A (£>12) J ; £/■ < 2 A p > 2A (I>13) in be calculated (£>14) (£>15)
; ejr < 2 A
m* A
nfl2d
( f-0
; « j >2A (£>16)
From these expressions, the Hall factors follow. For ep < 2A net] 1 (^>17)
Ad]m; {c” ‘ i1 - ¥) - (l - Ì ) [¥ (2 “ ¥)] 7 }
» ’ { [ « ( i - t ì r + t ì f-*)«--1 ( » - * ) }
and r = m = r" 1 r*1» a J (D19) results are r xyM = 1 (£>20) A J 'rn ; /<, r" * 2d1 U V (D21) r . , , = r , " i (£>22) 84<
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