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固体物理笔记

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Solid state phyiscs

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1 �N¤�nØ

�N�SàåA�Ü8Ïu>f�K>Ö��fØ�m�·>�p�

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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 =

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

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