晶体的能带理论

1、n1a1 n2a2 n 3a3h i 2 V0 V(r)= H?0 #' 2mh22m2屮0(r) Vo,o)(r)二E(0)(k)? (0)(r)宇 r(0)(r ) kr reikr,E(0)(k)h2k22 me'k(r)二XU) ik(1)(r)二k0r：H k k'-(0)k' E(0)(k)-E(0)(： Jk®(r)第23讲晶体的能带理论续一.在周期势场中，晶体电子的能量本征值1.弱周期近似V(r)=V(r+ RJ, Rn h22 rc ' V(r)二 2mV。，H?、vrh 22m二川甲 r，orvr严 ror*dvE(k)二

2、E(0)(k) E(1)(k) E(2)(k), (QE(k)二 0)|2rrrh2kE(k)二 E(0)(k) E(2)(k)二2-+ Z.2mek' E(0)(k) E(0)(k')AHkk'E(k)g k k=±色2一兀/a7t/aFree electron gasFree electron 雪吕編 and difTraction at BriUouiti zones罟爭牛0晋警罟一图豹 一维晶格申能和破矢女间的关采Probability densityiiStanding v/ave 1'Standing wave Z /厂Distributi

3、on of probability density in the periodic potential for sta nding wave 1 and 2The standing wave 2 piles up electrons around the positive ion cores, which means that the average pote ntial en ergy will be lower than for a free traveling wave (constant probability density). The pote ntial en ergy corr

4、esp onding to sta nding wave 1 will have higher pote ntial en ergy tha n a free traveli ng wave, si nee it piles up electr ons betwee n the ion cores (not compe nsated by positive ion s). The en ergy differe nee betwee n the sta nding waves is the origi n to the energy gap E g.2.紧束缚近似 Tight Bin di n

5、g Approximate ndelocalised statePutting a lot of atoms together: solids : atom ； molecule ； solidEnergyantibonding sbondi瑚 |ibonding santibondingbondi瑚 |ibondingantibonding pZ :antibonding pZ :valence band from p tending orbitalsconduction band from antibonding s orbitalsconducTonbani from anlitondl

6、ng p orbitalsvalence band from 耳 bonding orbitals紧束缚近似也称为原子轨道线性组合法LCAO )Lin ear Comb in atio n of Atomic Orbitals (ENERGY LEVELSELECTRONSHO. 5NO. 4NO. 3NO. 2NO.1ISOLATED ATOMELECTRON'SENERGYV(r) = V(r 十 Rn),R. =+ W3:梟2皿Hhrh2 ' 2Vatomd)2mVALENCE BAIIOCONDUCTIOH BAIW =FORBDDEN BANDATOM IH A SO

7、UD4 H?atom H?'H?atom =ht2m2Vatom(r),H?'= V(r)V(r)-Vat°m(r)H?atom i(J 2 Vatom(r )IL 2m")=ENERGY LEVELSIr)Rmam i(r - Rm) where *(r-Rr) i(r - Rm) dv 二Q v j(r) = uj(r) eik rk '， k '，_1_* NeikRmr 1Y(rneik(r-Rm)-eik®(r+RnRm)r r 1.S(r RJ= N e1ikRmr r reik (Rm -Rn): eiNeikRnle

8、ik Ri(r - R)=eikk(r)一2mr 1 r r r rr 1 r r r rV(r) N jkRm""E(k) NpikRmi(Rm)ZrR,r reikRmr reikRmr reikRmrir-E(k)+V(r)®i(r-Rm) = 0h 2IL 2mh 2r r rrI-7+ Vatom (r) E(k)+V(r) Vatom ()-2m一rr ri -E(k) V(r)-Vatom(r) 1(r -Rm)= 0l(r-Rm) = 0*(r)： e'旳Rmr r'.、ik RmerRmr rik Rmmrri -E(k) 7(：

9、)宀如(：Rm)i(r - RJdOi® - Ek川件rtr - RmW 川件rvr-Vat°mr - RmFir - Jdv=000eRm-'0rrr 2i -Ek +川7宀如肿心| dv+召JR * r rr r r rm 1(r)V(r)-Vatom(r-Rm) i(r-Rm)dvrr . r rrr r 2i -E(k)-J(0) + 瓦J(Rm)eikRm=0, where J(0) =-川V(r)-Vatom(r)pi(r)| dv, a ndm :nearest*、：ineighbersJ(RJ(r)V(r)-Vatom(r - RJ ' (r

10、 - Rm)dv called asoverlap integrals.cos kza)E(k) = ；j - J(0)-、J(RJeikRmFor S states in a simple cubic lattice?E(k)八s 一 Jo 一 2JMcoskxa coskyan.n.Fig. 7.6 Comparison between the LCAO bands for Ge, computed with 肯n sp basis, and the free electron bands. From Harrison (1980).<原子轨道线性组合法得到的 锗的能带图与自由电子能

11、带图的比拟 >二.在周期势场中，晶体电子的能量本征值图示1.能带的结构的构成r r r r r r r o h2 -2 rV(r)=V(r+RJ,尺=山印十 “282+ “383, R+V(r)2muk(r)二 E(k)u：(r) =F?k(rE(k(r), 一亡 V+ZlTh+ + V(川.2m2m一2. 2V(rr)、 hkv )2mr r ro r r rQu：(r)二 u：(r &)= To confine H?'(k, ,r) uj(r)二 E(k) u：(r) in a cell.r r rrThe solutions could be u,k(r), u2k

12、(r),u3k(r),L ,un k(r) and their eigenvalues? r X r r r r G(kYr)uk(r) = E(k)ukr(r)Where H?'(k,v,r)=h22 rC 2ik2mE1(k), E2(k), E3(k), L , En(k), where n is the indexrof the energy bands.The energy bands are En( k)f for different n ¥ and different ofk = -nrb1 + 匹b? + 匹b3. For the samen and diff

13、erent k, they are just in one energy band NiN2N32.能带En(k)1的对称性(stop here)巴(&在倒空间中的周期平移对称性：巳()二En(k _Gh)也(/的时间对称性：En(k) = En(-k) r r r r rr-A)B)H?'(k, Jr) u【(r) = E(k) u：(r)IL 2mh2 、 2-c2IL 2mh22IC_ 2m2ik-2ik2ikrv)2 2+ h.k_ +2m2 22mrV(r) %(；) = E(k)ujr)+ V(r)'uk(r E(k)uk(r)+ 乎+V(r)"u

14、(r)二 E(-k)u.(r)2m斗斗u：(r)二 u(r)E*)二 E(-：)二 屮 n k(r) and 宙 n j; (r) aredegenerate.B' /的时间对称性的推论：在布里渊区边界处有警=0For 1-D lattice, E(k)二 E( k)1dE(k) dE(-k)d(-k) dE(-k) 与2d(-k)dk d(-k) dkdE(k)dkdE(-k)d(-k)k Gh2E(k) = E(k 厲戸 dE(k)-dE(k Gh)k dkdk2Gh , dE(k)dkdE(k)少 d(k)k 亠 2GhdE(-k) d(-k)_ dE(k)“匕 d(k)2dE(

15、k')d(k')1k'/dE(k)d(k)k 亠；GhdE(k) d(k)dE(k) kGhd(k)2kGhdE(k)dk=dE(k) d(k)dE(k)d(k)C)En(k)1在倒空间中的点群对称性：En(k) = En(?k)，其中:?代表晶体所属点群的任意对称操作。r r rrF?(r)甲 n,k(r) = En(k)Jk()rrLet n,?(r) = n k(°?r), as°? is symmetric operationrRd) n,?(r)= En(k) n,?d)；)?叩希"rkn,k5+凡冷=_:?(rRn)Q k :?

16、R is a number, so ：?_1(k :?Rj 二 k :?R 垐 垐rat垎 k R r rr r r r?'(k ?Rn)?*k :?一?Rn = 7枭 Rn? r( :?f：?operatiofrn,kn,k：-Rei k Rn 心=eik症农空 nk°?i<?£ Ab “c =e n,kr(?r):?(rRn)(r)= ?= ?1kThe conclusion is 空 n,k (为卜)= n(r), and En(k)= En(bk), or En(°?k)= En(k).r r n, k-n,?(r%) 一 en,?n,3.在

17、倒空间的布里渊区中表示能带lEn(k)?的方法由于：A) fEn(k)1在倒空间中的周期平移对称性：End) = En(k _Gh)B) lEn(k)：的时间对称性：En(k)二 En(k)C) En(k)1在倒空间中的点群对称性：En(k)二Enc?k)First Brillouin Zone2 -/a4 /a6 二 /aLabelCartesian CoordinatesLattice CoordinatesRangerpOO)0PointA(0,2 x/a , 0 )x(S + b3)T0 < x < 1X(0,2 TT/a , 0 % (b. + bjJPointz(x j

18、v a R 2 ir/a r 0 )% b1 + V； x b? + % (2+x) b30 < x < Iw(ff/a p 2 n/a : 0 )bd + ba + b3PointQ(n/a , (2 - x) n/a , x /a )+ x) b3 + %3-x)b30 < x < 1L(, tt旳,?r/a )% b1 + b2 + % b3PointA(x n/a T x ?r/a T x n/a )4 x bd + 4 x b2 + 14 x b30 < x < 2 1E(2 x/a . 2 jt x/a , 0 )16 x b1 + % x b2

19、 + x b30 < X <K = U(3/2 jr/a t 気 ju/a b 0 )% b1 + % bj + b,PointS(2 ?i, 2 n x/a , 0 )4 x bd + 4 x b2 + x b3%< x< 1(2 n/a , 2 ir/a T 0 )必十墟场+ X)PointrErx wru x(b)Figure 7+20 A comparison of the empty-lattice energy bands (a)日nd detailed calculations forAl ibk Again the nearly free electro

20、n character of Al is confirmed. (Harrison. Wv Pseudopotentials in 噪 Th&ory of Metals, 196&, Ad dison-Wes ley Publishing Co.r Reading. Massachusetts, Figures 3.19 and 3,20. Reprinted with permission.)Electr on bands in Al金属铝的能带图简约表示3*11 First Brillouin zone for mnicri带h crysultizing in the di

21、amond am I Afler BULcmorc. ' RepnnteJ cih pn fkisMuit )identifier the zone center (JI - U)X . L denotes the zone end ftlong a : 1 tX): dircuon. ndL -denotes the zone end along a (Hl? direct ion-wave vector4.在倒空间，面心立方 空晶格(Empty Lattice )能量图示FCC in real space r 1 a厂 § a 球+ ?r 1a2 = 2a?+ ?B C

22、C in reciprocalspaceb12a? ?刃1牛2a? ?1 4 二r 1a 2*?十?b32a?TS3 4-a-n JQd 巴 gtlJH口Free-electron bands in a face-center-cell crystalH?rJ,kr=Enkmrh2 f 2For an empty lattice,H?(r)=- +V0(r), where V0(r) = V0(r + Rn) = V0 = 0 2mik r ,n,k5(；)=川叫,卅),where(：)=叫丄(：R)嘉Gh r= ei(kGh)r(Z n,k(r)= En(k)中 n, k(r) ,wher1

23、h2 J 2ei(k (Gh)f2m二 En(k)ei(k Gh)r = En(k)二h2 k +2mr h2k + Gh En(k) =2h22mGhz)2,0点的Enk曲线:做010轴上(从 0,0,0点到 X0,过 :0,0,0点白线轴，Eik曲线：Gh=0，在第一布里渊区内 简并度为1。Ei(lf>0; Ei(k:X2_ (0 0)22 二(a0)2 (0 0)2 佥¥)2E2k曲线：Gh二占M -八、La2- ? 2 二?k； - k； a a2 二E2(k :M)E2(k:N)?kz，在第二布里渊区内从2 二到 N ,0, aa-2、2 一 乙)2 © 三

24、)2c 1 “2 h、2aa2 二Ja2m (0t)2 e2m _ aa深黄线轴，简并度为4。E3k曲线：Gh2 ?Kxa2 二(ayr；2m aJ2 (0 J2 =2(aa2m1 ,2辿)2a2 二？k；a2 二?，从点PJ a2:一到aQ-a ar hE3(k:P)二E3(k:Q) = h绿线轴，简并度为4。a2 /c2二、2/八2二、2s2二、2-12- h(0)(0 )(0 )2m IL aaa2 s 2 o 22(0)( 2m L a a: 2-)2 (0a兰)2 ,丄(禺22m aa? 4 二? E4(k)曲线：Gh = 0kx - ky a?°k；,W0,- ，0到aH

25、 0,0白线轴，简并度为1aE4(k:W)E4(k:H)=(0 0)22m _占(0 0)22m IL(0-(兰a4 2212 二 h 2)(0 0)2 =4 ( )2； a2m a牛 2212 h 2)2 (0 0)2 T()2 a2m arr ? E5(k)曲线：G 0k； 0ky蓝线轴，简并度为4。- 2 2m (0 0)2 f (0 0)24 二?kz，从点 J 0,0,a4 二2 二 4到 K0, aa aE5(k: J)二E5(k:H)二h2+ (0+ 0)2 + (0 +竺)2 Ia 一(0)2 (0 Jaarr ? Ee(k)曲线：Gh 二 0k；线轴，简并度为1。h224 2

26、2厂 |(0+0)2 + (0 + )2+(0 + 0)2 2m Lah2c 24(0 0)2 (-2m _a aE6(k:S)E6(k:T)ky a0?，从点 S0,2=禺一 2m a4-a，0到"TO 蓝.1 ,2 h 2 =4();2m a二 4 2212- h八)2 + (0十0)2-w)2Euq ±ll*!EA9/32(A2/m/)lrxlrx何(b)Fig* 7.6 Comparison beiween the LCAO bands for G© compu馆d with an sp3 basis, and the free electron band

27、s. From Harrison (1980).在空晶格上放上离子实，空晶格的能量曲线在布里渊区边界处，出 现能量断裂，产生带隙。，E(k)-0d kk429)Figure 7+20 A companson of the empty-lattice energy bands a) and detailed calculations forAl ibk Again the nearly free electron character of Al is confirmed. (Harrison, Wv Pseudopotentials in 噪 Theory of Metals, 1966 Ad

28、dison-Wes ley Publishing Co, Reading. Massachusetts, Figures 3,19 and 3,20. Re卩rinted with permission.)piiEq uoylanpuow3/32 估/)r(a)Fig, 7.6 Comparison between the basis, and the free electron bands.wave vector*E(k)=0dkk总色2LCAO bands for G离 computed with an sp3 From Harrison (1980).5.费米面在布里渊区边界处的断裂B&

29、#39; En(k)的时间对称性的推论：在布里渊区边界处有狂亦)=0ckFermi Surfaces3D Free electron gas:IJSurface of constant energy in reciprocal space: 日幻=钉=> onh in inehils!=> determines electrical propertiesFermi Surface is a sphere with radius:2D Fermi Surface: empty lattice用、tH?D Fermi "circle'*:/3b21st zone2 n

30、d zoue3皿 zone(a)(b)(cl(d>Fi呂tir电 7.T1 rhe presence or more than one periodicity may cause overlap in the integrated density of states.(J7TFflJT a:b'卜 iqiw 11stales and band strnclures giving 固 an insiikitor. (b)直 metal or a seining tu!bircaiLsr of band oerkipb suhI (e) aberaiisc of cJwtron cyncentratiGn. In (h) the overlap neednot 仇迟 th炽 satne directions in the Brilloiiin AOtie. If the overlap is sma|T with rclativek few slutrs involvi-d, wc speak of <)senihnetal.(a)<b) in 仏冲由电产的G正方豪搗中正启的班自由电子那能纯()国W1(b)二维止方晶格的尊能蛻示倉图 儀束缚近似“近自由电f