固体物理笔记
DESCRIPTION
Solid state phyiscsTRANSCRIPT
�NÔn)P
1 �N¤�nØ
�N�SàåA�Ü8Ïu>f�K>Ö��fØ�m�·>�p�
^"^åéSàå�k�fK�§�kÚå�±�Ñ"ÏLØÓ�â�µ�
�U!��uå±9�d�"¤¢SàU§´�3ýé"Ýeòò¬N©
)��åá��!·��¥5gd�f¤I��Uþ"3lf¬N¥¬
�U´�ò|¤¬N�lf©)��åá��!·��gdlf¤I�
�Uþ"
¬N(Ü�Ì�a.µ
1. ����d.µ·��§äk4Ü>f���¥5�fÏL�>Ö©
ÙÞák'�����då�f/(Ü3�å"
2. lf.µÅz?§>fdwx�f=£�®x�fþ§dd/¤�l
fòÏL�Klfm�·>áÚå (Ü3�å"
3. 7á.µ?§d>følwx�f§/¤úkz>f/°0§�lf
ÑÙuÏm"
4. �d.µ7f�§¥5�f´ÏL¦��>f©Ù��UÜ© (Ü
3�å�"
1.1 lf�£±NaCl�~¤
lf¬Nd�Klf|¤§lf�d·>�p�^�)"
1
1.2 �d�£±�©f�~¤
1.3 7áThe Drude Theory of Metals
Kinetic theory of gases treats the molecules of a gas as identical solid
spheres,which move in straight lines until they collide with each other.
Drude applied the kinetic theory of gases to a metal,considered as a gas of
electrons.Assume that there are two kinds of particles in metal:positive par-
ticle and negative particle.Drude assumed the positive charge was attached
to much heavier particles,considered to be immobile.
Drude @�|¤7áü���f§Ù�>fil37álf|¤��>
Ö�µe§¿��>Ö�þ���ud>f§�±@�Ù�±ØÄ"
When isolated atoms condensed to form a metal,the core electrons remain
bound to the nucleus to form the metallic ion,but the valence electrons are
allowed to wander far away from their parent atoms.In the metallic contex-
t,they are called conduction electrons.
The ”gas” of conduction electrons of mass m move against the background
of heavy immobile ions.
1. • Õá>fCq£the independent electron-electron approxima-
tion),�Ñ-EL§¥�>f->f�p�^
• gd>fCq(the free electron approximation)�Ñ-EL§¥
�>f- £�¤lf�p�^
Õá>fCq3Nõ�¹e´éÐ�Cq§�´gd>fCqI��
�ï"
2. >f-lf-EÓÌ�/ "-EÌ�8Ïu>f-lf-E Ø´
>f->f-E§¯¢þ>f->f-E´¤kÅ�p¡�Ø���
�"
3. • ü �mSu)-E�VÇ 1τ§τ¡�µþ�m§½ö-E�m§
½ö²þgd�m
• 3�ãá���mSu)-E�VÇ�dtτ
• �Å]À��>f§lT��m©§>f²þ��1�mτ�â¬u)e�g-E"½ö`§3u)�g-E�§²þ�²
L烪2gu)-E"
2
4. >f�±��¸��9²ï����ªÒ´-E"-E���)��
Ý���´�Å�§��ÚÛ��§Ý��"
7á�>{
E = ρj (1)
>6�Ýj´��²1u>Ö6Ä���¥þ§���ü �mSR�
ÏLü ¡È�>Ö��"XJþ!>6IÏL�ã�Ý�L§�¡È
�A���§@o>6�ÝÒ´j = I/A.��üà�>³��V = EL,dª
£1)�V = IρL/A§¤±R = ρL/A.ü NÈSkn�>f§�ܱ�Ýv$
ħ@o
j = −en(vdt)A
Adt= −nev (2)
ª(2)¥�j¢Sþ´À>6�Ý">f�$Ä�,ÏÃÙ�§bXØ\>
|§�����$Ä�p-�§À>6�ÝÒ´0.3\>|¥§>f�²
þ�ÝØ´"§���Ú>|����"
3>|¥�>f¼��>|\��
−eEtm
§ >fdu-E¼���Ý´�Å�§¤±oNþ-E�Ý�p-�§
vk�z"¤±>f�²þ�Ý´− eEtm �²þ"ùp�tlþ�g-Em©
O�§¤±Ù²þ��τ .
vavg = −eEτm
j =
(ne2τ
m
)E (3)
½Âσ = 1/ρ,
j = σE σ =ne2τ
m(4)
ª(4)Jø�«��µþ�mτ��{§
τ =m
ρne2(5)
p(t) −→ total momentum per electron at time t
current density −→ j = −nep(t)
m(6)
3
t −→ t+ dt,the probability of colliding before t+ dt is dt/τ
the probability of suffering a collision is 1− dtτ
external force −→ f(t) will give an additional momentum f(t) + O(dt)2 to
electrons which experience no collision.
�u)-E�>fÓo>fê8�'~�1− dt/τ§¦��z�Äþ�
p(t) =
(1− dt
τ
)[p(t) + f(t)dt+O(dt)2
](7)
�Ñp��§ª(7)�±�¤
p(t) = p(t)−(dt
τ
)p(t) + f(t) +O(dt)2
u)-E�>fÓo>fê8�'~�dt/τ , �3-E�>f��Ý�
�´�Å�§¤±z�-E��>féoÄþ��z=�lf(t)¥¼�
�Äþ§Ù��< f(t)dt§¤±XJ�Ä-E>f��z§�¬\þ�
�(dt/τ)f(t)dt§´�����þ§�±�Ñ"¤±·��±�e
p(t+ dt)− p(t) = −(dt
τ
)p(t) + f(t)dt+O(dt)2 (8)
ª(8)¥�ÄþOþ�)¤k>f��z§�¤�©�/ª
dp(t)
dt= −p(t)
τ+ f(t) (9)
ª(9)`²>f-EÚå�o�J´O\��{Z�"
1.3.1 ¿��AÚ^{
Figure 1: Schematic view of Hall’s experiment
4
XFig.1¤«§>fÉ�âÔ[å�K�§¬÷-y¶�� £§>fæ
È3��ý¡þ§/¤-y���>|"
ExÚjx�'´^{
ρ(H) =Exjx
(10)
Halluy^{Ú>|Ã'"
î�>|EyÚâÔ[å²ï§¤±§AT�'u\^|HÚ÷x���>
6�Ýjx"¤±½Â¿�ÏfXe
RH =EyjxH
(11)
16f´K>Ö��¹e§RH´K�¶16f´�>Ö§RHÒ´��"e
¡O�RH .Äké�3>|©þEx, Eyäk?¿�,^|3z����¹e§
éA�>6�Ý©þjx, jy"
f = −e(E + v ×H/c) (12)
dp
dt= −e
(E +
p
mc×H
)− p
τ(13)
3ð>6¥>6Ø��mCz§@opx, py÷vXe'X
0 =− eEx − ωcpy −pxτ
0 =− eEy + ωcpx −pyτ
(14)
Ù¥
ωc =eH
mc(15)
3ª(14)ü>¦±−neτ/m§¿�5¿�j = −nev§,�ÒU��
σ0Ex = ωcτjy + jx
σ0Ey = −ωcτjx + jy(16)
Ù¥σ0´ª(4)3Ø\^|�¹e��"-ª(16)¥�1��ªf¥�jy�
"§��
Ey = −(ωcτ
σ0
)jx = −
(H
nec
)jx (17)
¿�ÏfÒ´
RH = − 1
nec(18)
ª(18)`²¿�Ïf��6uá��16f�Ý"�´¯¢þ¢�¥uy¿
�ÏfÚ^|§§ÝÚ�¬�XÝÑk'X§ù´DrudenØ)ºØ�"
5
1.3.2 7á�6>�
b½k����mCz��6>|
E(t) = Re(E(ω)e−iωt) (19)
ª(9)AT�¤dp
dt= −p
τ− eE (20)
·��éª(20)���½)§äk±e/ª
p(t) = Re(p(ω)e−iωt) (21)
rEê/ª�(21)(19)�\(20),·���
−iωp(ω) = −p(ω)
τ− eE(ω) (22)
>6�Ý�±�¤j = −nep/m§|^ª(22),k
j(t) =Re(j(ω)e−iωt)
j(ω) =− nep(ω)
m=
(ne2/m)E(ω)
(1/τ)− iω
(23)
P�6>��
σ(ω) =σ0
1− iωτσ0 =
ne2τ
m(24)
K
j(ω) = σ(ω)E(ω) (25)
3�3®�>6�Ý��¹e§·��±�eð�d��§∇ ·E = 0
∇ ·H = 0
∇×E = −1c∂H∂t
∇×H = 4πc j + 1
c∂E∂t
(26)
37᥷��±ÏLª(25)rj^EL«
∇× (∇×E) = −∇2E =iω
c∇×H =
iω
c
(4πσ
cE− iω
cE
)(27)
½ö���¤
−∇2E =ω2
c2
(1 +
4πiσ
ω
)E (28)
6
ª(28)äk��ÅÄ�§�/ª
−∇2E =ω2
c2ε(ω)E , ε(ω) = 1 +
4πiσ
ω(29)
ª(29)¥0>~ê´��Eê§XJ>|ªÇvp§÷v
ωτ � 1 (30)
3��Cqe§ª(24)(29)�Ñ
ε(ω) = 1−ω2p
ω2(31)
ω2p =
4πne2
m(32)
ª(32)¡��lfªÇ"XJε´¢ê��u"(ω < ωp),�§(28)�)3�
m´�êP~�§ØUD¶�ε�u"�§�§(28)é�)´��f�
§§7á´�Bß�"
1.3.3 7á9�
½Â96
jq = −κ∇T (33)
kïÄ���¹§UìDrude�b�§>fÏL-E5���Û�����
§Ý" ux?>f��´lx− vτ?L5�§äk(n/2)vε(T [x− vτ ])�U
þ¶��´lx + vτ?L5�§äk(n/2) − vε(T [x + vτ ])�Uþ§üö�
\�Ñ
jq =1
2nv [ε(T [x− vτ ])− ε(T [x+ vτ ])] (34)
jq = nv2τdε
dT
(−dTdx
)(35)
Ï�gd§l = vτ�~�§éª(34)3x?�Ðm��ª(35) qÏ�⟨v2x⟩
=⟨v2y⟩
=⟨v2z⟩
=1
3v2
ndε
dT=
(N
V
)dε
dT=
(dE
dt
)1
V= cv
�±r96��
jq =1
3v2τcv (−∇T ) (36)
7
éìª(33)§�±��
κ =1
3v2τcv =
1
3lvcv (37)
(37)/(4)§��κ
σ=
1
3
cvmv2
ne2(38)
rcv�¤3nkB/2,mv2/2�¤3kBT/2£�ân�íN²;nؤ§,���
κ
σ=
3
2
(kBe
)2
Tκ
σT=
3
2
(kBe
)2
(39)
ª(39)ÚWiedemann-Franz Law¬Ü"
>f��Uþ3§ÝFÝ��^e/¤96§�´>f´�>�§dÞ7
,¬/¤>6§du·�?Ø�Ø´4Ü£´§>f�ª¬Èà3�à§l
/¤��>|§�>|��Ú§ÝFÝ����"ù�>|Ï~��
E = Q∇T (40)
���Q§·��IJïG�e�:x?��¹§T:?du§ÝFÝÚ
u�²þ�Ý�
vQ =1
2[v(x− vτ)− v(x+ vτ)] = −τv dv
dx= −τ d
dx
(v2
2
)(41)
-v2 −→ v2x�⟨v2x⟩
=⟨v2y⟩
=⟨v2z⟩
= 13v
2§¤±
vQ = −τ6
dv2
dT(∇T ) (42)
>|Úå�²þ�Ý�
vE = −eEm
(43)
²ï^�e
vQ + vE = 0 (44)
Q = −(
1
3e
)d
dT
mv2
2= − cv
3ne(45)
1.4 ��u壱ý5íN¬N�~¤
ý5íN�f�>f�����W÷§3gd�f¥>f>Ö�©
Ù´¥é¡�"Xd�5§>f>Ö�·>³3¥5�f±��fØ>
Ö�·>³-�"bX�f�>Ö©Ù´f5�§K�fmØ�3Sàå"
8
• áÚ�p�^�´§�f�pa)ó4ݧù«ó4Ýò���f�m�áÚ�p
�^"�����.§�Äü��å�R��Ó�5��f1Ú2§z
��f�k���>Ö(+e)Ú(-e)§XFig.2�K>Ö�m�ål©O
�x1Úx2§âf÷x¶�ħÄþ©O�p1 Úp2§å~þ�C"3Ã
�6��¹e§TXÚ�M�îþ�µ
H0 =1
2mp21 +
1
2Cx21 +
1
2mp22 +
1
2Cx22 (46)
Figure 2: ü��f���I«¿ã
b½�u)ÍÜ�z���fäk����ªÇω0§ù�Au��{
��fªÇ"l kC = mω0"-H1L«ü��f�m�¥Ô�p�
^U§AÛ /XFig.2¤«§Øm�I�R"u´
H1 =e2
R+
e2
R+ x1 − x2− e2
R+ x1− e2
R− x2(47)
3|x1|, |x2| � R�Cqe§ò(47)ªÐm§B���$?CqL�ª
�
H1∼== −2e2x1x2
R3(48)
ÏL{��C�
xs ≡1√2
(x1 + x2) xa ≡1√2
(x1 − x2) (49)
)Ñx1Úx2
x1 =1√2
(xs + xa) x2 =1√2
(xs − xa) (50)
9
Ó��H1�(48)ª�Ñ�Cq/ª§K�±¦XÚ�oM�îþé�
z"Ù¥§eIs Úa©OL«$Ä�é¡�ªÚ�é¡�ª"? ·
��±���ùü«�ª�éX�ÄþpsÚpa§=k
p1 =1√2
(ps + pa) p2 =1√2
(ps − pa) (51)
3?1ª(50)(51)�C��§o�M�îþH0 +H1�±��
H =
[1
2mp2s +
1
2
(C − 2e2
R3
)x2s
]+
[1
2mp2a +
1
2
(C +
2e2
R3
)x2a
](52)
�ª(52)§��ÍÜ�f�ü�ªÇ§§�´
ω =
[(C ± 2e2
R3
)/m
]∼= ω0
[1± 1
2
(2e2
CR3
)− 1
8
(2e2
CR3
)2
+ · · ·
](53)
Ù¥ω0 = (C/m)1/2§3ª(53)¥§®ò²��Ðm"TXÚ�":
U�12~(ωs + ωa)"du�3�p�^§ù���'�ÍÜ��2 ·
12~ω0$∆U ,
∆U =1
2~(∆ωs + ∆ωa) = −1
8
(2e2
CR3
)2
= − A
R6(54)
w,§(54)ª´��áÚ�p�^§§Uìü��fmålR�K6g
�Cz"T�^¡�����d�^§5 uó4f- ó4fÍܧ
¿Ø�6uü��f�>Ö�Ý��U"
• ü½�p�^ü��f�p�C�§>Ö©ÙòÅìu)�U§��f��vC
�§du�|Ø�N�n§ü��f�þfêØU���Ó"éu4
���f§XJ���)�U§7Lk>f�-u���Óâ�pU
�â�±"Ïd§ù«�Uò¦XÚ�oUþO\§ é�p�^�
Ñü½�z"ù«ü½�^�±^��²�úªB/R125£ã"éÜ
ª(54) �Ñ�áÚ³§·��±�ÑLennard− Jones³§
U(R) =B
R12− A
R6(55)
10