Condensed Matter Physics 2016 Preliminary Planning 24/11

Condensed Matter Physics 2016 preliminary planning 24/11 – 15/12

24/11 Phonons 25/11 Electron-phonon interaction 29/11 Temperature-dependent transport 2/12 Superconductivity: BCS theory 6/12 Superconductivity: Ginzburg-Landau theory 8/12 Quasicrystals (Stellan Östlund) 9/12 Topological superconductors (Jan Budich) 13/12 Phase transitions and broken symmetries 15/12 Magnetism (note: 4 hours)

All lectures start 13:15. See TIME EDIT for lecture room. Condensed Matter Physics 2016 Lecture 24/11: Phonons

1. Lattice dynamics 2. Quantization: Phonons 3. Phonon band structure 4. Density of states 5. Heat capacity 6. Phonon interactions

References: Ashcroft & Mermin, 22, 23 Taylor & Heinonen, 3.5-3.9 Mahan*, 7.1-7.4

* Condensed Matter in a Nutshell (Princeton University Press, 2011) 1. Lattice dynamics 1 1

Crystal Hamiltonian H = He + Hc + Hec

H = He + Hc + Hec

Hc = K + U (1) 2 2 p 1 ZnZn e Hc = K + U = n + 0 = K + U (1) (2) 2 22Mn 2 rn rn0 p 1 Z Z ne n,n0 | | = n + n Xn0 = K +XU (2) 2M 2 r r n n n n0 X n,nX0 | | Recall the Born-Oppenheimer approximation, treating the cores as ﬁxed (providing a static periodic potential in which the electrons move). We shall now remove this simplifying assumption, and study the effect of including also the dynamics of the cores. 1

H = He + Hc + Hec 1

H = K + U (1) c H = H + H + H e c ec 1 p2 1 Z Z e2 = n + n n0 = K + U (2) 2Mn 2 rn rn 1 1 1 n n,n 0 H = K +0U| | (1) 1 X cH = HX+ H + H e 2c ec 2 pn 1 ZnZn0 e = + H = H + H + H = K + U (2) H = He + Hc + Hec H = He + Hc + Hec e c ec 2Mn 2 rn rn0 n n,n0 1 X X | H =| He + Hc + Hec xj = x0,j + yj Hc = K + U 1 (1) Hc = K + U Hc = K + U 2 Hc = K + U 2 (1) (1) (1) 2 2 2 pn 12 ZnZn0 e2 2 p 1 ZnZn e =p 1 Zn+Zn e pn H =1=KK++ZnUUZn0 e (2) (1) = n + =0 = Kn ++U 0 = K=x+jU= x0,j+c +(2)yj = K + (2)U (2) H = H2eM+nHc +2Hec rn rn 2 2 2Mn 2 rn rn 2Mn 2 rn rn 2Mn0 2 rn rn0 n 0 n n n,n 0 n,n0 n n,n pn 1 ZnZn0 e X n,nX0 | X| X X0 | |X | X | =X0 | +| = K + U (2) 2M 2 r r n n n n0 X n,nX0 | | H = He + Hc + Hec x = x +1y Nj xj = x0,j +Hycj = K + U j 0,j j xj = x0,j + yj (1) Template: 1D monatomicxj = latticex0,j + y j 1 Nj p2 1 Z Z e2 xj = x0,j + yj = n + n n0 = K + U (2) 2M 2 r r n n n n0 1 N X 1 Njn,nX0 | | 1 Nj Hc = K + U 2 1 Nj (1) 2 1 Njpj 2 2 H = + V (y ,y ,...,y ) xj =pxj0,j + yj 1 2 N pn 1 ZnZn0 e 2M = + H = = K + U+ V (jy1,y2,...,yN ) (2) 2M 2 r r X p2 n n n n0 2M j n,n0 j H = + V (y1,y2,...,yN ) | | 2 2M X X pj j XH =1 Nj + V (y ,y ,...,y ) X 2M 1 2 N j X ⇤ xj = x0,j + yj ⇤ p2 H = j + V (y ,y ,...,y ) 2M 1 2 N j harmonic⇤ approximation,y ,p X Fourier transformedq collectiveq variables yq,pq 1 Nj ⇤ 1 1 2 H = p p† + M! y y† , ! = V /M 2M q q 2 q q q q q q p2 X ✓ ◆ q H = j + V (y ,y ,...,y ) 2M 1 2 N j X

blackboard ⇤ 1 1 1 1 1 1

H = He + Hc + Hec H = He + Hc + Hec H = He + Hc + Hec H = He + Hc + Hec H = He + Hc + Hec

Hc = K + U Hc = K + U (1) (1) 2 2 p2 1 Z Z e2 H = H + H +n H n n0 Hc =pnK + U1 ZnZn0 e e = c + ec =(1)K + U (2) = + 2 2 = K + U 2Mn 2 rn rn0 (2) n n,n0 pn 1 ZnZn0 e | | 2=Mn 2 + rn rn0 = K + U X X (2) Hc = nK + U n,n0 | | (1) X 2MnX2 rn rn0 Hc = K + U n n,n0 | | (1) X 2 X 2 2 p 12 Z Z e xj = x0,j + yj p 1 nZ Z e n n0 n= xj =nx0+,jn+0 yj = K + U (2) = + xj = x0,j + y=j K + U (2) 2Mn 2 2Mrnn r2n rn rn0 n Hn c = K +0 n,nU 0 (1) X Xn,nX0 | |X | | 1 Nj 2 2 1 Nj1 Nj pn 1 ZnZn0 e 2 = + pj = K + U (2) H = + V (y ,y ,...,y ) 2M 1 2 N xj = x0,j + yj 2Mn 2 rnj rn0 2 x2n= x + y n,n0 X p pjj 0,j j | | H = j X+ V (y1,y2,...,yN ) X H = +2MV (y1,y2,...,yN ) 2Mj j X X ⇤

1 Nj ⇤ xj = x0,j + yj yq,pq ⇤ 1 Nj 1

yq,pq 1 1 2 H = pqp† + M! yqy† , !q = Vq/M yq,pq 2M q 2 q q 2 H = He + Hc + Hec q ✓ ◆ pj X q H = + V1(y1,y21,...,y2 N ) H = pqp† + 2M! yqy† , !q = Vq/M 2M 2M q 2 q q 1 j q ✓ pj ◆ 1 Nj ~!q(aq†aq + ) 1X 1 2 q 2 X Hc = K + U q (1) H = H p=qpq† + M!q yqy+q† V, (!yq1=,y2V,...,yq/M N )X 2M 2 p2 1 Z Z e2 q 2M n n n0 X ✓ j = ◆+ 1 q = K + U (2) ~!q(2aMq†anq + 2) rn rn0 n 2 n,n0 | | a 2a = b Xq X X X 1 ⇤ ~!q(aq†aq + ) 2 x2 = x + y q j 0,jp j a 2a = b j M1 M2 X H = + V (y1,y2,...,yN ) ⇤ 2M j 1 Nj a 2a = b yq,pq M1 MX2 2. Quantization: phonons 2 pj 1 M1HM=2 + V (y1,y2,...,yN ) aq = (M!qyq + ipq†) (3) Introduce bosonic annihilation2M and 2M ! yj q,pq ~ q 1 1 creation operatorsX (1D monatomic lattice) 1 2 aq† = p (M!qyq† ipq) (4) H = pqp† + M! yqy† , !q = Vq/M⇤ 2M~!q q q q 1 2M 2 a = (M! y + ip ) (3) q q q q ⇤ q† p ✓ 2M◆~!q q X nq 1 (aq†) 0 = nq! nq aq† = p (M!qyq† ipq) | i | i (4) 1 2M~!1q p 1 y2q,pq H = pqp† + M! yqy† , !q = Vstateq/M with nq phonons aq = qp 1(M!qyq +qipq†) q with wave number q (3) 2!M(a a2M+!q )2 q ✓~ q q† q ~ ◆ 1 2 q Xq 1 1 2 a† = Hp= (Mp !p†q+y† Mip! qy)y† , ! = V /M (4) X q 2M q q q2 q q q q q 2Mq ~!q X ✓ ◆ q p 1 nq a 2(aaq†=) b0 =~p!nqq(naq†aq + ) | i | qi 2 q X 1 Eq = ~!q(nq + ) 2 M1 M2 a 2a = b

M1 M2

1 aq = (M!qyq + ipq†) (3) 2M~!q 1 aq† = p (M!qyq† ipq) (4) 2M~!q p 1 aq = (M!qyq + ipq†) (3) 2M~!q 1 aq† = p (M!qyq† ipq) (4) 2M~!q p 1

H = He + Hc + Hec

Hc = K + U (1) p2 1 Z Z e2 = n + n n0 = K + U (2) 2M 2 r r n n n n0 X n,nX0 | |

xj = x0,j + yj

1 Nj 3. Phonon bandstructure p2 H = j + V (y ,y ,...,y ) 2M 1 2 N j X

⇤

yq,pq

1 1 2 H = p p† + M! y y† , ! = V /M 2M q q 2 q q q q q q X ✓ ◆ q

1 ~!q(a†aq + ) q 2 q X

a 2a = b

M1 M2

1 aq = (M!qyq + ipq†) (3) 2M~!q 1 aq† = p (M!qyq† ipq) (4) 2M~!q p

nq (a†) 0 = pnq nq q | i | i 1D phonon bands 1

H = He + Hc + Hec

Hc = K + U (1) p2 1 Z Z e2 = n + n n0 = K + U (2) 2M 2 r r n n n n0 X n,nX0 | |

xj = x0,j + yj

1 Nj

1D phonon bands p2 H = j + V (y ,y ,...,y ) 2M 1 2 N j X

⇤

yq,pq

1 1 2 H = p p† + M! y y† , ! = V /M 2M q q 2 q q q q q q X ✓ ◆ q

a 2a = b

M1 M2

Diatomic chain with different masses (and ”spring constant” ) 1

H = He + Hc + Hec

Hc = K + U (1) p2 1 Z Z e2 = n + n n0 = K + U (2) 2Mn 2 rn rn 2 n n,n 0 2 X X0 | | 2 2 1 1 Eq = ~!q(nq + ) Eq = ~!q(nq + ) 21 2 xj = x0,j + yjEq = ~!q(nq + ) 1 2 Eq = ~!q(nq + ) 2 Phonons in 2D, 3D latticesl = n1l 1 + n2l2 + n3l3 l = n1l1 + n2l2 + n3l3 l = n1l1 + n2l2 + n3l3 3D Bravais1 latticeNjl = n1l1 + n2l2 + n3l3 1 i j ij V 1 y y iVj ij ⇡V 2 l yl0 yl,lV0 1 jl l0ij l,l0 ⇡l,2l0;i,j i Harmonic approximation: V X, ;i,jy y V (monatomic lattice) ⇡ l Xl0 l l0 l,l0 1 2 V l,Xl0y;ii,jyj V ij ⇡ 2 l l0 l,l0 l,l0;i,j ij X iq (l l0) ij 2 Vq =ij e ·iq(l Vl0) ij Dynamicalp matrix elements: V = e · llV0 j ij q ij ll0 l,l0 iq (l l0) Vq =X ,e · V H = + V (y1,y2,...,yXl l0 N ) ll0 ij li,ql0 (l l0) ij 2M Vq = eX· V j ll0 Xl,l0 X Diagonalize the dynamical matrix V qVq! (2x2 in D=2, 3x3 in D=3) Vq Vq q ⇤ q q q 1

H = He + Hc + Hec

Hc = K + U (1) p2 1 Z Z e2 = n + n n0 = K + U (2) 2M 2 r r n n n n0 X n,nX0 | |

xj = x0,j + yj 1 Eq = ~!q(nq + ) 2

l = n1l1 + n2l2 + n3l3 1 Nj

1 V yiyj V ij ⇡ 2 l l0 l,l0 l,Xl0;i,j 2 p2 ij iq (l l0) ij j Vq = e · V H = ll0 +1V (y1,y2,...,yN ) l,l0 Eq = ~!q(nq + ) X 2M 2 j X V q l = n1l1 + n2l2 + n3l3

1 Phonons in 2D, 3D lattices withV a p-basisyiy j V ij q ⇡ 2 l l0 l,l0 l,Xl0;i,j ⇤

ij iq (l l0) ij Vq = e · V !(q)q ll0 Xl,l0

D(p-1) optical branches Vq

q D acoustic branches

!(q)q

D=3, p=2 1

H = He + Hc + Hec

Hc = K + U (1) p2 1 Z Z e2 = n + n n0 = K + U (2) 2M 2 r r n n n n0 X n,nX0 | |

xj = x0,j + yj

1 Nj

2 2 pj 22 H = + V (2y1,y2,...,yN ) 1 2M Eq = ~!q(nq + ) 112 j EE1qq==~~!!qq((nnqq++ )) Eq = ~!q(nq + ) 22 X 2

l = n1l1 + n2l2 + n3l3

ll==nn11ll11++nn22ll22++nn33ll33 l = n1l1 + n2l2 + n3l3 1 V yiyj V ij 4. Phonon⇡ 2 densityl l0 l,l0 of states 11 j ij V l,Xl0;i,jyiiyj Vij 1 jV ij y ly V V yiy V ⇡⇡22 l ll00 ll,l,l00 ⇤ ⇡ 2 l l0 l,l0 ll,Xl,Xl0;0i,j;i,j l,Xl0;i,j van Hove singularities ij iq (l l0) ij V = e · V q ll0 ijij l,l0iqiq((ll ll0)0) ijij VV == Xee ·· VV ij iq (l l0q)q ij ll0 V = e · V ll0 q l,l0 ll0 XlX,l0 Xl,l0 Vq VVqq Vq

q qq q

DOS, diatomic chain!(q)q !!((qq))qq !(q)q

D(!) (! !,q) ⌘ ,q D(!) X(! !,q) D(!) ⌘ (! !,q) D(!) (! !,q)⌘ ,q ⌘ X,q ,qDebye approximationX (2D) DOS for Al X ! D (!)= , ⌘ = LA, T A 2D,⌘ 2!⇡v2 D (!)= ! ⌘, ⌘ = LA, T A !D22D,D,⌘⌘(!)= 2, ⌘ = LA, T A 2⇡v2⌘ D2D,⌘(!)= 2 , ⌘ = LA,2⇡ Tv A⌘ 2⇡v⌘ 2 !D(⌘)=v⌘qD, ⇡qD = area of 1BZ 2 !D(⌘)=v⌘qD, ⇡q2D = area of 1BZ !D(⌘2)=v⌘qD, ⇡qD = area of 1BZ !D(⌘)=v⌘qD, ⇡qD = area of 1BZ 1

H = He + Hc + Hec

2 Hc = K + U (1) 2 2 pn 1 1 ZnZn0 e = E+q = ~!q(nq + ) = K + U (2) 2M 2 2 r r n n n n0 X n,nX0 | | l = n1l1 + n2l2 + n3l3 2

1 V yiyj V ij ⇡ 2 l l0 l,l0 1 xj =l,Xl0;i,jx0,j + yj Eq = ~!q(nq + ) 2

ij iq (l l0) ij V = e · V q ll0 Xl,l0 l = n1l1 + n2l2 + n3l3

Vq 1 1 Nj V yiyj V ij ⇡ 2 l l0 l,l0 l,Xl0;i,j q

ij iq (l l0) ij V = e · V q ll0 Xl,l0 2 pj !(q)q H = + V (y1,y2,...,yN ) Vq 2M j D(!) (! !,q) ⌘ ,q X X q ! D2D,⌘(!)= 2 , ⌘ = LA, T A 2⇡v⌘

5. Phonon!(q)q heat capacity 2 ⇤ !D(⌘)=v⌘qD, ⇡qD = area of 1BZ

D(!) (! !,q) ⌘ ,q CV (T ) 3R, kBT ~! (Dulong-Petit) X ⇡

! D (!)= , ⌘ = LA, T A 2D,⌘ 2 3 2⇡v⌘ C (T ) const T V ⇡

2 !D(⌘)=v⌘qD, ⇡qD = area of 1BZ

CV (T ) 3R, kBT ~! ⇡

C (T ) const T 3 (Debye) V ⇡ 6. Phonon interactions

To correctly predict the thermal expansion or heat conductance of a solid, one must take into account the interaction between phonons. For this, one has to go beyond the harmonic approximation (somewhat cumbersome…).