H eterostructures such as single heterojunctions and quantum wells w ith sem i-insulated substrates may be approxim ated by an overlayer w ith a semi-infinite su b strate. In this case the elastic constant tensor and m aterial density of the h etero stru ctu res can be described in the form
C « H (r) =
C{jue ( z
3) + ( C S H -C*u )e(x
3 - Zo) (B.25)Pm(r)
= pi<ä(x3) + (/>B - p i)@ (x 3 - ■ (B.26) Now th e differential operators can be expressed as follows;L<°>( V M = 0 (*3)L(^ ) + 0 ( x 3 - zo)(L(B) - L(y,)) (B.27) where
< u? -
vf kj +
0ik t(vf -
v?)£-3
^
0 ) = 0 a;2 -v2k2
+v2£?
0 vik*(vt - v? ) j k
0 * R T 1 cs = 1 (N 3 / 0ik
((B) =
0 u;2 -v,2k2
+ v,2£
s 0 0 a.2 - i f fc2-
i f andB*0*(fc||ai|x3) = <S(
x3)B<a> + 5(
x3 -
z0)(B<b > -
B<A>)(B.28)
where B ^ ) = 0 i v 2t kI, \ ° \ *(w? - 2 w 2 )fc || 0 0 / B (ß) = « ;2a fe 0 i v ? h \ 0 * K 2- 2 i f ) & , 0 v 'l 3 7 7 / A ,4 R R 2 C11 2 C44 2 _ £ u /2 £ 4 4 v l = — , h - " V P m P m r m » = P m r m w ith
B. Elastic Green’s Function
104From the equations
(B.10)
and the boundary condition operator(B.28),
we can obtain the Green’s function as follows(1) 0 <
x3 < zq,
0 <
x'3 < z03n(fc|h^ I x 3, x 3) = g [ f ( k h u> | x 3,x'3)
+ A te
~a,x3 +
B tea , X 3 + + B,e°“X3(B.29)
3 i3(fc|l,w
I
x3, x 3) = g [ f ( k „ , wI
x 3,x3)+ G te~°“X3 + H,ea,X3 + G,e-°“X3
+
H,ea'X3(B.30)
g - n i h , wI £
3,^
3) =
Q22\ k 11,w I
x 3,x3) + E te~a,X3 +(B.31)
gsi(kv u I x 3, x ’s) = g i i ’(kt, u \ x 3, x 3)
-
Yt(A ,e-°“X3-
B tea,X3) - r ,(A ,e- “ ' 1 3-
B ,e a‘X3)(B.32)
g33(k t ,uj
I
x3,x3) = f f3 3)(fc||,ojI
x 3, x 3)- r,(G ie -0'13 -
H tea,X3) - T ,(G ie-a,X3 - Hie°“X3)(B.33)
(2 ) 0 < x3 < z 0 , x3 > z0
gu(h,u>\ x 3,x'3) = R,(e~°‘x3- A2e - “'13 + Aie a‘X3)
+
Si(ea,X3+ Aje- “'13 -
A 2e°“X3)(B.34)
<713(fc„w I x3,x'3) = X ,(e- “ ' 1 3 - A2e- “ ' 1 3 + Aie “‘13)
+
Y,(e°m+ A1e"“'13 -
A 2ea,X3)(B.35)
g22( h
, I
x3,x'3)=
Vt(e~°“X3 +e“'13)
(B.36)
<73i(fc|,w
I
x 3,x'3) = R , ( Tt (A2e - ° ‘X34-Aie“<13)
-r,e~ “'*3)
+ 5i(r,e“'13 - r,(A 1e“a,X3 + A2e“*13))
(B.37)
S33(fc|,w I
x3,x'3) = X , ( r t(A 2e-°“X3 + A- r , e- “‘13)
+ Vi(r,e“'13 - r i f A ^ - “113 + A2e“113))
(B.38)
(3)
z0 < x3,0 <
x 3 < z0g n ( h ^
I S 3 ,4 ) =
R',e~a',X3 + ^ ' <X3(B.39)
B. Elastic Green’s Function
105S22(fc||,W I * 3 , 4 ) = vt e “ 'I3 (B.41)
S 3i(*i,w I * 3 , 4 ) = - ( r j Ä j e - “ !** + r jÄ je - “ !*’ ) (B.42)
g33(kh u> I 13,4 ) = - M e - i ' + r ;x ,'e - “ ^ ) (B.43)
(
4)
< ^
3,
-
2-0<
40ii(fc|l?w I x3,x' 3) = g [ i \ k h u I
z3,4) +
A't e~a>tX3+ Aje""»®3
(B.44)# i3(k|l,u; I z3, 4 ) = flfi3 )(^ihw I ^ 3 ,4 ) + G'te~atX3 + Gje“ ®«*3 (B.45)
^22(^11, w I z3, 4 ) = flf22)(fcihw I ® 3,4) + E[e~a^ 3 (B.46) ^31(4 ^ I ^ 3 ,4 ) = s4?}( % ^ I ® 3,4) - (r ^ i e_a!/tX3 + rJAJc""**3) (B.47) <733(4 , “ I * 3 , 4 ) = < 7 ^ (4 ,“ I * 3 , 4 ) - ( 4 G ; e - ^ 3 + r j G J e - ^ ) (B.48) w here
^ ( fc l.W I * 3 ,4 ) = - ( 4 ? )le" “ ‘|13^ ' “
I * 3 , 4 ) = - ( ~ > 9 n ( x 3 - 4 ) [ e - “ 'l" - '» l - e-« .l-3 -^l] g ^ \ k ^ \ x 3,x'3) = - ( ~ ) e - a‘ ^ - ^ Zatvtd f ß i ’ V I *3,4
) = - ( ^ ) Äsn(*3 - 4 ) [ e_“ ‘|I3_lil - e_“ ,|X3_i3i]
gii\k„ w I *3 ,4 ) = - ( ^ ) [ e' “ ,|X3" X31 -
ji]
5<f»(fc|ha; I * 3 , 4 ) = - ( 4 4 e~“ ;N " 13' - r e - “ :'13^ 1] ZttjLJ*
d l V i h ’ UI *3 ,4 ) = - ( ^ ) 3 f f « ( * 3 - 4 ) [ e-“ ‘1*’ ” *’ 1 -
S,22)(fc| , “ I * 3 ,4 ) = - ( 4 ^ e_0':|l3” lt31
I * 3 , 4 ) = ~ ( ^ ) s g n ( x 3 - 4 ) [ e - “ i l - 3 - ; i _ e-« ;|x3 -x;|]I *3 ,4 ) = - ( ^ 4 e' “ :|l3~131 - f e-“ «’1**-'»1]
( k j —io2/ vf ) ^ i i k ^ > u 2/ vf —i(u)2/ v2 — k2) 2 if k2 < ijJ2/ v2 <4 4
“ )B . E la s tic G re e n ’s F u n c tio n 106 T t
r,
= { { k«(- ■ f ! : ‘ ]Ll
if)2
o. (m = { a't { k t ui) = ( =~ kE
( f c j - oj2/v't2) i - i ( u2/ v't2 - k \ ) la ,(i||w )a i(A ||U ))
fc?
r =
i f k 2 > lo2/ v 12 i f < tv2/ v ' 2 i f > to2/ v 2 i f A:J < u>21v2 i f k 2\ > to2/ v ' 2 i f k 2^ < to2 /v 12 a t ( k ^ W i ( k ^ ) k t ' u 2p j + c jAot2 - c j k 2\ v a «(c12 + C44)^|| / + p/ 0 /( ^ 2 + C44)k \\ ) ’A = i f — — — 1
A'
1 2 ^ r t n j ’ 1 = i — i a _ i ff2 _ 2 t, + nA
A', =
. Z ' + cf4 « i2 - cfl^N <*t(c?2 + « ! (c?2 + c £ )fc |Ir l; _ n{
2 Tj
n:1
1(11 + 51]2 tj n;J
T ^-fa .-ffcu A ),
t; = -(<*; - tfc„r;)
t; = -(aj - tfc,rj)
n,
= c f2fc|| - i c ^ a , r , , n ; = c ® /,■„ - h:® ft',!’', 11/ = c^2^h — ic ^ a /F /, flj = c \ 2 ^ w ~T h e coefficients in equations (B .2 9 ) to (B .4 8 ) w ill be d e te rm in e d fro m th e fo llo w in g b lo c k m a tr ix e q u a tio n w h ic h was o b ta in e d fro m th e b o u n d a ry c o n d itio n s fo r the G re e n ’s fu n c tio n s : / A 0 0 0 0 0 ^ ( Xl ^ ( Yl \ 0 A 0 0 0 0 X2 y2 0 0 H 0 0 0 X3 y3 0 0 0 Q 0 0 X4 y4 0 0 0 0 Q 0 X5 y5 \ o 0 0 0 0 K j \ X * J e ) (B .49 ) w here A is a 4 x 4 m a tr ix , H is a 2 x 2 m a tr ix , Q is a 6 x 6 m a tr ix , K is a 3 x 3
m a tr ix , Xi, X 2, Yi, and Y2are sin g le-colum n m a trice s w h ic h have fo u r elem ents, X3
and Y3are sin g le -co lu m n m a trice s w h ic h have tw o elem ents, X4, X5, Y 4, and Y5are
B. Elastic Green’s Function
107
matrices which have three elements. Their elements are give by
4(1,1) = c £ , [ T - T i(A2e-»'*» + A ^ “' 20)]
4 ( 1,2) = —c^4[Tie“|J° - T ^ A je -“1« + A2e“« )]
4(1,3) = - c f 4T ;e -“!« ,
4(1,4) = - c ^ T J e - “!2»
4(2,1) = n (e -a'2° - n,(A 2e -“12» - Aie“'2”)
4(2,2) = Ilje“'20 + n ((Aie- “' 20 - A2ea,2°)
4(2,3) = —Ilje- “*20,
4(2,4) = - n j e " “!20
4(3,1) = e"“'20 - A2e -“‘2» + Aje“'2»
4(3,2) = e“120 + A1e -“,2° - A2e“12°
4(3,3) = - e ““!20,
4(3,4) = - e ““!20
4(4,1) = r,(A 2e -“‘20 + Aje“'20) - r ,e - “‘2°
4(4,2) = C e“12» - r t(A1e - “,*> + A2e“|2°)
4(4,3) =
H{ 1, 1) =
H ( 2, 1)<3(1,1)
<3 (1, 4)
<3(2,1)
<3(3,1)
<3(3,3)
<3(3,5)
<3(4,1)
<3(4,3)
<3(4,5)
<3(5,1)
<3(5,4)
<3(6,1)
<3(6,4)
* ( 1, 1)
K (2
,1
)# (3,1)
- r ; e - “'>2" ,
4(4,4) = —r(e- “!20
= c44a,(e“*20 — e~“,2°),
# (1 ,2 ) = cf4a'(e_“>20
= (e“,2° + e - “*2»),
H( 2,2) = - e ~ ° >
= T ( ,
<3(1,2) = —T(,
Q (l,3) = T ,,
= - T , ,
<3(1,5) = <3(1,6) = 0,
= <3( 2,2) = n (,
Q(2,3) = <3(2,4) = n,,
= 4 T , e - “120,
<3(3,2) = —c44T (ea,2° ,
= 4 T , e - “‘2»,
<3(3,4) = - 4 T , e “'2»,
=
- c f 4r 'te~a‘z°,
<3(3,6) = - cf4r;e -« i2” ,
= n ie - “12»,
<3(4,2) = n 4e“‘2»,
= n ,e - “'20,
<3(4,4) = II(e“120,
= — n'(e- “>20,
<3(4,6) = —Ilje- “!20,
<3(2,5) = <3(2,6) = 0
— ß~atz o
— gQflZO<3(5,2) = e“,2° ,
<3(5,3) = e - “‘2“ ,
<3(5,5) = —e - “'>2° ,
<3(5,6) = - e - “!2» ,
C e - “*2»,
<3(6,2) = - r te“'2» ,
<3(6,3) = r ,e - “'2» ,
- r , e “‘2» ,
<3(6,5) = -T 'te~a> ,
<3(6,6) = - I V “!2» ,
1,
# (1 ,2 ) = —1,
# ( 1,3) = 0,
c44o te““‘2" ,
A (2 ,2) = - c 44o (e“'20,
A'(2,3) = - c f 4o(e_a«20,
e"“*20,
A'(3,2) = e“,2° ,
# (3 ,3 ) = - e " “!*»,
B. Elastic Green’s Function
108* i(l) =
-§ e-»;W-*o)]
^ ( 2 ) = -sV K cfi -
- (cf, - ^ rcf2)oJe-“:W-*o)]
Ki(3) = - 2 ^ ? ( e' “'(^ ' zo) -
yi(4) =
- e"ai^3-^o))
K2(l) = icf4[(2 Ä )e-« i(4 -") _ ( ^ L ) ( l + | r ) e - “!(*3-o)]
V2(2) = - ^ [ ( 4 - cf2)e-«;W -o) + (cs _ ^ c f O e - iW - » ) ]
y2(3) = - i - ( e - a't^3-2o) _ e-a|(*'-zo))
y2(4) = i ( e ' " H Ir J:o) - f ,e“ori(*3-*o))
>s(l) =
,
y3(2) = - ^
e' a'tK_2o)
*i(l) =
*
4(
2) = s ! r M ^ 4 - 4 i ) e '“|I3 +
< * ( (4- 4 ) e_“,li]
^ ( 3 ) = 4 [ ( ^ K
c" (20“lJ) - § e - “'(4o-xJ)]
>4(4) = ^ r [ a i ( 4 - c^2)e -“,(*o-^) _ a / ( 4 - ; | 4 ) e~“'<20~l2)]
F4(5) =
- ^ e " 01^20-^ ))
F4(6) =
L(e- at^ °-x3) _
e ~ M z0-^ ))
y5(l) = z i [ a t(l + ^ ) e ~ “^3 _ 2 aie" a«*i]
n(2) = -J*[(4 - c&)e-^ + (4 - ^cft)e—'l]
F5(3) = * 4 ( ^ ) M 1 + ä ) e-«.(3o-xJ) _ 2 a,e-“'*2»-^)]
n ( 4 ) = « ^ [ ( 4 - 4 ) e - “-(2»-ri) + ( 4 - ^ 4 ) e - “‘(30-x;)]
r5(5) = i^L.(e-“,(2o-*3) _ e-«i(«-*j))
K5(6) =
~
^ ( l ) = - 5 ^ e - “^ ,
r 6(2) = | | e - “-(2» -^ ),
^ ( 3 ) =
.
Xi = ( Ä , ,S ,, 4 , A j ) ,
X2 = (X,,Y
X4 =
(At, B u Ah B i X tX i ) , X5 = (Gt, H t, Gh Hh X't, X[),
X6 = (Et, FtX ) -
APPENDIX C
Effective Source Force Density
The source force due to electron-phonon interaction depends on the direction of propagation of the phonons. Therefore the effective force for a specific m ode can be obtained using coordinate transform ations. It is easier to use the effective force in (x',y',z') coordinates (as shown in Figure 5.1) where the elastic G reen’s function can be decoupled into LA and TA phonon parts. F irst of all, we need to decouple the force into LA and TA p arts using the phonon coordinates (x,y,z) where the phonons are propagating along th e ^-direction [see Figure 5.1]. In th e phonon coordinates, th e x- ,y~, and com ponents of the force density F" are the effective force densities for LA, TA i, and TA2 phonons, respectively. Using the transform ation m atrix U ,
th e source force densities in the crystallographic coordinates (X,Y,Z) and in the coordinates ( x \ y ' , z ' ) are related by
F'(qh x 3,uj) = U F (q ,|, £ 3 ,c j) (C .l) and using the transform ation m atrix R, the relation between the source force den sities in th e phonon coordinates (x,y, z) and in th e coordinates (x', y \ z') is
F '(9:„ x3,a ;) = R - I F " (q ,X 3 ) u)) (C.2) where F, F', and F" are th e source force densities in th e (X,Y,Z), ( x ' , y \ z ' ) , and (x , y, z) coordinates, respectively. Here F" is
r
I _p(TAi) \ f ( t a2) y JP(LA) ^
(C.3)
where is the effective force density for LA phonons, and F lTAll and F^TA2) are th e effective force densities for P- and H- TA phonons. The source force density
C. E ffe c tiv e Source Force D e n s ity 110
F its e lf can be easily o b ta in e d in th e c ry s ta llo g ra p h ic coordinates. T h e n th e source
force d e n s ity F" in th e phonon coordinates is
F " = aF = RUF =
^ cos 9 cos (j)Fi + cos 9 sin <j>F2 — sin 9 F 3
— sin (f>F\ - f cos (f)F2
y sin 9 cos (f)F\ - f sin 9 sin (f)F2 - f cos 9 F 3
(C .4 )
O nce th e e ffe ctive source force densities fo r each phonon m ode in th e phonon coor
d in a te s are o b ta in e d , th e source force densities fo r each phonon m ode in ( x ' , y ' , z')
co o rd in a te s can be evaluated using e q u a tio n (C .2 )
F' = R_1F"
/ cos0F(TAl) + sin6»F(LA) \
i?(TA2)
^ — sin # F (TAi) + cos #F1l a ) /
(C .5 )
In o th e r w ords, th e effe ctive force densities fo r each phonon m ode in (x', y' , z')
co o rd in a te s are / sinl9F(LA) \ p/(LA) _ 0 (C .6 ) \ cos 6>F(LA) / cos #jFdTAl) pfT A x) = \ — sin 0F^TAll F'(t a2) _ ^ (t a2 0 (C .7 ) (C .8 )
T h e re fo re we can ca lcu la te disp lace m e n t vectors fo r each phonon m ode using equa tio n (5.56) and th e above re la tio n s.
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