references - springer978-3-662-09940-7/1.pdf · references 1. there are several series publications...
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References
1. There are several series publications devoted to solid state physics and collections of data on solids and their properties:• Solid State Physies, edited by F . Seitz, D. Turnbull, H. Ehrenreich, F .
Saepen, since 1955 (Academic Press, New York)• Springer Series in Solid State Sciences , edited by M. Cardona, P. Fulde,
K von Klitzing, H.-J . Queisser (Springer, Berlin)• Springer Series in Surfaee Scienees, edited by G. Ertl, R. Gomer, D.L. Mills
(Springer, Berlin)• Springer Series in Materials Sci ences, edited by U. Gonser, A. Mooradian,
KA. Müller, M.B. Panish , H. Sakaki (Springer, Berlin)• Semieonduetors and Semimetals (eds. R.K Willardson, A.C. Beer) since
1966 (Academic Press, New York)• Springer Tracts in Modern Physies, G. Höhler (managing editor) (Springer,
Berlin)• Topies in Applied Physies, Series eds . C.E . Ascheron , H.J . Koelsch
(Springer, Berlin)• Landolt -Börnstein, Numerieal Data and Funetional Relationships in Sei
enee and Teehnology, New Series Group III: Crystal and Solid StatePhysics, editors K-H. Hellwege (Vols. 1 to 9), O. Madelung (Vols. 10 to23) , W. Martienssen (since Vol. 24)
• Festkörperprobleme/Advanees in Solid State Physies, editors F. Sauter(1962~1966), O. Mad elung (1967-1972) , H.J. Queisser (1973-1975) ,J . Treusch (1976-1981) , P. Grosse (1982-1987) , U. R össler (1988~1992) ,
R. Helbig (1993-1998) , B. Kram er (1999-present) (Vieweg, Braunschweig)2. Eneyclop edie Dietionary of Condensed Matt er Physies edited by C.P. Poole
(Elsevier, Oxford 2003)3. F . Seitz: The Modern Th eory of Solids (McGraw-Hill Book Company, New
York 1940)4. Ch . Kittel: Quantum Theory of Solids (John Wiley & Sons Inc. , New York
1963)5. D. Pines: Elementary Exeitations in Solids (W.A. Benjamin Inc., New York
1964)6. R. Kubo, T. Nagamiya: Solid State Physies (McGraw-Hill Book Company,
New York 1969)7. ,I.M. Ziman: Prineiples of the Theory of Solids (Cambridge at the University
Press 1969)8. W .A. Harrison: Solid State Theory (McG raw-Hill Book Company, New York
1970)9. W . Jones, N.H . March: Theoretieal Solid Stat e Physies, Vol. 1: Perfeet Lattiee
in Equilibrium, Vol. 2: Non-Equilibrium and Disorder (John Wiley & Sons ,London 1973)
294 References
10. H. Haken: Quant enfeldtheorie des Festkörpers (B.G. Teubner , Stuttgart 1973),Quantum Field Theory of Solids 2nd printing (No rt h-Holland , Amsterdam1983)
11. J . Callaway: Quantum Theory of the Solid State (Acad emic Press, Sa n Diego1974)
12. N.W. Ashcroft, N.D. Mermin: Solid State Physics (W .B. Saunders Company,Philadelphia 1976)
13. W . Ludwig: Festkörperphysik, 2. Au flage (Akademische Verlagsgesellschaft,W iesbaden 1978)
14. O. Madelung: Introduction to Solid Stat e Th eory, Springer Serie s in Solid StateSeiences 2 , eds. M. Cardona, P . Fulde, H.-J. Qu eisser (Springer-Verlag, Berlin1978)
15. S.V . Vonsovsky, M.1. Kat snelson : Quantum Solid State Physics, Sp ringer Series in Solid State Seiences 73, eds. M. Cardona, P. Fulde, K. von Klitzing,H.-J . Qu eisser (Springer , Berlin 1989)
16. A. Isihara: Condensed Matter Physics (Oxford University Press 1991)17. P.W . Anderson: Concepts in Solids (World Scientific, Singapore 1997)18. P.M . Chaikin , T .C . Lu bensky: Principles of Condensed Matter Physics, 1st
paperback edition with correct ions (Cambridge Univers ity Press , Cambridge2000)
19. P.L . Taylor , O. Heinonen : A Quantum Approach to Condensed Matter Physics(Cambridge University P ress, Cambridge 2002)
20. E.P. O 'Reilly: Quantum Theory of Solids (Taylor & Francis, New York 2002)21. E. Kaxiras: Atomic and Eleetronic Structure of Solids (Cambridge University
Press , Cambridge 2003)22. P. Phillips: Advanced Solid State Physics (Westview Press, Boulder Colorado
2003)23. http ://www.ccmr .comell .edu/brouwer/p654/notes .ps.
http ://www.fys.ku .dk/flensberg/book2003.pdf .http ://www.thp .uni-koeln .de/~alexal/qf t_ssOl .ps
24. Ch . Kittel: Introduction to Solid State Physics, 7th edit ion (John Wiley &Sons Inc., New York 1996)
25. K.H. Hellwege: Einführung in die Festkörperphysik, 3. Auflage (Springer,Berli n 1988)
26. H. Ibach , H. Lüth: Solid State Physics: An Introdu ction to Th eory and Experiment, 3rd edition (Sp ringer, Berlin 2003)
27. U. Kreibig, M. Vollmer : Optical Properties of Metal Clusters, Springer Ser iesin Material Seiences 25 (Springer , Berlin 1995)
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212. O. Heino nen (ed .) : Composite Fermions (World Scientifie, Singapore 1998)213. P. Streda, J . Kucera, A.H. MacDonald: Phys. Rev . Lett. 59 1973 (1987)214. M. Büttiker: Phys. Rev . B38 9375 (1988)215. S. Nakajima, Y. Toyozawa, R. Ab e: Th e Physics 0/ Elementary Ex citations,
Springer Ser ies in Solid State Seiences 12 (Springer, Berlin 1980)216. P. Vogl: Th e Electron-Phonon Interaction in Semi coruluctors in Physics 0/
Non linear Transport in Semiconduciors, D.K. Ferry, J .R. Barker, C. Jacoboni(editors) , NATO ASI Series B: Physics Vol. 52 (Plenum Press, New York 1980)
217. G.L. Bir , G.K P ikus: Symmetry and Strain induced Effeets in Semiconductors(J . Wiley & Sons , New York 1974)
218. KL.Ivchenko, G.K Pi kus : Superlattices and Oiher Heteros tru ciures , SpringerSreies in Solid State Seiences 110 (Springer , Berlin 1995)
219. H. Fröhlich, H. Pelzer , S. Zienau: Phil. Mag. 41 221 (1950)220. J .T . Devreese (ed .) : Polarons in Ionic Crystals and Polar Semiconductors
(North-Holland Publ. Comp., Amsterdam 1972)221. L. Regg iani (ed .) : Hot Electron tromspori in Semiconductors, Topics in Applied
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truciures, Springer Series in Solid St ate Scienees 115 (Springer, Berlin 1997)225. K Seeger : Semiconductor Physics, Spr inger Series in Solid-State Seiences 40,
8th ed ition (Springer, Berlin 2002)226. V.L. Bonch-Bruevich , S.G. Kalashnikov: Halbleit erphysik, translated from
Russian (VEB Deutscher Verlag der Wissenschaften , Berlin 1982)227. M. Lundstrom: Fundamentals 0/ Carri er Transport , 2nd edition (Cambridge
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229. B.K Rid ley: Electrons and Phonons in Semiconductor Multilayers (Cambridge University Press 1997)
230. H. Kamerlingh On nes: Leiden Comm. 120b, 122b, 124c (1911)231. P.G. DeGennes: Superconduetivity 0/ Metals and Alloys , (W .A. Benjamin Inc.,
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236. J . Bardeen , L.N .Cooper, J .R.Schrieffer : Phys. Rev . 108 1175 (1957)237. J .G . Bednorz, K.A. Müller : Z. Physik B64 189 (1986)238. A.G. Leggett : J . Phys.(Paris) C7 19 (1980)239. C.A. Regal,C. Ticknor, J .L. Bohn, D.S. Ji n: Nature 424 47 (2003)240. W. Kohn in : Solid State Physics Vol. 5 257 (1957)241. S. Pantelides: Rev . Mod . P hys . 50 797 (1980)242. M. Lannoo, J . Bourgoin: Point Dejecis in Semiconductors I: Theoretical As
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245. W . Schröter (ed .): Electroni c Struc ture and Properti es of Semicondueto rs, Materials Science and Technology Vol. 4 (VCH Publichers, New York 1991)
246. E.F. Schubert : Doping in 111- V Semiconductors (Cambridge Univers ity P ress,Cambridge 1993)
247. J .-M. Spaeth , H. Overhof: Point Defeets in Semiconduetors and Insulato rs.Determination of Atomic and Electronic Structure from Paramagnetic Hyperfin e Int eractions, Springer Series in Materials Science Vol. 51 (Springer , Berlin2003)
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250. U. Kaufmann in: Advances in Solid State Physics 29, U. Rössler (ed .) (Vieweg,Br aunschweig 1989) , p. 183 and B.K. Meyer , K. Krambock, D. Hofmann, J .M. Spaeth ibid. p. 201
251. R.J. Elliot t , J .A. Krumhan sl, P.L. Leath: Rev . Mod . Phys . 46 465 (1974)252. J . Rammer : Quantum Transport Theory, Frontiers in Physics 99 , ed . by
D. P ines (Perseus Books, Reading Mass. 1998)253. B. Kr amer, A. MacKinnon: Rep. Progr. Physics 56 1469 (1993)254. E . Abrahams, P.W . Anderson , D.C. Liciardello, T .V. Ramakrishnan: Phys.
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(2001)258. J .S. Blakemore: Semicondu ctor Statistics (pergamon Press, Oxford 1962)259. I.M . Gradsteyn, I.S . Ryshik: Table of Integrals, Series and Products, tran slated
from Russian (Academic P ress , New York 1965)
Solutions
Solutions for Chap. 11.1:
poin t lattiee:
reeiproeal lattiee:
W igner- Seit z eell:
set of lat t ice vecto rs R n = 2:.~= 1 n ia i, n i int eger ,a , linear indep endent (d dim ension of the system)
G m = 2:.:=1 m j bj , mj integer,bj linear indep endent , an d a ; . bj = 2'mSij
contains all point s which are closer to a given R nt han to any ot her R n' # R n
(Lst.) Brillouin zone : W igner-Seitz cell of t he reciprocal lat t ice
d = 2, square latt iee: a l = a (l , 0) , a 2 = a (O, 1)-> b l = 21l-ja(1, 0) , b2 = 21l-ja(0 , 1)
d = 3, simple eubie(se) , body eentered eubie(bee), f aee een tered eubie(f ee)se: a l = a(l , 0, 0) , a 2 = a(O, 1,0) , a 3 = a(O, 0,1)-> bl = 27l' /a (1,0,0) , b2= 27l' / a (0 ,1 , 0) , b3 = 27l' / a (0,0,1 )
bee: a l = a/2 (1 , 1, - 1) ,-> b l = 27l' /a (1,1 ,0) ,
a 2 = a/2 (1, - 1, 1) ,b2 = 27l' / a(1 , 0, 1) ,
a 3 = a/2(- I, 1, 1)b3 = 27l' /a (0 , 1, 1)
f ee: a l = a/2 (0 , 1, 1) , a 2 = a/2( 1, 0,1 ) , a 3 = a /2(I , 1,0)-> b l = 21l-ja (-1 ,I ,1 ) , b2 = 27l'/ a (I , -1 ,1 ) , b3= 21l-ja (I ,I,-I)
1. 2 : Create Fibonacci sequence by replacing LS -> L and S -> L [21]:
LS
LSLLSLLS
LSLLSLSL
LSLLSLSLLSLLS
LSLLSLSLLSLLSLSLLSLSL not periodic
Rep lacing in t he last line LS -> L' and L -> Si gives t he configur ation of t he secondbu t last line (self-similarity) . For the Four ier transform see [30].1.3: Given two vectors a l ,a2, wit h la l l = al , la 21 = a2, and al . a 2 = cos o ,spanning a plan e. T he following 5 cases can be distinguished :
a1 = a2 , a = 7l'/2 square
o = 7l'/3 tri angular or hexagonal
304 Solutions
a #- 1r12, 1r13
ar #- a2, o = 1r12 recta ngu lar
o #- 1r12
1.4: T Rn is t he t ranslat ion operator. It acts on a funct ion accord ing to
and commutes wit h t he sys te m Hamiltonian , [TRn , H ] = O. Therefore, t here existsimultaneous eigenfunctions of H and T Rn with t he pr oper ty
T Rn <pk( r ) = eik.Rn <pk (r )
i.e., t he wave eigenfunct ions in different Wigner-Seitz cells differ only by aphas efact or wit h wave vector k from t he 1st Brillouin zone.
1.5: Count nearest neighbors (n . n .) an d spheres per cube :
41r (a) 3 3 1rsc 6 n. n. , 1 sphere -> 3 2 [a ="6 = 0.52
41r (V3a) 3 3 V31r .bee 8 n. n. , 2 spheres -> 23 - 4- [a = - 8- = 0.68
41r ( a ) 3 3 tt[cc 12 n. n. , 4 spheres -> 43 2V2 [ a = 3V2 = 0.74
. 41r (V3a) 3 3 V31rdiarnond 4 n. n. , 8 spheres -> 83 - 8- [a = 16 = 0.34
1.6: A mass density n(r) = 8( r - r i) and using 8(r-ri ) = L q exp (iq · (r - r i ))I Vgives for t he scat tering amplit ude
F (k , k' ) = F (k - k' ) = L eiq.r i~1ei(k- k' +q).r = L eiq.ri8q ,k_ k'
i ,q v i, q
and wit h r i -> R n + T for a crystalline solid
.,. n
where t he last sum vanishes exee pt for q = G and Ln eiq.Rn = N 8q ,G . T hus theseattering amplit ude, which equals t he statie st rueture fact or (up to a faetor N)has peaks for the reciproeal lat ti ee veet ors.
Solu t ions for Chap, 2:
2 .1 : T he matrix representat ion of the eommutator [H,p] = 0 reads
n'
For eigens ta tes of H one has H mn, = E m8m,n' an d
Solutions 305
which means for Ern =1= En or m =1= n for non-degenerate states t hat pmn = O. Thesa me result is found by starting from t he statistical operator of t he eanonical (orgrand canonica l) ensemble.
2 .2 : Denote t he ground sta te by Itlio} and show t hat (TrßA)T=O= (tlioIAltlio) . With
ß = exp (-ßH) /Z write
Tr(ßA) = ~ L (tlimle- ßHItlin } (tlinIAl tlim) = ~ L e-ßEm(tlimI Al tlim) .m.n rn
With Z = e-ßEoL e- ß(Em -Eo) we ean wri t e
ttt
• ""' e- ß(Em-Eo) (tli IAltli )Tr (ßA) = U m m m
2:m e-ß(Em-Eo) T ---> O
becau se with Ern - Eo > 0 for all m =1= 0 all exponent ial factors vanish for T ---> 0except t he one for the ground state Itlio }.
2.3 : T he t hermal expectation value of the nurnber operato r is IV = 2:a na where
na is t he opera to r counting t he par t icles in an eigenstate of H. The gra nd eanonicalpar t it ion funct ion ean be written
00
ZG = L L e ß 2:o (JL-Eo )n o = Il L e ß (JL - E o )n o ,
N=O {no }N .o n o
where {na }N denotes t hose sets of particle numbers na whose sum is N . Forferm ions: n a ean be 0 or 1 and
while for bosons n a can be any non-n ega tive int eger and
ZG = rr (1+ eß(JL -Eo) + ... ) = rr (1_ eß(JL - Eo»)- 1o
Now make use of
(IV) = ~ 8 ln ZG = ~ _l_ 8ZGß 8 f-t ß Z G 8Jl
to obtain for fermions (IV) - '"(n ) _ '" 1- L.J o - L.J exp (- ß(f-t - Ea» + l
a o
and for bosons (IV) = L (na) = L exp (- ß( f-t~ Ea» _ 1a a
The function
f ± - 1o - exp (-ß(f-t - Ea» ± 1
is t he Fer mi-Dirac dist rib ution (upper sign) and t he Bose-Einst ein distribut ion(lower sign) . For large kBT » f-t we have exp ßf-t ~ 1 and f;; ---> exp (-ßEa).
306 Solutions
2 .4: Ohm's law can be written jA = (> AlL El" thus the observable to be measured is(a component of) the elect rica l cur rent dens ity
N
A --> JA= e L Vl, A1= 1
with the velocit y op erator Vl,A = PI ,A/m. It can be written also as
JA(r) = 2~ L (ppJ(r - rl) + J(r - r l)Pl,A) .I
The op erator of the kinetic energy in the presence of an elect ro-magnetic field (hererepresented by the vector potential A(r, t)) is for the l-th elect ron
1 ( , ) z pr e ( ) z-PI- eA(rl ,t) =---2 pl ·A(rl ,t)+A(rl ,t) ,pl +O(A).2m 2m m
Neglect ing the last term on the rhs, we identify the perturbation as
Vext(t) = - L 2~ (PI ' A(rl ' t) + A(rl ' t) · PI) orI
= - Jd3r 2~ L (P1J(r - rl) + J(r - r l)Pl) ·A (r , t )
I, J
and t he observable iJ as anot her component of the elect ric current density. Theelectric field component is given by
EI' = - BAI' = -iwA IJ- and we can writeBt
Vext (t ) =-±L Jd3r3IJ- (r )E IJ- (r )e
iw t
I'
which in the long-wave length limit , when t he dependence of the vector pot enti alon r can be neglect ed , gives
Vext(t) = -±L Jd3rj lJ- (r )E lJ- e
iwt
I'
and we obtain the elect ric conduct ivity
as a correlation function for the components of the current density.
2.5: Wi th L1p(t) = L1PI (t) + L1pz(t) + . . ., where the ind ex refers to different ordersof Vext , we can write the equa t ions
(1) [Ho , L1pI] + [Vex t , po] = inpl first order in Vext
(2) [Ho, L1pz] + [Vex t , pI] = inpz second order Vex t .
The solution of (1)
Solutions 307
LlPI(t) = i~ [too
dt'e - iH o(t- t ' ) / h [Vex t (t') , po]eiHo(t -t' )/h
is to be used in (2), which can be solved in the same way as (1), to yield
t t 'Ll (t) = ~ [ dt ' [ dt" - iHo(t- t ' )/ h iH o(t - t")/" [ [ ]] iHo (t -t') /hP2 in e e ... , . .. , . . . e
- (Xl - 0Cl
wh ere the double commutator
[ [ ]] - [V. (t') - iHo(t- t " )/h [V. (t") ]].. . , . . . , . . . - ex t , e ext , po
indicates the structure of the second order response function
(LlA2)t = ... [B (T), [B(T') , -4.(0) ]] .
It is a two-time correlation function.
2. 6 : Evaluate the principal valu e integral
1+ 00
ReX(w) = 2. P ~ (8(wo - w') - 8(wo+ w')) dw'1r W ' - W
- 00
whi ch yields
R () - Xo (1 1) _2Xowo ---n_1---::-ex w - - ---+--- - - - -tt Wo - W Wo +w 1r w5 - w2
Solutions for Chap.3:
3 .1 : The potential energy of the linear chain (Fi g. 3.16) is
(1 ) (2 )
here (1) is the contribution due to the relative d isplacements of MI and M2 in theunit cell n and (2) that due to the relative displacement between MI in unit celln + 1 and M 2 in unit cell n .Distinguish the force constants
, (nn) (nn)n = m , T = T : <p 1 1 = <p 2 2 = 2f
for the restoring force acting on MI (M2) if the neighbor atoms are kep t fixed ,
, (nn)n = m , T = 1, T = 2 : <p 1 2 = -f and
, (m+1m)n = m + 1, T = 1, T = 2 : <p 1 2 = - f .
Due to actio = reactio we have
308 Solutions
for T = 1 : cP ( 77) + cP ( 7~ ) + cP ( 7n; 1) = 2f - f - f = 0
for T = 2 : cP ( ~ ~ ) + cP ( ~ 7) + cP ( ~ n i 1 ) = 2f - 1 - 1 = 0 .
Tr anslation invar iance allows to shift the cell index.Considering the equilibrium positions X nl = na for M I and nn 2 = na + ro for
M 2 along the chain , we find the eleme nts of the dynarnical matrix
1 (nn) 21D ll(q) = M l cP 11 = MI '
and
For n = m , X nl - X n2 = - ro and for n = m + 1, X m +l ,1 - X m ,2 = a - ro, the forceconstant equa ls - 1 and with a = 2ro we have
D I2(q) =~cos(qro) = D21(q) .M 1M2
T he eigensolutions of
There ar e two solut ions for each q. For q = 0 t he squ ar ed frequencies are
2 2fM 2w+(O) = M
1M2' and w_ (O) = 0
and for q ':::'. 0 with 1 - 2 sin2 qro = cos qa ':::'. 1 -la2/ 2 (for q « 'Ir ja)
w2(q) ':::'. _1_ [M ± {M2 _ MIM2l a2}1/2] ':::'. L [1 ± {1_Lla2}]M 1M2 J-l 2M
with the reduced mass J-l = M 1M2/M and
2 'Ir 21w+(q« -) ':::'. - independent of q and
a J-l
2 'Ir 1 2 2 ('Ir rTw_ (q)(q« ; ) ':::'. 2M q a ....... ta .: q «;) ':::'. V2Njaq .
For q = n f a and cos qa = -1 t he solut ions are
w~(~)= M!M2 (M±{Mf+MJ-2MIM2r/2) or
Solutions 309
w(q)
o rrJ2a q rrJa
Fig. S.1. Disp ersion for the linear chain with twodifferent masses per unit cell (sol id lin es). Thedash ed curves show the result if the two massesare equa l
A plot of t he two branches is shown in Fig. S.l : the lower br anch with t he lineardependence around q c::= 0 for the acoustic phonons and the ftat upper branch forthe optical phonons are separate d by a gap which results from the different massesMI =f M2. For MI = M2 = M the gap closes and we have a chain with period aand the dispersion
w(q) =~Sinqaextends to 11"/ a (thin dash ed line in Fig. S.l) , which is the limit of the Brillouin zonefor the chain with lattice const ant a.Solving the eigenvector equat ions, we findfor q = 0 : w- (O) = 0 e- (O) rv (VfV[;,yM';") move with same phase
w+(O) =,fiJ e+(O) rv (yM';",-VfV[;) move with opposite phase
for q = ~ : w _ ( ~) = 1ft e ., ( ~) rv (1,0) M 2 in rest
w+ ( ~ ) = jIi; e+ (~ ) rv (0,1) MI in rest .
3.2: For cent ra l forces t he adiabat ic potential dep ends on r nt, n IR~ + U nt -
R~ - U n l and t he force constants can be written
with k =m-n .
•7
•3
•8
•2
•o
•4
•6
.x
•5
Fig. S .2. Sketch of the two-dimensional qu adratic latticewith numbers to address the individuallatt ice point .
310 8olutions
Nearest neighbors to mass in the center (see Fig. 8.2) are
R~ = aex, Rg = aey , R~ = -aex, R~ = -aey
with force const ant s
(10)_ (20)_ (30)_ (40)_ (kO)_O1> x x - 1> y y - 1> x x - 1> y y - 1>1 ,1> x y -
next nearest neighbors are
R~ = aex - e y , R~ = -aex + ae y
with force const ant s
(50) (50) (70) (70) 11> x x = 1> y y = 1> x x = 1> y y = 2 1>2
(50) (50) (70) (70) 11> x y = 1> y x = 1> x y = 1> y x =-21>2
and
and
R~ = ae; + aey with (60) (60) (60) 11> x x = 1> y y = 1> x y = 2 1>2
Rg = -aex - aey with 1> (~ ~) = 1> (~ ~) = 1> (~ ~) = ~1>2 .
The force constant
1> ( ~ ~ ) follows from ~ 1> ( ~ ~) = °
---+ 1> (~~) = - t 1> (~ ~) = -2(1)1 + 1>2)Oi,j .
The elements of the dynamical matrix ar e
D xx(q) = ~ ~ 1> (~~) e- iq. R~
2= - M (1)1(1- cos qxa ) + 1>2 (1 - cosqxacosqya))
2Dy y(q) = - M (1)1(1 - cosqya) + 1>2(1 - cosqxacosqya))
D xy(q) = - ~ 1>2 sinqxasinqya .
The secular problem is
11
D xx(q) - w2
Dxy(q) 211 = 0 .D xy(q) Dyy(q) - Mw
ForT-X : O:::::qx :::::1r /a , qy=O , sinqya=O , cosqya= 1 wehavetwobranches
Solutions 311
2 2 2 2Wl (qx) = M«(Pt+ 1>2)(1 - cosq xa) and W2 (qx) = M 1>2(1- cosqxa)
with Wl,2(qx) rv qx for qx «1'i/a but different slopes.For T - M : q = (q, q)/V2 , 0 ::; q ::; V21'i /a the secular problem yields
with the solutions
M w2
= 2a ± 2b : M wi = 21>1 (1 -cos ~)
Mw~ = 21>1 (1 -cos ~) + 21>2 (1 -cos J; )Again we find two branches with w(q) rv q for «< n ]a and different slopes.For X - M : q = (1'i /a ,q) , 0::; q ::; n Ia t he off-diagon al terms of the dynam icalmatrix vanish and one has
wi = ~(21)1+ 1>2(1+ cOSqa)) and w~ = ~(1)I+ 1>2 -(1>I - 1>2) cOSqa) .
The two bran ches have always finit e frequenci es and connec t t hose already obtain edfor the X and M point .
For each of these directions one eigenvector is longitudinal (1Iq) and one tran sverse (-1 q) .
3 .3 : Periodic boundaries account for the fact t hat the physics of a solid rep eat sover macroscopic dist ances, i.e., t he Bloch phase faet or equals one for a t ranslationRN = N i o., + N2a2 + N3a3 over a macroscop ic length (Ni » 1 , i = 1,2, 3:)
ei k.R N = 1 or k· RN = 21'i x integer.
This implies (t ake a simple cubic lattice with V = L1L2L3 , Li = Ni a as example)that the components of the wave vector t ake the discrete values (particle in thebox) ki = 21'ini/Li , i = 1,2,3 and 0 ::; n i ::; Ni - 1. Thus N = N 1N2N3 is thenumber of k in the first Brillou in zone and each k takes a volume (21'i) 3/V. Thiscan be exploite d in replacing a sum over k by an int egral acco rding to
L" '=(2~) 3 J...d3k.
k
Similar considerations hold for syste ms with reduced dimension.
3.4: For t he commutator [as(q) ,a~,(q')l evaluate
[ws(q)Qs(q) + i1\(-q) ,ws,(q')Q s,(-q') - iPs,(q')] =-iws(q) [Q s(q) , Ps'(q')] +iws'(q') [Ps(-q) ,o. (-q')] = 2hws(q)6s,s,6q ,q '
'---v--' ' v "
ih oS,s , oq ,q ! - ili c5 s ' S, oq ,q !
to find [as(q) ,a ~ ,(q')] = 6s,s'6q ,q " The two other commutation relat ions follow inthe same way (no te that ws(q) = ws(- q» .
312 Solutions
3.5: In the Schrödinger picture we have
[as(q) , Ho] = hws(q)as(q) or as(q)Ho = (Ho + hws(q))as(q) .
Thus, for any power fun ction f(Ho) we ca n write
as(q)f(Ho) = f (Ho + hws(q))as(q)
and obtain
( t) t ( , 0) iHot/h ( 0) - iHot/h t ( , 0)a; q, as' q , = e a; q, e as' q ,
= e-iW8( q)t as (q , 0)a~ / (q' , 0) .
Using the commutation relation and taking t he t he rmal expectat ion value gives
(as ( q, t ) a~,(q' , O)) = e- iws (q)t (ns (q ,T ) + l)Os,sIOq ,ql .
Similar we have
a1(q)Ho = (Ho - hws(q)) a1(q)
and by following the sa me steps we find the second relation . For the third relationwe have, afte r extracting t he exponential with the ti me-de pe nde nce , t he thermalexpectat ion value of a product of two annihila t ion operators, which van ishes.
3.6: The displ acement is a time-d ep endent operator in t he Heisenberg picture. St artby writ ing (with [po , Ho] = 0)
((q . un(t)) 2) = Tr (po eiHot /h (q . Un(O)) 2e-iHot/h)
and obtain by cyclic permutation under the t race
which is inde pende nt of t . Formulate the latt ice displ acem ent with (3.23) and (3.39)
( )
1/ 2
Un = L 2N:~s(q) (a1(-q) + as (q) ) es (q)eiq ' R~s,q
in te rms of ph onon op erators and evaluate
(q 'Un )2= N~ L (a~/(-q')+as/(q'))(a~"(-q")+ as"(q")) xs ' ,q !
e" .a"
(
2 ) 1/ 2h i ( q' +q") . R~ ' "X 4w
s, (q') w
slI(q") e (q.es/(q))(q .esll(q )) .
After mul tiplying ou t the op erator terms, the t hermal expectation value of thisexpression follows with t he formulas of problem 3.4 as
For T -+ 0 we have ns(q,T) -+ 0 wh ich leaves only the contr ibution of t he zeropoint motion
Solu tions 313
For T > 0, employ the Debye model by writing Ws (q) = vq for all s and indepe nde nt
of the direction of q . Summation over s ' yields a factor 3. Wri t ing (q . e s , (q , )) 2 =q2 cos 19 2 where 19 is the angle between q and e s, (q ') , the sum over q' can be carriedout in spherica l pol ar coordinates with the cut-off at qt: = wo /v (with the Debyefrequency wo ) giv ing
( (q . U n ) 2) = ~~61WD lu» ( exp (nw/~BT)_ 1 + ~) dw .
For high temperatures, kBT » lua , t he distribution fun ction afte r expanding t heexponent ial yields knT[Iu» and by neglecting t he term 1/2 we find for the DebyeWaller fact or
W - Y k T lwD
d _ 3lkBT- 3 B w - 2
M wo 0 M wo
which is always posit ive.For low temperatures, knT « lua , the int egral over w reads afte r substituting
nw/kBT = x
l
wD _ nw6 (knT)21XD xdx. . . dw - + " 1
o 4 u. 0 eX-
For T ---> 0 the upper lim it goes to 00 and the int egral take the value 11"2 /6 (seeAp pendix). Thus we may writ e
w = 3l . (nw6 + 11"2 (knT)2) = 3n2l (! + 11"2 (.I-) 2)
M W6 4 6 n MkBeO 4 6 eowith t he Debye temperature eo .
3.7: a) The point group of a cubic lattice consists of 48 elements (24 rotations,each can be combined with t he inversion). Unde r these op erations, wh ich can berepresented by orthogon al 3 x 3 matrices Sai , t he coordinates x, y , z are interchangedand (under inversion) chan ge their sign . Likewise the components of the elas t ict ensor tran sform according to
Ca ß"I1j = Sa i S ßjS"Ik SIj/ Cij k l (double ind ex summat ion) .
The invari ance of the elas t ic te nsor under these t ransformations leaves only t hosecompo nents d ifferent from zero , for which pairs of indice s are identical and of thenon-vanishing compo nent s all those are identic al which t ransform into each otherby the symmetry operations . T hus, there are only 3 ind ependent te nsor components
CXXXX= Cy y y y = Czz z z = Cu
Cx y x y == Cx z x z == Cy x y x == Cz x z x == C12
Cx x y y == Cy y z z == Cz z x x == Cx x z z == Cz z y y == C44
which are written here in Voigt notation.b) Using ui(r , t) = u; exp (i(q · r - wt)) the wave equat ion for the elastic displacement field lead s to a set of coupled homogeneous linear equations for t he componentsui , which has solut ions if
Ilpw2oi! - Cij k / qj q k 11 = O.
314 Solutions
For r - X or q = (q ,O,O) it reads
11
pw2 ~Cll q2 pw 2 ~CI2q2 2 ~ 2 11=°° ° pw - CI2q
and has solutions
WL =~ q , e r, = (1,0,0) longitudinal
WT =~ q , e T = (0,1 ,0) t ransverse
= (0,0,1) transverse .
For r - Kor q = (q , q , 0) /v'2 we have Cijklqj qk = (CiXXI +Ciyyl +Cixyl +Ciyx!)q2/ 2and the secular problem
11
pw2 - ~(Cll + C12)q2 -~(CI2 + C44 )q2 ° 11
- ~(C120+ C44)q2 PW2_~( COll+ CI2)q2 ° =0 .pw2 _ CI2q2
One solut ion is immediate ly found to be
WTl =~ q , eTl = (0,0,1) transverse .
The other two follow from
11 <; A pw-:;~ A 11 = (pw 2 - A)2 - B2
= °wit h A = (ci r + C12)q2/ 2 and B = ( C12 + C44 )q2/2 and read
J Cll - C44 r:WT2 = q , eT2 = (I ,-I ,0) /v2 t ransverse2p
2CI2 + Cll + C44 ~ . .WL = q , er. = (I,I ,0) /v2 longitudinal .
2p
These resu lt s can be compared with the phonon disp ersion (e.g. those given inSect. 3.6 wh ich are all for lattices with cubic symme t ry ). The slope of the aco usticbranches for given p can be t aken to determine the elast ic constants.
3 .8: The cubic anharmonic ity Ll(at + a)3 is first writt en in norm al order
(at + a)3 = a t3 + 3at2a + 3(at + a) + 3ata 2+ a3
and t hen truncated by replacing at a ---> (ata) = n(T) and omitti ng t he terms at3
and a3• Thus the Hamiltonian reduces t o
H = nwoata + Ll(T)(at + a) , with Ll(T) = 3Ll(n(T) + 1) .Calculate now the correct ions t o t he oscillator ground state In) with n = °due tothe anharmonicity, which by making use of (Ola II)( I la t IO) = 1 reads in BrillouinWi gn er perturbation theory
c = Eo _ nwo = Ll2(T)
.2 c -nwo
Solutions 315
The smaller solut ion of the qu adratic equat ion in e
E« = .!.hwo _ Ll2{T)
2 hwo
expresses a zero-point energy which decreases as Ll {T ) increases due to thermalph onon excit at ion with t he temperature. This is the beh avi or of a soft mode.
Solutio n s for Chap. 4:
4.1 : The electrostatic pot ential of a homogeneou s positive charge density +en {r)with n{r ) = N / V is
if>{r) = r e n{r ' ) d3r' .Jv 41rcolr - r' l
l t s interaction energy with the homogeneous elect ron density - en{r ) is
r 3 ( N)2 J 3 J 3 I e2
'He l - io n = -e Jv
n{r) if>{r) d r = - V d r d r 41rcoJr _ r/ l .
With (see Appendix)
e2
_ L iq .( r- r ' ) e2
- vqe , v - - -41rcolr - r /l q - coV q2
q
the double integr al is evaluated
giving 'Hel-ion = -N2 vo. Similarly t he interact ion energy of t he hom ogeneous electron and ion sys tems can be ca lculated wh ich each give the sa me result up to afactor - 1/ 2. Thus t he sum of all t hese divergent interact ion energ ies vanish for t hejellium mod el.
4 .2 : The elect ro n density n determines via the density of states D {E ) and theFerrn i-Dirac distribution fun cti on the che m ical potential p,{T ). Für a 3D elect ro n
sys tem we may write D {E ) = 3nVE/ 2E:/2
and define a fun ction G{ E ) with
1E E 3 /2
o D{EI )dE' = G {E) = n (EJ
to express t he particle density as
n = G{E)j{E ,p" T) I: -100
dEG{E) Öj{~:, T) .
The first t erm vanishes and the integral can be eva luated by using the fact , t hatthe derivative of j{E, p" T) is strongly peaked at E = p, (für kBT « p,). ExpandG {E) in a power seri es around E = p,
G{ E) = 11" dE' D {E' ) + f (E - p,)n dnG(E ) I .O n dE E= I"
n= l
The int egra l gives G {p,). Because ö j / ö E is an even fun ction only the eve n powersof t he expansion contribute and we find as t he two lead ing t erms
316 Solutions
100 81 3n 1 100
2 81n = - G (/-L ) dE dE - ~ r.; dE(E - /-L) dE (*).
o 8E F y/-L 0
T he first term gives n(/-LI EF )3/2 . With
81 eß(E -p, )
dE = - ß ( e ß ( E -p, ) + 1)2 ' ß = kBT and subst it ut ing x = ß (E - /-L)
100 2 81 1 100
2 eX
dE(E-/-L) dE= - ß 3 dx x x 2 'o 0 (e +1)
The value of the integral is Jr2 /6. Thus (*) reduces to the relation
(/-L ) 3/ 2 ( ( k BT) 2 Jr
2)1::= - 1+ - -
E F /-L 8
which can be solved to give
This result corresponds to the Sommerfeld expansion (see Appendix).
4 .3 : For 2D the number of st ates (p er unit area) D(k)dk in a circular ring withradius k and thickness dk is
D( k )dk = (2~)2 2Jrkdk (the factor 2 counts the spins) .
Use the dispersion relation for free elect rons Ek = h?k2 / 2m to substitute k by E
D(E)dE = D(k) dk dE = ~ ~VE ~_1_dEdE JrYh? Yh? 2VE
mor D(E) = - 2 = con st .
Jrn
For 1D the corresponding number of st ates per un it length is
2 dk 1 [2:;;; 1D(k)dk = 2Jr dk and D(k) dEdE = ;y h? 2,jEdE
~1
or D(E) = - 2 - 2 rr;; '2Jr ti y E
For a zero-dimensional system the spect ru m is discrete (with energies Ei) and thedensity of state s is given by
D(E) = 2 L 8(E - Ei) '
4. 4 : The condition to fill n elect rons into the lowest (spin-degenerate) Landaulevelfollows from (4.43) or in simplified form from EF = luu; and reads
which can be solved to give
Solutions 317
Note, that B is rela ted to t he number of elementary flux quanta. Take the valuefor nie= 0.65810- 15 Tm2 to obtain for a met al n = 1023 em- 3 a magnetic field ofB c::: 1.37 105 T and for a doped semieonduct or with n = 1014 em -3 a field strengthB c::: 0.137 T . The latter is easily achieved in a laboratory.
4.5: The Zeem an energy for free eleetrons is ±/-lBB. It shifts the density of statesof up and down spins (Landau qu antiza tion is no t eonsidered here) against eachother
1 1 1 dD( E)D±(E, B) = '2D( E ± /-lBB) c::: '2D(E) ± '2/-lBB~ which yields
D (E B) c::: .!. _1_ (2m) 3/2 .JE(1 ± /-lBB) .± , 221T2 n2 E
The nu mb er of spin up and down elect rons N±(E , B) is obtained by inte grati ng thedensity of stat es mult iplied with t he Ferrni-Dirac dis tribution function and multiplying with t he volume V. The first term of D±(E, B ) gives N (Jt )/2 indep endentof B . The second term is evalu at ed by employing the Sommerfeld exp an sion (seeAppendix) lead ing to
=±.!. B{J1l dD(E)dE 1T2(k
T)2d2D(E) I } .
. .. 2/-lB d E + 6 B dE2 E=ll- 00
The int egral gives D(JL) and with D(E) ~ ,JE the second te rm ca n be rewrittenusing d 2 D(E) /dE2 = -D(E) /4 E 2 to obtain
1 V {1T2(kBT) 2}N±(E B) ~ - N(/-l) ± -/-lBBD(/-l) 1 - - - ., - 2 2 24 /-l
The magn etiz ation follows as
2 (k T)22 1T BM = /-lB(N+ - N _ )/V = /-l BBD(Jt) { 1 - 24 ---;- } .
For T = 0 with D(Jt c::: EF) = 3n/2EF this is identical with (4.50) .
4.6: T he HF approximat ion is better for t he elect ron system with th e smallerdensity par amet er T s . According to Tab le 4.1 the Ts-values of doped semiconductorsare smaller t han those of rnet als . On the other hand the elect ron density (per cm- 3
)
is higher in met als. Note, that T s is given in the length seale (effective Bohr radius )of the material.
4 .7: Second order per turbation yieIds a cont r ibut ion
a) Applying the int eraction operator to the Fermi sphere IlJio) gives non-vani shingcontributions only if the states with p , q are inside and those with p - k , q + kouts ide of th e Fermi sphe re . Con sider therefore the corresponding excite d statesIlJirn ). These st ates with Ern - Eo = n2k · (q - p + k) /m eont r ibute to E2.b) The electron t aken from p to p - k is put back to p in the direct process but toq in the exchange process. For the former eva luate
(lJiol L Vk I C~ / _k l pC~ / +k l pI Cq l p l eplpllJim ) (lJirnl L VkC~_kaC~+kaI CqaICpa l lJio)p 'q Jk ' pqk
pp' o o '
318 Solutions
For the direct process the intermediate st ates have to fulfill the conditions
p ,o' = p' - k' ,p p - k;« = p' ,p
q.o = q' + k' ,p' q + k ,« = q' ,p'
or
0' = p , 0" = p' , p' = p - k , q' = q + k, k' = -k .
It remains to det ermine
(lJioI c~o- C~o- ,Cq+ko-l(~P -ko- C~ _ko-C~+kO-' cqo-' Cp o 11]/0 ) = .. .which can easily be rearranged as for k =I- 0 all fermion operators anti-commuteand one obtains
.. . = (lJioIc~o-Cpo-c~o-' Cqo ' Cq +ko- ' C~+ko-,Cp-kaC~ _ko- IlJio )
= npo-nqo-' (1 - nq+ko-' )(1 - np -ko-)
where tl-qo = B(kF - q) is the Fermi-Dirac distribution function for T 0 K.Summing over spin indices (factor 4) gives for the direct process
Eg ir = -4~ V~ 2 (m ) n po-nqo- ,(l - nq +ko-' )(1 - np -ko-) .c: n k · q -p+kp,q,k
c) For small k , i.e., excit at ion elose to the Fermi surface
np +k :::::: n p + k: >V'pnplkF = n p - k· ep8(kF - p) and
n p(1 - np +k) = n p{ l - n p + k · e p8(kF - p)} = npk · ep8(kF - p) '" k .
Replace now the denominator für small k by 2kkF and per form the sum over p andq in polar coordinates. Finally the sum over k is to be perfo rm ed over an expressionwhich contains 1/k4 from o», l / k from t he denominator, and k 2 from the numeratorwhich together with k 2 from integration in k-space leads to Jdk /k = lnk.
4.8: a) The meaning of cl (c,,) of creating(annihilating) a fermion in t he state owith the probability amplit ude 'IjJ" (r) implies, that lJit (r )(lJi(r)) creates(annihila tes)a fermion at r .b) Write
[lJi t (r), lJi(r')] = L 'IjJ: (r )'IjJ" ,(r') [cl, c",]= L 'IjJ: (r )'IjJ" ,(r') = 8(r - r'), '-v-"
O: , Q /)0. ,0'
Q
and simil ar for [lJi(r),lJi(r ')] = [lJit(r) ,lJit(r')] = O.c) For free elect rons t he density operator is
n(r) = lJit(r)lJi(r) = L e- i(k'-k ).r~Ck ' Ck and with q = k' - k
k ' ,k
no = 2:k cl ck /v = n is the op erator of partiele density (its eigenvalue being n)
while 2::k C~+q Ck = nq describes density fluctuations.d) The Coulomb interaction can be rewritten with
t t _ t ( " " t)Cp+ko-Cq_ko-,Cqo-'Cpo- - -cp+ko- Uo- ,o-'Uq-k ,p - cpo-cq_ko-' Cqo '
Solutions 319
by applying t he fermion commutat ion mies and by usin g t he number opera tors t heCoulomb int eract ion becomes
1",", t t _ 1 ",", ( " " )2" L VkCp+kuCq_ ku,C qu' Cpu - 2" L Vk N kN_ k - Nk # O,p ,q k # O
G , a l
4 .9: Replace in t he given express ion
([Nq(T), N _q(O)])exact ---> ([Nq(T),N- q(O)])o and Nea ---> Ns« + (Nq)
wit h t he induced number fluctua t ion (Nq ) . From Sect . 4.5, t ake
!im eV
2
2 .~ 100
dTeiWT-rT([Nq(T) , N_q(O)])o = V q7l"o(q,w)r -o cO q In 0
to write
or (Nq) = NextVq7l"o(q ,w)1 - Vq7l"o(q ,w)
and identi fy with
1 _ 1 (Nq ) RPA- - - - - - =:- c (q ,w ) = 1 - V q7l"o(q ,w) .c(q ,w ) Next
4 .10: Using relat ions given in Sect . 4.6, the lhs of t he given equat ion can be written
Aft er showing (by insert ing a complete set of eigenstates IlJim) of Hjell) t hat
-L 21iwm ol(lJioINqIlJim)12 = (lJio l[[Hjell, Nq],N- qlllJio)
one has to evalua te t he do ub le commutator . T his is done wit h Hjell wri tten in t er msof density fluctuations (see (4.105) and problem 4.8). Rea!ize first with Nk = N_ k
and [Nk, N d = 0 t hat t he int eract ion ter m commutes wit h N q . Evaluate
wh ich by applying fermion commutat ion mies yields
[Hjell, N q] = L Ek (4ck+ q - 4_qCk ) = L (Ek - Ek+q) ct Ck+qk k
and with Ek = 1i2 k2 12m
" 1i2 q2 " fj,2 "'"' t
[HjelI, N q] = - 2m N q - m L k · q Ck Ck+qk
T he first term rv Nq does not cont ribute to t he double commutato r which, t he refore,reads
and with k + q --> k
320 Solutions
Evaluating t he commutator with t he ru les for fermion operators lead s to
~ ~ _ n,2~ ( t t )[[Hj e l\, N q], N _ q] - - m L-- k · q CkCk - Ck+qCk+q
k
n2 n22= --~ ct Ck = --q N .
m L--k mk
Conside ring the factors on e has
1=dwwlm c(q~w) = - ;r~q ( - 21n)(-~ lN)= -~w; .
4 .1 1: Evaluate for w = 0
Cl (q, 0) = 1 - _ c2 L fk+q - i» b . . c 0c. y wr it ing ror q --> :
CoV q2 Ek+q - Ekk ,a
Thus on e has
c2
cl(q) ':::'. 1 + - V 2 L 8(Ek - E F)cO q
k ,a
c2
2V 1= 2= 1+ COV q2 (2;rp 4;r 0 d k k 8(Ek - E F)
= 1 + L _l_ (2m)3/2E l/2 andcOq2 2;r2 h.2 F
e2 1 2m kF2
T 2 3ne2
Cl(q) ':::'. 1 + ------kF = 1 + - with kFT = -- .cOq2 2;r2 n2 q2 2coEF
The meaning of kFT becomes evident when looking at t he screened Coulomb interaction Vq/Cl( q, O) whose Fourier transform is ofthe form exp(-kFTr) /r : l /kFT isthe Themas-Fermi screening lengt h .
4.12: Use the pair-distribution fun ct ion (1.11)
ger) = 1 + ~ L eiq.r (S (q ) -1) and use
q
S(q) = ~ Lnpa(1 -np+qa) towritep,a
and
ger) = 1 + :V I >iq.r [ ~ L Tl-pa (1 - n q+q,a) - 1]q pur
Solutions 321
The su mmation can be carried out by writing
_ ""' - ip.r ,,", i(p+q)·r _ 2""' - ip.r ,,", iq' .r. . . - L n pe L n p+qe - L n pe L n q,e .p ,a q p q'
This do uble sum with the occ upation factors was ca lculated already in Sect . 4.4 forT = 0 K by integrat ing over the Fermi sphere and yields ger) = 1 - pHF j en (seeFig.4.13) .
Solutions für Chap. 5:5 .1: Taking spin into account , the expe ct at ion value of HN with the Slater determinant rJtN is written
(rJtN lllN lrJtN) = t Jdx1/J~(x) ( ;:~ + v(r) ) 1/J,, (x )0 =1
N
+~ L JJdxdxl'l/)~(x) 1/Jß(x')v(r - r/)
1/J,,(x)1/Jß(x')o,ß= l" 'l'ß
N
-~ L JJdxdxl1/J~(x) 1/Jß(x')v(r - r/)
1/Jß(x) 1/Jo,(x') .o ,ß = l" 'l'ß
Carrying out the summation over spin variables , t he first two terms b ecome ident icalwith (5 .8) , while the third term (which appe ars because the Slater deterrninant isan antisyrnmetrized product of N single-particle wave fun ctions) contributes on lyif 1/Jo. and 1/Jß ar e states with the same spin . The variational principle leads for t hefirst two terms to the Hartree equations, which become modified by a contributionfrom the third term , t he exchange term (5.11) .
5 .2 : For free elect rons with 1/Jk(r ) = exp (i k · r) /VV the averaged exchange densityreads
- H F ( ') 2n r , r = - Nk k '
Ikl ,l k 'l :SkF
L I 1 1 i(k' - k+q).(r - r ' )d 3 I-- e rq2 V v
k k ' q '- J
Ik l ,lk ' l :S kF v
k ,k '
Ikl.l k' l:SkF
2e2
---coN V
and with t he Fourier transform of I /Ir - r/l the exchange potential be comes
2e2
Vx.Slater(r) = - coNV
W it h t he exchange energy Ex(n) = - 3e2kF/ 167l'2co from Sect . 4.4 on e finds
LDA 4 LDA 2Vx (r) = 3"Ex (n(r)) = 3"Vx,Slater(r) .
5.3: The nu mber of discrete k = (k 1 ,k2,k3 ) with k, = 27l'ni/Li, i = 1, 2, 3 with0::; n , < Ni in a Br illouin zone is N = N1N2N3, which is t he nu mber of unit cells
322 Solutions
in the erystal or periodicity volume. Thus, for eaeh elect ron in the unit eell withgiven spin t here is one state in the energy band, i.e., eaeh band ean aeeommodate2N eleet rons .
5.4: A point at the Brillouin zone boundary is eharaeterized by the relation k' =k - G. T he eondit ion of degeneraey is k2 = k,2, thus (k - G? = (k ')2 beeom es2k . G = G 2
, whieh is the eondit ion for Bragg refleetion.
5.5: The primitive reciproeallattiee veetors of the square lattiee are bl = (1, 0)21l" ja,and b2 = (0, 1)21l" / a. Wri te t he free eleetron energies
E(k) = ~(k + G)2 = ~ (21l") 2 ",22m 2m a
for the sm allest G at the points r,M , and X and eonnect eorresponding point s byparabolas defined by G .
If 1/ = 1,2,3 is the number of eleet rons per atom then , for one atom per uniteell, n s = 1/ / a2 is the areal eleetron density. The radius of the Fermi circle is givenby kt: = v21l"n s and t he Fermi energy by
E F = ~k~ = ~ (21l") 2 s:2m 2m a 21l"
or "'} = 1/ / 21l" whieh is 0.159 for 1/ = 1.
5.6: Proeeed as in problem 5.5 and see [111] for the free-eleetron bands along T - L.
5.7: In the almost free-e lect ron pieture, the energy bands of AI, Si, and GaAsderive from the free-eleetron bands of the fee lattiee (see problern 5.6) . Due to thedifferent eryst al struetures (Bravais lattiee for AI, diamond strueture for Si, zineblende strueture for GaAs) the energy gaps are det ermined by different Fouriereomponents of the pseudo-potential :
v. () ""' (T)( RO) V. (G) - ig'T (T)(G )psp r = L vpsp r - n - T ---> p s p = e vpsp .n,T
For AI with T = 0 the strueture factor S(G) = LT
exp (- iG . T) equals 1 for allG . For diamond and zine blende with T = ±T' with T' = (1,1 , l)a/8 one has
Vpsp(G) = e- iG ' T' v(+)(G ) + eiG ' T
' v( - )(G )
= eos (G · T') VS(G) - i sin (G· T') VA(G) ,
where
vs (G ) = v(+)(G ) +vH (G ) and VA(G ) = v(+)(G ) - v(-) (G ) .
In Si t he anti-symmetrie potential VA(G) vanishes. Thus, Fourier eomponents atdifferent reciproeal lattiee veetors determine the energy bands of AI, Si, GaAs.Especia lly, for G = (2,0, 0)21l"[a we have eos (G . T) = 0 but sin (G . T) = 1 andthe ant i-sy mmetrie poten tial present in GaAs removes the degeneraey of t he levelXl in Si (see Fig.5.1O).
5.8: The erystal field split t ing is det ermined by t he matrix formed by
Kv ' v = L Jd3r
4»(r)v(r - R~) 4>v(r)n
with 1/ = x y , yz , zx , 3z 2- r2, x 2- y2. The point group op erations of t he eubie latt ieeturn t he eoordina te t ripIe x , y , z into any other permutation including sign changesof x ,y , and z , while leaving Ln v(r - R~) invariant . Thus, the groups of orbitals
Solutions 323
dxy,dyz, dzx and d3z3 _ r 2 , dx2 _ y2 form invariant sets under the cubic po int gra up,which are classified by the irreducible representations r25, and r12, respectively,and the matrix with t he elements K v'v has block-diagonal form. Further inspectionshows, that each of the diagon al blo cks is itself diagonal for t he given basis withidentical diagonal matrix element s, Ums, the crystal field split t ing gives a 3-fold(r25, ) and a 2-fold (n2) state as ca n be seen at t he r po int of t he band st r uct uresdep ict ed in F igs.5.14 and 5.15. There is no differenc e between sc, bcc, and fcccrystal st ruct ure becau se they have the sa me point gro up .
5.9: T he overl ap matrix Sv 'v(k) is hermitian and ca n be diagonalized by a unitar ytran sformation U: U SU - 1 = S ' with S~''" = S~8," , ,"" The diagonal element s S~ ofthe transformed overlap matrix rep resent t he norm s of the new basis states wh ichare always positi ve. T hus the eigenvalue equat ion ca n be rewri tten
U IIU - 1 UC = E U SU- 1 UC ."-v--'~ '-v-'~
H ' C' S' e r
One can multiply this equat ion with the inverse square raot of the diagonal matrixS ' to ar rive at the eigenva lue equat ion
S, - 1/2 II ,S, - 1/2S I1 / 2C ' = ES' l /2C'
and with ij = S, -1 /2II' S, -1 /2,C = S,l /2C' one has t he standard eigenvalue problem with t he secular equat ion
with iI = S, - 1/2UH 'U - 1S ,- 1/ 2.
5.10: T he near est neighbors in the sc crystal structure are
R~ : a(±I,0, 0) , (0, ±1 , 0) , (0, 0 ± 1)
leading to the disp ers ion
Es (k ) = Es + 2Jss(a)(cos (kxa) + cos (kya) + cos (kza))
which for k = (k ,O,O) becomes Es(k) = Es + 2Jss(a) cos (ka) and for k =(k ,k,k) , Es (k ) = Es + 6Jss(a) cos(ka) with band wid ths E (O) - E((rr /a ,0,0)) =4Jss(a) and E (O) - E ((7r/a,7r/a,7r/a)) = 12Jss(a) , respec t ively.For the bcc crystal st ruct ure one has
o aRn : 2"(±1 , ±1 , ±1) , (Cfl , ±1 , ±1) , (±1 , Cfl , ±1), (±1 , ±1 , Cfl)
leading to t he disp ersion
Es(k) = Es + 2Jss (~) {COS(~(kx +ky+ kz)) +cos(~(-kx+ky+kz))
+ cos ( ~ (kx - ky + kz)) + cos (~(kx + ky - kz)) }
which for k = (k ,0, 0) becom es Es(k ) = Es + 8Jss(a /V3) cos (ka/2) and for k =(k , k, k ), Es (k ) = Es + 2Jss(a /V3)(cos(3ka/2) + 3c os (ka/2)) with band widthsE (0)-E((27r/a,0,0)) = 16Jss(a /V3) and E (O)-E((7r/a, 7r /a , 7r /a)) = 8Jss(a /V3).5.11: For a solut ion see P.R. Wallace, Phys. Rev. 71 , 622 (1947) and the ar t icle byS.E. Loui s in [104].
5 .12: Use the Peierl s subst itut ion E(k ) -t lI(p - eA? with the vector po t enti alA = (0, B(x cos 0 + z sin 0, 0)) corresponding to B = B(sin 0,0, cos O) to write
324 Solutions
2 1 2
H = 2PX +-- (py - eB(x cos(} + z sin(})f+.EE..-.m t 2m t 2m l
T he equations of mo ti on for t he compo nents of t he mom ent um are (up to te rms'" Pu whieh vanish la ter due to py = 0)
i e2 B 2
Px = -·d px , H ] = - - - (x cos {} + a sin e) cos {}fL mt. 2 B 2
pz = - ~ [Pz , H ] = __e - (x cos {) + zsin {}) sin e ,n. mt
Take t he derivati ves of t hese equation with resp ect to t and replace ± = Px/mt , z=pz/ml t o obtain
- Px = wZ cos2 {}Px + WtWt sin ()cos {}pz
- Pz = wZ sin {} cos {}px + WtW/ sin2 (}pz
with w/,t = eB/m /,t. With px,z '" exp (- iwt ) t his becom es a set of homogeneouslinear equations and the eigenfreque ncies follow from
11
w; cos2
{} - w2
WtW/ sin ocos o 11 = 0w; sin {} cos {} WtW/ sin 2
{} - w2
with t he non trivial solution
2 2 2 . 2 2 2 ( cos2
{} sin2 o)W = Wt cos {} + WtW/ Sill {} = e B --2- + - -mt m tmt
The express ion in t he bracket is t he squared inverse cyclotro n mass for t heanisotropie ene rgy sur face if t he magnet ie field includes t he ang le {} wit h the zax is. See [4] .5.13: For k = (kx , ky, 0) and "VV II (OOI) t he interface spin-orb it (or Rashba) termreads
Hso (k ) = al "VVI(kyax - kxa y)
and wit h k± = kx ± iky = kexp (±i~) and a ± = (ax ± ia y)/2
Hso (k , ~) = ia l"VVI (k+a _ - k-a+) = ia' (ei'Pa _ - e-i'P a +) .
T he subband Hamilt oni an wit h spin-orbit inter act ion becom es
(
Ek ia ' e-i 'P )H(k , ~) = . I -i'P .
- l a e Ek
It s eigenvalues E±(k) = Ek ± a l"VV lk do not de pend on ip and are two parabo lasshifte d ag ainst each other . Use the eigenvect ors
Ik ± ) - _1 1 1 ), - J2 =fie' 'P
to calculate t he expec tation value of the vector of Pauli spin matrices:
(k , ± Iu lk , ± ) = ± (ex sin rp - e y cos rp) .
T hus , t he spin is always or iented perpendicular to t he wave vector k = (kx , ky , 0)and rotates wit h ip , Note, that t he states on eac h par abola form a Kr amers pair .See [141].
Solut ions 325
Solutions fo r C hap.6:
6. 1 : Choose t he quantum numbers for t he Bloch states
0= nka, ß = nkjj , 0 ' = n'k 'IJ' , ß' = n'k'jj' .
After carrying out t he summat ion over spin variables t he mat r ix element reducesto
Vaßß'a' = Jd3r Jd3 r' 'l/J~k u (r ) 'l/J~ kü (r ' ) Kir e~ r ' l 'l/Jn' k' u (r )'l/Jn'k' ü (r ' ) .
Decompose t he Bloch fun cti on into plan e wave and lat t ice per iod ic par t s, expandt he products of period ic par ts wit h t he sa me arg ume nt in a Fourier series
u~k u(r)Un ' k'u(r) = L B nnkku (G )eiGor ,
G
and use the Fourier transform of t he Co ulomb interaction to perform the integrationover the space vari ables to find
Vaßß' a' = L B nn'kk ' u (G) B nn'kk 'ü( G')G,G' ,q
e2 Jd3 - i(k- k' - q- G)or Jd3 , - i(k - k' +q- G') .r'
X - V 2 r e r eEO q
For t he single-band approximation set n = n = n' = n' and with only leading termsin t he Fou rier series, G = G ' = 0, write
B nn'kk' u (G ) --> Bkq , B nn' kk ' ü (G' ) --> B kq
and obtain
Vaßß' a' = L v( q) V2
Bkq B kqÖk' ,k_q Ök ,k' -q .q
6. 2: To check normalizat ion and ort hogonality write
L I d3r<p~u (r - R )<Pn' u, (r - R ' )
1 J: """""" - ik ·R- ik' ·R ' Jd 3 ,I,' (),I, ( )= N Va ,u' Z:: e r "Pn k a r lf/n ' k 'a rk ,k'" ,v
_ 1 L - ik .(R-R') J: _ J: J:-- e Un n' - Unn ' UR R 'N " ,k
Thus locali zed fun ction s, represented by Bloch funct ions , are orthogonal whe n cen tered on different sites. In sofar t he Wannier representation d iffers from t he LCAO.Int rod uce fermion operators for t hese locali zed Wannier st ates by
1 """""" ik .RCn H o = r;;r L...J e Crvk a
yN k
and ca lcula te the anti-commutator
326 Solutions
_" s: 1 """" ik .( R - R' ) _ s: , s:- Un ,n ' Uu ,u ' N L...t e - Un ,n 'U u ,u 'U R ,R ' .
k
T he othe r ant i-commutato rs yield
{ C~Rcr ' c~, R ' er' } == { Cn R u , Cn' R ' q ' }
6 .3: Use t he represent ation (6.7) of t he Bloch fun ctions to write
1N2
x Jd3r Jd3r ' </J*(r - R 1 )</J* (r ' -R2 ) t;; l r e~ r 'I </J ( r - R~ ) </J ( r ' -R; ) .
Use (6.8) with the expo nent ials and sums over the wave vectors to replace thefermion operators of Bloch states in the inter action term by t hose of Wannier state sto find (6.10) .
6.4: T he commutator between c+C! and cl Cr
[ t t ] _ t t t tcrcl , Clcr - CrClCl cr - cl cr crcl
is eva luated by using t he ant i-commutator for fermion operators and writ ing
chcl cr = c+(1 - clcdcr = c+cr - c+clcLcr
= c+cr - cl (1 - CrC+ )Cl = c+cr - ClCl + clC!cr c+
where the las t t erm ca ncels in t he commutator . T hu s we have
[ t t ] _ t tcrC!,cl cr - crcr- cl cl '
T he ot her commutat ion rela tio ns
follow in a similar way. T hese t hree commutators are t he same as t hose betweenS ± and SZ and corr espo nd to t he operator algebra of t he Ca rte sian component s oft he angular moment um S X, SY, and SZ.
6. 5 : Evaluate the commutator
[Sf S~, sj] = SfS~Sj - SjSfS~
= - iExß-ySj S~5i,j - S f iExßßj5j ,k ,
to write t he commuta tor of Sj with t he first te rm of t he Heisenberg hamiltonian
for i = j , bl j
for k = i , k i= i
= - i L Jij (S i X S j )xi h
Solu t ions 327
where for ob t aining the last !ine use was mad e of i =I k and of the meaning of theLevi-Civ it a symbol. For the commutator with the sec ond term of the Heisenberghamiltonian evaluat e
"'[SZ SXJ - ", . Sßs: . . - ·sy6 i, j - 61c zxß j U' .J - 1 j .
i . ß
Putting together both con tributions find
dSj i [ X) 1dt = r; 'H. spin , Sj = -r;
6 .6: Eva luate the commut.ator
[CI'k 'O{ ,) = [ukb1k - vkb1k,Uk,b;k' -vk, b2k, )
=UkUk' [b1k,b;k,) +VkVk' [b1k,b2k,) = (uk - vk)Ok.k"'"-v--"" '"-v--""
0k ,k ' - 8 k ,k '
which for (u~ - v~ ) = 1 is a boson commutation relation . In the sam e way, calculate
[CI'k , ßk' ] = [1Lkb1k - vkb1k ,Uk,b2k, - vk, b;k' )
= - UkVk' [b1k,bik ') - VkUk' [b1k,b2k, ) = 0 .
The remaining commutat ion relations are obtain ed in a similar way.
6.7: Using t he new boson operators for coupled magnon-phonon modes, t he hamilt onian becomes
Hp -rn = L { Cl'kCl'k (liw~ cos2 Bk + liWk sin2 Bk - 2Ck sin B k cos B k )
k
+ ßkßk (liw~ sin ' Bk + liWk cos2 Bk + 2Ck sin Bk cos Bk)
+ Cl'k ßk [(liw~ - liWk) sin Bk cos Bk + Ck (cos2 Bk - sin2 Bk))
+ Cl'kßk [(nw~ - ttWk) sin Bk cos Bk + Ck (cos2 Bk - sin2 Bk)) } .
It can be diagon a!ized with
(hw~ - nwi,n) sin Bk cos B k + Ck ( cos" Bk - sin ' Bk) = 0
or
- 2Cktan 2B k = -:---;c-----,,....-,,,-
nw~ - liwi,n
For w~ = wi,n = Wk we have cos2 Bk = sin' Bk = 1/2 and find for the eigenenergiesof the coupled modes
liwJ: = nWk - Ck and liw~ = liWk + ci;
and for the correspo nding boson operators
and
6 .8 : With t he inverted t ran sform ation relations (6.59) write t he t erms of (6.56) :
328 Solutions
bikb1k = akak u~ + ßkßk v~ + (ak ßk + a kßk ) UkVk
b~kb2k = ßkßkU~ + ak akv~ + (ßkak + ßkak ) UkVk
blkb~k = (a ka k + ßkßk ) UkVk + ak ßku% + ßka kvk
b1kb2k = (a kak + ßkßk ) ue v» + a kßku% + ßka kvk
to obtain afte r arranging operators in nor mal order
'Hspin ::::: e, + 2JavS L { (o:kak + ßkßk ) (u% + vk + 2,kUk Vk)k
+ (o{ ßk + a kßk ) (2Uk Vk + , k(uk + vk) ) + 2(,k Uk Vk + vk )} .
Add and subt ract u% to t he las t ter m und er the sum to find with 11% - vk = 1 theexpression (6.60).
6 .9 : Extend (6.53) by t he te rms d ue to t he an isotropy field HA and t he extern alfield Hext
'Hs p in = J a L s.: S 2j - Y/lB L ((HA + Hext) S :i - (I-lA - Hext) S~i)n .n .i j
and rep lace t he sp in op erators by boson operators on each sublattice using t heHolst ein- P rim akoff transfor mation to obtain
'Hs p in = - 2JavNS2 - 4g/lBHANS
+ 2JavS L {blkb 1k +b1kb2k+'k (blk b~k +b1kb2k)}k
+ 2Y/lsHA L (blkb1k + b1kb2k) + g/lBHext L (blk b1k - b~kb2k )k k
Elimi nate the coupling between the sublat t ices using t he Bogoliubov t ransfor mation(6.57) and writ e with t he abbrev iations
C = - 2Ja vN S 2 - 4g/lBHANS , A = 2Ja vS , B = 2911BHA , C = g/lBHext
'Hs p in ::::: C + L { A,k (2ukvk (ak ak + ßkß k + 1) + (u% + v~ )(akßk + a kßk ) )k
+ (A + B) ((uk + vk )(a ka k + ßkßk ) + 2UkVk (ak ßk + a kßk) + 2vk)
+ C (akak - ßkßk ) } .
Diagon alize with A,k(uk + vk ) = - 2(A + B )UkVk. Square t his rela t ion and use
uk - Vk = 1 to get a b iquadratic equation for Uk. Take its solut ion to find
2 2 { (A + B ? }1/2 1{ A2,k }1/2Uk + Vk = ± (A + B F _ A2,k ' UkVk = =F 2 (A + BF - A2,k
Ident ify t he prefact or of t he magnon nu mb er ope rators as t he magnon energy
nWk = {( A + B) 2 _ A2, k} 1/2
which wit hout anisotropy field (B = 0) reduces to t he disp ersion relation for theant ifer romagne t ic magnons which for sm all k is linear. W ith anisotropy field one
Solutions 329
find s a finite magnon energy for k = 0 and a qu adratic dispersion for small k ascan be seen in F ig. 6.8.
6 .10: Evalunte for low tempe rat ure (T « Tc ) the magnet ization (per unit volume)
M(T) = sue (NS -~(blbk )) or
where the qu ad rat ic dispersion for small k is used. The int egral (it is a Bose integral)can be solved as described in t he Appendix to yield Bloch 's T 3
/2 law:
M(O) _ M(T) = ((~) g/lB(~) 3/2 T 3/2 .2 M(O) 41rD
See [154].6.11: Calculate the expec tation value in (6.106) for T = 0:
m
Replace the op erators M ± by t he fermion op erators an d extract the exponentialswith the time dependence. The energy differen ce E m - Eo of exact eigenstatesbecom es in HF approximat ion t he energy difference of single-p article energies forparticle-hole excit at ions with spin-flip across t he Fermi energy. By manipulationsas in Sect . 4.5 one arrives at the spin suscepti bility (6.108).
In order to evalua te (6.108) , use
n2 n2
Ek +q T - Ek! = - .1 + -k . q + Eq , Ek T - Ek - q ! = - .1 + -k . q - Eqm m
with Eq = n2l /2m and perform the sum over k with the subst it ut ion k . qkq cos e, cos e= x . The first term can be written
A ! = "" 1~ hu: + Ek + q T - Ek !
Ikl :Sk F I
V l k F
! 1+1
1= -- dkk2 dx.(21r)2 0 - 1 Iu» - Ll + Eq + n2 kqx / m
The int egral over x can be solved according to
1+1~ = ! la
+b
d z = ! In 11 + b/a I .
- 1 a + bx b a - b Z b 1 - b/a
Looking for collective exci t ations at small q one has b/a « 1 and can use theexpansion
330 Solutions
!lnI1+b/a! ~ ~ {~ +! (~) 3} = ~ {I +! (~) 2}b 1 - b/ a b a 3 a a 3 a
to obtain
l kF { 2 ( 2 ) 2 ( ) 2 }A - ~ 1 dk k2 1 +~ !!:...!!. 1! - (27f)2liw - L1+€q ° 3 m liw - L1+€q
After integration and corresponding evaluat ion of t he second term A T one arr ivesat (6 .108).
Solutions for Chap.7:
7.1: Evaluate t he commutator [Ck ,iT ' c~+qerct ' _qer,Ck 'erlckerl by successively interchangi ng C;;; ,iT with the four operators appearing in the interaction t erm. Each stepleads to a change in sign and in add it ion gives a Kronecker 6 for exc hange with thecreation operators, t hus
In terchange in the seco nd term t he las t two op erators and repiace C7 by C7' and k byk ' . Repl ace in t he first term q by - q and cons ide r t he prop er ti es of the int eractionmatrix eleme nt t o find t he ex pression for [C;;;, 'Hint] (7.20) . The commutator [4,'Hintlis eva lua ted by analogous steps,
7.2: T he equat ion of motion of t he full Gree n fun cti on G( k C7 ;t - t') (7.21) ca n bewr itten wit h (7 .23) as
(in:t - € ker ) G( kC7;t - t') = 6(t - t') +Jdt" E(kC7 ;t - t")G(kC7;t" - t') .
Rep lace t he Gree n fun ct ion and t he self-energy by t hc ir Fourier t ransform withrespect to t he ti me arg uments and identify t he in tegr ands to obtain
(liw - € ker ) G( kC7 ;w) = 1 + E (kC7 ;W)G(k C7 ;w) .
After mult ip lication with t he Green function of t he non-interacting system
GO(kC7; W) = (nw - € ker )- l
one arrives at t he Dyson equat ion
G( kC7;W) = GO (kC7 ;W) + GO( k C7 ;W)E(kC7;W) G(k C7 ;W) .
7.3: The commut at or of ni-erCier with t he single-p article part Ho yields three t ermsdue t o interchange of creat ion and annihilat ion op erators:
[ni- er Cier , Ho] = L tmj (CL er Ci - er Ci er C~er , Cjer ' - C~er, Cjer'CLerCi -erCier )m i o '
= L tmj (6im6er er ,ni-er Cjer - 6 i m 6 er- er, c L erCi erCj - er
nvi o '
- 6ij 6 er- er, C;"_er Ci - erCi er)
= L tim ( n i - crCm cr + C;_ erCm-crCi cr - C;" _ erCi - crCicr ) ,
rn
Solu tions 331
where the last line is obtained by prop er choice of the summation ind ices andrearrangin g the ope ra tors. Similarl y the commutator with the interaction term isevalua ted
[ni-aCia, H1 ] = ~UL (ni -aCian ma,nm- a' - nma ,nm-a,ni-aCia)rno '
~UL ni-a (Cia C~a, Cma, nm-a' - nma,nm-a,Cia)rnc"
= ~ UL ni-a (ÖimÖaa'Cian i- a + ÖimÖa-a ,ni-aCia) ,-mo '
wh ich using n7- a = n i-a becomes (7.55) .
7.4: Calculat e t he derivative of the self-energy (7.70) and obtain the spec t ral weight
(E - 100 - U(l - (n_a) ))2 + U2(n _a )(1 - (n-a) )
For the lower Hubbard band with E = 100+ 2t cos ka this becomes
(2t coska - U(l - (n _a) ))2Z'ower = 2
(2tcos ka - U(l- (n - a) )) + U2(n _a)(1 - (n - a) )
and reduces for k = rr/2a t o Zlower = 1- (n - a ) . For the upper Hubbard band withE = 100+ U + 2t cos ka the resul t is
(2t cos ka + U(n_ 0' ) ) 2
Zupper = 2(2t coska + U( n - a) ) + U2(n _a )(1 - (n - a) )
and Zupp er = (n - a) for k = tt / 2a. Note that the spec t ral weights for k = rr/ 2a aret hose of the atomic limit (see Fig. 7.3).
7.5: This problem is solved in some detail in [157,189]. It leads to the so-calledt - .J model, where t is the hopping integral and .J '" e/U is the effective exchangematrix element .
7 .6 : Describing the delocalized electrons by fermion operator s cL , Cteo in a bandwith dispersion Ek and the localized electrons with energy IOd by ferrnion operatorsdt , da , t he Hamiltonian is forrnulate d as
H = L EkcLcka +L E dd~dak ,O" o
k,a
Here th e third term describes t he hybridi zation between t he locali zed and delocalized elect rons. This is the Anderson impurity model. It can be ex tended byconsidering inst ead of a sing le impurity a periodic configuration of sites i with delect rons. For this case t he fermion operators for d elect rons becom e dIa' dia andsummation over t he sites i is to be considered . See P.W . Anderson: Phys. Rev . 12441 (1961) and [55]Sect . 6.2, [112, 178].
7 .7: For t he solution see [112, 178]. It uses a contour int egration in the complexfrequency plan e t aking into account the po sition of t he pol es for qu asi-particlesand holes.
332 Solu tions
7.8 : Start from the expression (4.134) for t he real part of t he dielectric fun ction,whi ch for T = 0 reads
( ) 1 L 2Ek - Ek+q - Ek- qc l q, W = + Vq .
(nw - Ek + Ek_q)(nw - Ek+q + Ek )I k l ~ kF
Making use of t he free electron disp ersion Ek = n2 k 2 12m t he numerator simplifiesto - n2l Im while t he denominator takes t he form
(n2 ) 2 (n2 2) 2(nw - Ek + Ek_q)(nw - Ek+q + Ek ) = tu» - m k · q - 2m q
and t he expression to be eva luated by integrat ion is
"" 1 ( 2m) 2 "" 1s: ( 1t2 ) 2 ( 1t2 2) 2 = f? c: ( 2 _ 2k . )2 _ 4 'Ik l9F lu» - -:;;: k . q - 2m q I k l ~ k F qs q q
with q; = 2mnwln2.For d = 1 the vectors becom e scalars and t he sum can be written as t he integral
(us ing the substit utio n x = 2kq )
L J kF
d k L 1 J XF d kI (q ,w )= 21T -kF q: - q4 - 4q; kq + 4k 2q2 = 21T 2q - XF x 2 - 2q; x + q: _ q4
T he int egral can be found in int egral t ables giving
L 1 I(2k Fq - 2l? - 4q: II(q ,w) = 21T 4q3 In (2kFq + 2q2)2 - 4q: .
For w = 0 (or qs = 0) t his integral diverges for q -> kF.For d = 3, consid ering from t he beginning t he simpler case w = 0, t he corre
sponding int egral is
I (q) = (L3)
3 21T l kFdk k
2J +l 4k 2 2
d co~ ~ _ 4 = 8 L2
33 1
kFd k In Ikk - q I'
21T 0 - 1 q cos q 1T q 0 + q
and yields by int egration
I (q) = 8~3q2 { ( k; _ 1) In (~ _ 1) _(k; + 1) In ( k; + 1)}For q -> kF t his int egral remain s finit e.
Solutions for Chap. 8:
8 .1 : The operator for linear elect ron- phonon int er action
'Hel - p h = - L V'lv (r l - Rn-r I R~ "T .Un"T
l ,n ,T'
can be wri tten in t erms of ph onon and electron op erators using (3.22) together with(3.39) and by rep lacing with (4.74)
L V'lv(rl - Rn "T I R~ "T -> L (nk l V'v l nl k')C~kCn' k "I n, k ,n ' ,k '
Solutions 333
The sum over the lattice sites n effects on ly the gradient of the potential and hasthe form of aBloch fun ction
'"' iq.R:;' I _ iq·rL e \Jv (r -Rn-r ) RO - e Uq-r(r)n.,.
n
where Uq-r (r) is a lattice periodic func tion. Aft er decomposition of t he electronBloch functions in t he matrix element into plan e wave and lattice per iodic partthe product U~k (r )un ' k' (r )Uq-r (r) can be expanded in a Fourier seri es. T hus theint egration over the crystal volume can be carried out with
Jd3 - i(k- k' - q- G)·r s:r e rvUk,k'+q+G
and yields t he relation between the involved wave vectors.
8. 2 : Using (3.76) one has OU;jOXi = Eii and can write
\J . u (r ) = L O~i ui(r) = L Eii = TrE .i i
For u (r) rv eiq.r follows \J . u = iq . u which differs from zero on ly for longitudinalphonons.
8 .3 : For transverse phonons with q = L " q"e" , u = L ß ußeß we have in general
q . u = L" q"u" = - iTr E = 0 but e.g. for the special case q = (q , 0, 0) , u =(0,U2, U3) the st ra in t ensor components are
i iEU = E22 = E33 = E23 = 0 , but El 2 = '2qU2 , El3 = '2qu3 ,
giving a nonvani shing cont r ibut ion to the elect ron-phonon interaction.
8 .4: Start with the classica l express ion for the interact ion energy (8.22) with
\J . P (r ) = - ~ e I4 1 6' ij k l (qiqkUj + qiqj Uk ) eiq.r double ind ex summation
= -2e14 (qyqzu x + qzq xUy + qxqyuz ) eiq
.r
.
Usin g (8.23) for the charge density, expressing t he displacement field by phononoperators, and integrating over the cryst al volume convert s Eint into the op erator
1t~- ph = _ 2e el4 '"'6'06' 00 L
s ,q
tiN qxqy e~(q) + C.p .
2Mws (q) q2
x (a1(- q) + as(q)) L Ci +qCk .k
8. 5 : To be calculated is the expression (T) (8.70) with t he given dependencies of T.The integrals are solved with the substitution E/kBT = x and we may write
1
00 3/ 2 - €/ knT d (k 1 ') 5/2 100
3/ 2 -xd T 5 / 2E e E= B X e X rv .
o 0
Sim ilar for T rv T - 1E- 1/ 2
100 3/ 2 - €/knT d T - 1 (k T)2 1 °O - xd TTE e E = B xe x rv
o 0
334 Solutions
and for T ~ T - 1E1/ 2
100 3 /2 -< /kBT d T - 1 (k T )31OO
-xd T 2T E e E = B xe x ~ .
o 0
Thus for deformation potential coupling J1 (T ) ~ T - 3/2 and for piezoelectric coupling J1 (T ) ~ T - 1/ 2. This explains t he different slopes of the correspo nding graphsin Fig.8.5. For more det ails see [4,227] .
8.6: For t he solution see [4], Cha p. 7. The calculat ion is essentially the same asfor t he Fröhli ch coupling bu t with the l /q dependence of the inter act ion po tenti alrepl aced by a .J(j dependence. The number of virtually excited ph onons being proportional to the squar e of t he effect ive mass of the elect ron t urns out to be mu chsmaller t han 1. Replacing the electron by the much heavi er ion wou ld increase thisnumber to a value much lar ger than 1 indi cating that the perturbation calculationis not appropriat e.
8 .7 : We have to evaluate the commutator
['Hel- ph, S] = L VqVq' [Cl +qCk (a~q + aq) , cl'+q,Ck' (Qa~q' + ßaq, ) ]k ,q
k ' .a'
and write
[... , ...] = [a~q + aq, Qa~ q' + ßaq,] cl+qCkCk'+q,Ck'
+ [Cl +qCk,ck' +q,Ck'] (Qa~q' + ßaq, ) (a~q +aq)
The elect ron part of t he first term ca n be rearranged giving
cl+ qCkCk'+ q,Ck' = ck+qCk' +q,Ck,Ck + 8k,k'- qTtk'
wh ere t he first te rm has t he structure of an elect ron-elect ron inter action (t his isthe one we are looking for ). The ph onon part of t he first term gives
[a~q + aq, Qa~q' + ßaq,] = (-Q + ß )8q,q' .
The seco nd te rm, being bilinear in t he ph onon operators , is neglected as weil asthe term with the elect ron number operator to obtain t he approx imate form of thecommutator ['Hel-ph, S].8 .8: Using (8.119) the ground state expectation value of H rela ti ve to the groundstate of the normal sys tem is
Eo = (H) + L IE(k)1k
and can be written with
(Ukz) = COSBk , (Ukx) = sinBk and ~lf L sin Bk' = E(k)tanBkk '
as
Eo = - L E (k ) ( COSBk + ~ t an Bk sin Bk) + L IE (k )1.k k
Wi th
2<12
E (k ) tan Bk sinBk = -v. andelf
Solutions 335
and by replacing the sum over k by an energy integral over the she ll with thickn essnwo at the Fermi energy, one find s
1!iWD { 2 }.12Eo = 2D(EF) 0 de I' - (1'2 +ELF)l /2 - Veff '
where D(EF) is the density of states at the Fermi energy. The int egrat ion can easilybe performed and yields
The last expression, obtained for 1 » D( E F)Veff , teils us that the superconductingstate is stable as long as the effective interact ion is positive.
Solutions for Chap. 9:
9.1 : Besides the poles of Go(E) the full Green function G(E) has addit ional po lesfor 1 - Go(E)U = 0. In site representation U has block-diagonal form with a nonzero block U1 = U{R[} only in the diagonal for lat t ice sites around t he impurity.With Go(E) written in t he cor responding block form (with Go,{Rd(E) = GÖ(E)and corresponding matrices for the other blocks) , the matrix mu lti plication can bepe rformed to give
_ ( GÖ(E ) G6(E) ) ( U O) _ ( GÖ(E )U1
0 )Go(E)U - G5(E) G6(E) °° - G5(E)U 1 ° .
The determinant of 1 - Go(E)U is the product of the det erminants of it s diagonalblocks or 111- Go,{Rd(E)U{Rd ll = 0.9. 2 : The average of the individual resis tances is
N
(r) = ~ L rii= 1
and thereforeN N
(R) = (L Ti) = L (ri) = N (r ) .i= l i= l
The variance of R is given by
VarR = (L rirj ) - N 2(r )2 = (L r; ) + (L rirj) - N 2(r )2i j i# j
and with uncorr ela ted ftuctuations of the r ,
The relative variance given by
VarR(R)2
1 VarrN (r) 2
vanishes for N -; 00 •
9. 3 : T he transver se response reads
x(O,w) = e2 L (ilvln (f ' lvli') * 1=dT([C;(T)ct(T) ,Cj,C;t] )o .
i,J,i',J' 0
The time dependence of the operators gives a factor exp (i( Ei - I' f )TIn) and theintegral over T yields
336 Solutions
i 100
dTek Cnw+c;-</+io)r = _ 1h 0 Iu» + 10; - 10f + io '
where t he small parameter 0 is introduced to regularize t he int egral. To eva luatethe remaining t hermal expectation value first calc ulate the commutator
With t he t hermal expectat ion value of (crc;' )o = f( 10;)0;,;1 the response functiontakes the form of (9.87)
X(O ,w) = _ e2~ 1(ilvl J)1 2 f(E ;) - f (Ef ) . .L....- lu»+ Ei - Ef + 10i ,f
9 .4: Write (for finit e w) the 0 function in the form
o(hw+ Ei - Ef ) = JdEo(E - Ei)O(E+ hw - Ef)
so that with Ei = E an d Ef = E + hw the conductivity is expressed by
O"(O,w) = -rre2 JdEf(E) - f~E + hw) L l(il vl f )120(E - Ei )O(E + hw - Ef) '
i ,f
The 0 functions can be replaced wit h (2.77) and E + = E + io by the imaginarypart of the corresponding single-part icle Green functions and give for the doublesum expression under the integral
L '" = L (i lv lf) (J lv li) ImGii (E+ )ImGff (E + + hw)i , f i , f
= :2 I > lvG(E + + hW)/J) (J lvG(E+) li ) ,i ,f
whic h for w -> 0 leads to Tr(vlmG(E+) vlmG(E+)) .
9 .5: Write ß (g) in the form
dg Lß(g) = dL g .
For 9 -> 00 use geL) = O"Ld-
2 and 0" indep endent of L to find
dg = a(d _ 2)Ld - 3 = (d - 2)2..dL L
which means
lirn ß(g) = d - 2 .g~oo
For 9 -> 0 use geL) '" exp (-L/,\) with dg/dL = - g/'\ and
lim ß(g) = - ~ .g~O /\
The sign of ß(g) is det ermined by dg /dL. For d ::; 2 and assuming a monotonousfunction it is always negative while for d > 2 t here is a sign change.
Index
absorption coefficient 56acoustie phonon 47adiabatic- approximation 37- potential 38, 44alkali halide 7alloy 263Anderson model 212anti-ferromagnet 171,175approximation- adiabat ie 21- Born 268- Born, self-consist ent 268- Born-Oppenheimer 21,223- coherent potential 268- continuum 52- Cooperon 274- effect ive-mass 144, 150,258- harmonie 39- Hartree 96, 118, 163- Hartree-Fock 95,97, 162, 176, 180,
199- local density 121,123- local spin density 158- mean field 98, 176, 248- molecular field 175- one-band 257- random ph ase 104, 183, 271- relaxation time 235- self-consistent field 104- t-rnatrix 267- Thomas-Fermi 135- tight-binding 139, 160, 195- virtual cryst al 267atomie limit 198
back scat te r ing 274band- conduction 134- heavy hole 149- ind ex 118,125- light hole 149
- structure 118, 124, 125, 131- valence 134, 143band gap 129, 143- engineering 152- problem 124band structure- free elect ron 131- of Al 132- of Cu 141- of Fe 158- of Si 134- of transition metals 141BCS theory 242Bothe-Salp et er equation 273binding- chemieal 4, 61, 75- covalent 16,61,64, 141, 144- het eropolar 16,61- homopolar 16- ionie 16- metallic 16- van der Waals 16Bloch- elect ron 124, 127, 180- equat ion 167- funct ion 125, 126, 137,257,271 ,281- representation 167, 169, 172- st ate 255- theorem 6,41 ,281Bogoliubov tran sformation 172, 208Boltzmann equat ion 234bond charge 61, 144- model 61Born series 260, 264Bose-Einst ein- conde nsat ion 252- distribution 47Bragg condition 129Brillouin- function 177- zone 6,41 ,43, 49,118
8047123, 158
338 Index
bulk modulus 70
carbon nanotubes 143,207center coordinate 85central-cell- correction 258, 259- potential 259characte r t able 281chemical potential 194cluster 4, 17collective- excit at ion 41- mode 41,43collision term 234compliance constant 52composite fermion 220compressibility 54conditions- periodic boundary 41,77conductance 277conductivity- dc 272,276- electric 25,32,223,270- heat 32configur at ion- average 265- equilibrium 18, 63- ofions 38const ant
com p lia nce 52elastic 51
- Madelung 19stiffness 51
Cooper pair s 246cooperon approximation 274correlation 12- density-density 31- effects 189- elect ron 82, 189- energy 108
hole 112spin-spin 30
coupling- exciton-photon 173
phonon-photon 173- piezoelectric 231- plasmon-phonon 173- spin-orbit 17crit ical- exponent 178- po int 47- t emperature 157, 175,241 ,251crystal
liquid 4- mixed 7, 263
molecular 7- momentum 5- photonie 8- quasi 4- structure 5crystal field split t ing 139cryst a lline structure 4Curie temperature 179Curie-Weiss law 179current density 271- diamagnetic 271
electric 25heat 25
- par amagnetic 271cyclotron frequency 83
de Ha as-van Alphen- oscillat ions 131, 133Debye- frequency 49,245,251
law 50- model 48, 170- temperature 49,251Debye-Waller factor 50,68,73defect- ant i-site 256, 262- Frenkel 256- point 256deformation potential 227degeneracy- Kramers 126- spin 126density- average 11- fluctuation 11,205,208- paramet er 78,97,109density functional theory 63, 118, 120density of states 151, 182,251,261 ,
275- elect ron- phononDFT-LDAdiagram- ladder 273- maximally crossing 274diamagnetism- Landau-Peierls 89diamond 49dielectric- function 25, 31- polarization 25- screening 30
dielectric constant- high frequen cy 58- st atic 58dielectric function 56- elect ronic part 101- longitudinal 101- transverse 101dimensionality 8, 149, 206, 207, 278dipole- approximation 56- moment 54- operator 55direct- exchange 165- t erm 96dis location 262disorder- compositional 7,67,255,263- diagonal 263- isotope 263- off-diagonal 263- st ruct ural 4,7,255,263,276- su bstitutional 276distribution- Bose--Einstein 47- Fermi-Dirac 80, 272doping 256Dulong-Petit law 48dynamical matrix 41Dyson equ ation 193,264
edge channel 221effect- de Haas-van Alphen 90, 131- qu anturn Ha ll 152effective- 9 factor 85- mass 85,89, 147- single-particle potential 117, 151Ein stein relation 278elastic- constant 51,53- modulus 52electric conductivity 25,223electron- closed shell 15- core 75, 136- gas 75- valence 15, 75, 136clectron system- low-dimensional 83- two-dimensional 150, 189, 215electron-electron interaction 161,279
Index 339
electron-phonon interaction 223elementary f1ux quantum 86energy- band structure 118- dissipation 57,233- fun ctional 122- gap 129,259- loss spectroscopy 103- relaxation 234energy bands- of Al 132- of Cu 141- of ferromagnetic Fe 159- of Ge 143- of Si 134- of transition metals 141energy gap 134, 143energy- Ioss- fun ction 31, 105, 107,271ensemble- canonical 23- grand-canonical 24- mean value 22equilibrium- configurat ion 18- position 18, 38- thermodynamic 20exchange- charge density 120- direct 165- energy 159- hole 100- Rudermann-Kittel 165- t erm 97excitat ion- collective 41, 107, 165, 183- elect ron-hole 103- elem entar y 43- particle--hole 106- spin-flip 182
Fermi- circle 130- contour 131- energy 78, 141, 144- gas, ideal 78
hole 100, 112- int egral 81,287- liquid 189
sphere 77,78,91 ,95,96, 103, 106,108,159
- surface 90, 131,247- temperature 79- velocity 79
340 Ind ex
- wavelength 79Fermi surface- of Ag 133- of Al 133Ferrni -Dirac- d istribution 81, 204fermion- annihilation operator 92- composit e 152, 220- creat ion operator 92- heavy 82,89,210- vacu um 92ferrom agnet 175ferromagnetic- ground state 163, 166- insu lator 162, 165ferromagnetism 182- strong 183- wea k 183field quantization 45filling factor 86 , 152fine st ructure constant 231fluct uation- density 101- number 101force constant 40,51,60four -ce nt er integral 196Fröhlich- coupling constant 231- polaron 239fractional- cha rge 220- filling 215- quantum Hall states 215free energy 70, 88, 204function- dens ity-density corr elation 9- dielectric 30- pair- distrib ution 8,9,100- Wa nni er 161,261
ga p- direct 145- ind irect 145- par ameter 251Gr üneisen- parameter 71- relat ion 71gradient expansion 123grand-canonical- ensemble 24- potent ial 88Gr een function 28
- retarded 33, 190- spectral representation 33- two-part icle 273ground st at e 4,77,95- energy 78,97- of t he lat t ice 38,45- theorem 109group- of the wave vector 6
po int 6,40, 229- space 7- trans lation 5
harmonic- approxi mation 39,44- oscillator 39Hartree- approximation 96- equation 119- self-energy 193Har tree approximation 163Hart ree-Fock- approximation 95- eq uation 120- self-energy 193Har tree- Fock approximation 199Har t ree-Fock approximation 176heavy fermion 82,89, 210Heisen berg- Hamiltonian 162, 164, 175- model 164het erostruct ur e 149,207hierarchy problem 191Hohenberg-Koh n t heorem 121hole 144Holst ein - P rimakoff tran sformation
168Hooke 's law 52hopping matrix element 161,212Hubbar d- band 200,269- Hamiltonian 196- model 195Hund's rule 166hybridizat ion 141,158, 212
impurity 255, 275- conce ntration 264- deep 259- intersti t ial 256- shallow 257,259- substitutional 256insulator 134, 195,276- anti-ferromagn et ic 171
- ferromagnetic 165- Mott-Hubbard 200interaction~ effect ive elect ron-elect ro n 243- electron-electron 117- elect ron- phonon 223- exchange 157
fermion-boson 226- van der Wa als 62interface 262irreducible- di agram 267- represen t a tion 141,281Ising model 164isot ope 67- disorder 263- effect 242,252
J astrow-typ e wave fun ction 219jellium model 19, 76, 94, 117
k.p theory 146Kohri-Sham equation 121, 123Kramers deg eneracy 126Kr amers-Kronig relation 57Kubo formula 28Kubo-Greenwood formula 273
Lame const ants 54Landau- level 84,269,276- qu antization 91lattice- body-cent ered cubic 6
Bravais 5- displacement 18, 38- dis tortion 259- dynamics 37, 42
dynamics, nonlinear 44face-centered cubic 6point 5reciprocal 5, 10
- thermal expansion 44- t ranslation 40- vector 5- with basis 5Laughlin wave funct ion 219LCAO method 259LDA + U 201lifetime 223, 232- qu as i-particle 269- single-part icle 274Lindhard fun ction 104, 206linear response 24
Index 341
liquid 8- Bose 202- clas sical 202
Fermi 202Luttinger 207
- qu antum 202localization 215- length 255- weak 274Lyddan e-Sachs- Teller relat ion 59
Madelung const ant 19magn etic- length 84, 217- order 157magnetism 160- itiner ant elect ron 180- theory of 164magnet ization 176, 180magnon 157,165,168,184,226magnon d isp ersion 185mass- effect ive 147, 151- op erator 123matrix eleme nt- hopping 161- transfer 161,263matter- condensed 4- soft 4mean field 176mean free path 274mesoscopic- phy sics 152- regime 265- syst em 17rnet al 16,49,60,108,195,276- alkali 82- noble 6,79,82, 131, 139- normal 6, 79, 131, 139- transition 6,82, 140, 165metal-insulator~ transit ion 276mobility 237- edge 276model- Anderson 212- bond charge 61- Hubbard 212- jellium 76- rigid ion 60- shell 60- St on er 180
342 Index
- Tomonaga-Luttinger 210, 221modulus- bulk 54- rigidity 54- Young 54molecu lar field 176
nanophys ics 8nonlinear lattice dynamics 44nonparabolicity 147normal coordinate 42,43, 55
occupation- nurnber 45, 93- nu mb er operator 46, 93operator- annihilation 44- creation 44- field 271- stat istical 23orbital- S p2 143- sp3 143- ant i-bond ing 143- bonding 143- directed 143- hybrid 143- localized 162order- fer rim agnetic 165- ferromagnetic 165- par ameter 178, 252oscillat ions- de Haas- van Alphen 90, 133- qu antum 131- Shubnikov-de Baas 91, 131overlap integral 138
pair- electron-hole 134,144pair-distribution- function 9,11 ,12,100,111par amagnetism- Pauli spin 89particle- independent 189particle-hole- cont inuum 106, 185- excit at ion 106parti t ion fun ction- ca nonica l 24- grand-canonical 24, 46periodic- boundar y condit ion 125, 285
phase- ferromagnetic 178- liquid 4- par amagnetic 178- solid 4- t ran sition 175, 178phase transition 179, 202phonon 19, 37, 45- acou stic 47- confined modes 66- disp ersion cur ves 60- energy 45- focusing 53- hot 233- lifet ime 57- longitudinal 56- op erator 55- optical 54- surface 65- t ransverse 56- vacuum 45- virtual 239phonon dispersion- of C\'-Fe 60- of AI 60- of Cu(100) 66- of GaA s 64- of GaAs(110) 66- of KI 60- of Si 62- of solid 4He 62phonons- acoust ic 227- optical 229photonie- band structurc 8- crystal 8piezoelectric- coupling 231- effect 231- tensor 231plasm a frequency 107plasmon 107plasmon dispersion 185plasmon-phonon modes 108point defcct 255point group 281pol ari zation- func t ion 104pol aron 224, 239- ene rgy 241- Fröhlich 239- mass 241
polymer 207potential- effective 118- adiabatie 22,51 ,62- chemieal 24,46,80, 88- effective 120- effective single-particle 18, 117
empty core 135- exchange-correlation 151- grand-canonieal 88- Hartree 119, 151- self-consistent 260process- normal 226- Umklapp 226propagator 190pseudo-potential 135, 137
quantum- liquid 202- oscillations 88- wire 207quantum Hall- effect 91, 152- effect, integer 153- plateau 153quantum limit- elect r ic 151- magnet ie 88, 152quantum weil 149qu asi-hole 194qu asi-particle 99, 127, 194, 199,202,
203, 205, 210, 211- correction 123- dispersion 213- distribution 206- effect ive mass 205- energy 123,204- lifetime 269- weight 206quasicrystal 4
reflection coefficient 58relation- dispersion 33- Kr am ers -Kronig 33,57relaxation- electron-Iattice 223relaxation time- transport 223, 270representation- occupation number 44-46, 92, 232,
248
Index 343
resonance- cyclotron 84- elect ron spin 84- paramagnetic 84- spin-flip 84response- fun ction 28, 35, 52, 55- linear 24,270- nonlinear 32- transverse 271Reststrahlen band 58, 65
scaling theory 278scanning tunneling mieroscopy 8scattering- back 236- backward 270- forward 236, 270- inelastic 59- neutron 59- rate 232, 234screening 101- Themas-Fermi 107, 135SDFT-LDA 158self-averaging 265self-consiste nt- harmonie approximation 73- solution 119, 124self-energy 193, 198,200,202,227,
232,266,273- exchange-correlation 123- Hartree 193- Hartree-Fock 193- singl e-p article 192semiconductor 7, 16, 79, 108, 131, 134,
143-145,256shell model 60Shubnikov-de Haas oscillations 131Slater determinant 92,119soft mode 73solid- amorphous 4, 8, 263- crystalline 4, 8Sommer feld- coefficient 82, 205, 210, 215- expansion 81,289- model 78, 96sound- propagation 48, 53- velocity 49sound propagation- longitudinal 53- transverse 53sp ecific heat 48,71 , 171,203,205
344 Index
- of elect rons 81- of phonons 48sp ectral funct ion 34spin 17- degeneracy 126- density 158- pol ari zation 123, 158, 180, 186- susceptibility 183, 215sp in polarized electrons 159spin suscept ibility 203spin waves 157, 165, 168- in ant iferromagnets 171- in ferromagnets 165spintronics 157split t ing- longitudinal-transverse 65state- ant i-bonding 143,259- bonding 143,259- extended 6, 255, 275, 276- localiz ed 7,255,275,276sti ffness constant 51Stoner- condition 182, 186- cont inuu m 183- model 180, 182strain tensor 52, 231structure- diamond 7
fact or 8- rocksalt 7- zinc blende 7structure fact or 70- dyn amic 11,68, 105, 109- static 10, 109sllb band 149super-exchange 165supe rconductivity 224, 241superconductor- high-T, 7superlattice 66, 131- isotope 67surface 8, 262- magnon 157- phonon 65susce pt ibility 28- dielectric 28, 55- magneti c 29, 89system- mesoscopic 17
t-J modelt-matrix
331264,270
temperature- crit ical 175- Curie 179- Neel 175term- direct 162- exchange 162theorem- dissipation-ftuctuation 35, 206- Hoh enberg-Kohn 121thermal- average 9- energy 48- expansion 44, 70- expec tat ion valu e 22,46, 190, 271- lattice expansion 44time reversal 126t ran sfer matrix element 161, 195,263transformation- Bogoliubov 172, 208- Holstein-Primaleoff 168- Schri effer-Wolff 243t ransition- metal-insulator 200, 255, 276tran slation- op erator 124- pr imitive 18translation group 281transport- linear 235- nonlinear 235- relax ation t ime 223, 235, 274tran sverse- effective charge 59truncation 191
vacuum- fermion 92van Hove singularity 47velocity- of sound 48- sound 49, 53vertex op erator 273Voigt notation 52,231
Wannier- representation 169Wannier function 161,261warping 53, 147, 149Weiss constan t 176Wigner crystal 112,220Wigner-Seit z cell 5, 125
XY model 164
Zeem an termzero poin t- contr ibution- motion 45
83
173
zone scheme- extende d- reduced- rep ea ted
128128128
Index 345
Also 0/interest:
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