QUANTUM THEORY of SOLIDS Here A+ Is Simply the Hermitian Conjugate of A

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QUANTUM THEORY OF SOLIDS Here a+ is simply the Hermitian conjugate of a. Using these deﬁnitions we can rewrite the Hamiltonian as: Version (02/12/07) ~ω H = (aa+ + a+a). (6) 2

SECOND QUANTIZATION We now calculate (aa+ − a+a)f(x) = [a, a+]f(x), f any function In this section we introduce the concept of second quantization. Historically, the first quantization is the = 1 · f(x) , since p = −i~∂x. (7) quantization of particles due to the commutation rela- Hence tion between position and momentum, i.e., [a, a+] = 1, (8) [x, p] = i~. (1) which is the Boson commutation relation. In addition, Second quantization was introduced to describe cases, [a, a] = [a+, a+] = 0. Using this commutation relation we where the number of particles can vary. A quantum field can rewrite the Hamiltonian in second quantized form as such as the electromagnetic field is such an example, since the number of photons can vary. Here it is necessary H = ~ω(a+a + 1/2). (9) to quantize a field Ψ(r), which can also be expressed in terms of commutation relations In order to solve the Hamiltonian in this form we can write the wave function as [Ψ(r), Ψ+(r0)] = δ(r − r0). (2) (a+)n ψn = √ |0i, (10) However, there is only one quantum theory, hence n! nowadays first quantization refers to using eigenval- and a|0i = 0, where |0i is the vacuum level of the system. ues and eigenfunctions to solve Schrödingerequation, We now show that ψn defined in this way is indeed the whereas in second quantization one uses creation and an- solution to Schrödinger’sequation Hψn = Enψn. To hilation operators to solve Schrödinger’sequation. Both show this we first calculate methods are consistent which each other and the choice √ + n+1 √ of technique is dictated by the type of problems consid- + (a ) n + 1 + n+1 a ψn = √ |0i = p (a ) |0i = n + 1ψn+1 ered. In general, when considering many particles second n! (n + 1)! quantization is the preferred technique and will be intro- (11) duced next. and a(a+)n aa+(a+)n−1 aψn = √ |0i = √ p |0i The Harmonic oscillator n! n (n − 1)! (a+a + 1)(a+)n−1 √ = √ p |0i = nψn−1, (12) The Hamiltonian of the harmonic oscillator is given by n (n − 1)!

p2 K + + n−1 + n−1 H = + x2, (3) where we used that a a(a ) |0i = (n − 1)(a ) |0i 2m 2 by using the commutation relation (n-1) times. Coming p back to the Hamiltonian we then have where ω = K/m. The eigenvalues of this Hamiltonian + +√ are then obtained by solving the Schrödingereigenvalue Hψn = ~ω(a a + 1/2)ψn = ~ω(a nψn−1 + ψn/2) equation Hψ = Eψ, where p = −i~∂/∂x is used and one = ~ω(nψn + ψn/2) obtains for the energy = ~ω(n + 1/2) ψ (13) | {z } n En = ~ω(n + 1/2), (4) En where n is an integer. In the second quantized form An important difference between the first quantization we introduce the following operators: method and the second quantization method is that we don’t need the expression for the wave functions in order ³mω ´1/2 to obtain the spectrum, which simply follows from the a = (x + ip/mω) ”anhilation operator” 2~ form (9). The form of the Hamiltonian in (9) is diagonal ³mω ´1/2 since the energy is simply given by counting the number a+ = (x − ip/mω) ”creation operator”. (5) + 2~ of states, since a a is just the number operator. Indeed, 2 a+a(a+)n|0i = n|0i, since we have n levels above vac- The expectation value of the operator O then becomes: uum. Hence, the whole task in solving problems with X ∗ + the second quantization method is to obtain an expres- Oψ = hψ Oψi = Oαβaα aβ. (17) sion of the Hamiltonian in a diagonal form, like (9). The α,β

BCS theory of superconductivity in the next section is + a nice example of this diagonalization procedure. How- If a and a are deﬁned as the creation and anhilation ever, before that we will see how we can transform any operators we then have O written in second quantized Hamiltonian in ﬁrst quantized form into second quan- form. tized form. We now generalize this to more particles: In this case the wave-function is written as

ψ(··· , ri, ··· , rj, ···) = ψ(··· , rj, ··· , ri, ···) bosons General form ψ(··· , ri, ··· , rj, ···) = −ψ(··· , rj, ··· , ri, ···) fermions −iθ ψ(··· , ri, ··· , rj, ···) = e ψ(··· , rj, ··· , ri, ···) anyons The general idea to second quantize a Hamiltonian or X = A φ (r ) ··· φ (r ). (18) operatorP is to start by choosing a complete basis, |φλ(r)i α1,···,αN α1 1 αN N α ,···,α with λ |φλ(r)ihφλ(r)| = 1. Here we are only interested 1 N in operators of the form O(r) or V (r −r ). For example, P i j Here A are simply the coefficients. A simple prod- O(r ) can be a single particle potential like a confine- α1,···,αN i i P uct like φ (r ) ··· φ (r ) has the correct symmetry for ment potential and V (r −r ) can be a two-particle α1 1 αN N i6=j i j Bosons but not for Fermions, hence it is useful to intro- interaction potential, such as the Coulomb potential. For duce an antisymmetrization operator for Fermions, de- N particles we write O in our complete basis as: fined as ¯ ¯ ¯ φ (r ) ··· φ (r ) ¯ ¯ α1 1 αN 1 ¯ XN Y 1 ¯ . . ¯ S− φα (ri) = √ ¯ . . ¯ , (19) O = O(ri) i N! ¯ . ··· . ¯ i ¯ φ (r ) ··· φ (r ) ¯ i=1 Ã ! α1 N αN N X X X = |φα(ri)ihφα(ri)| O(ri) |φβ(ri)ihφβ(ri)| which is the Slater determinant. Then we can write for i α β Fermions X X Y = |φα(ri)i hφα(ri)|O(ri)|φβ(ri)i |hφβ(ri)| ψ(··· , r , ··· , r , ···) = A S φ (r ). | {z } i j α1,···,αN − αi i i,α,β α1,···,αN i Oαβ X X (20) = Oαβ |φα(ri)ihφβ(ri)|, (14) Now we don’t have to worry about symmetry anymore. α,β i The next step is to introduce the occupation number R so that we can rewrite the wave function as where hf(r)i = f(r)dr. X

Similarly, we can write: ψ = |nα1 , ··· , nαN i with N = nαi , (21) i

where nα is the number of particles in state αi. The 1 XN i V = V (r − r ) occupation number operator is then deﬁned as 2 i j i6=j n |n i = n |n i (22) 1 X X |{z}αi αi |{z}αi αi = V |φ (r )i|φ (r )ihφ (r )|hφ (r )| 2 αβγδ α i β j γ i δ j operator number α,β,γ,δ i6=j and Vαβγδ = hφα(ri)|hφβ(rj)|V (ri − rj)|φγ (ri)i|φδ(rj)i The number of states for fermions is limited to 0 and 1, whereas for bosons, n can take any integer value ≥ 0. (15) αi In similarity with the harmonic oscillator described in the previous section, the Boson creation and anhilation This is all very general. Now the idea is to write (14,15) operators are then deﬁned as in second quantized form. If we had only one particle we p could write the wave-function as + bαi | · · · , nαi , · · ·i = nαi + 1| · · · , nαi + 1, · · ·i X √ bαi | · · · , nαi , · · ·i = nαi | · · · , nαi − 1, · · ·i (23) ψ = aβ|φβ(r)i and β X and ∗ + ψ = hφα(r)|a . (16) α [b , b+ ] = δ . (24) α αi αj αi,αj 3

+ + + + + n + n All other commutators ([bi, bj] and [bi , bj ]) are zero. where we used [b, b ] = 1 and b b(b ) |0i = n(b ) |0i For fermions we have (the occupation number operator).P The lastP term in (33) + is obtained p times from the i, hence i δβ,αi = p. c | · · · , nα , · · ·i = | · · · , nα + 1, · · ·i if nα = 0 αi i i i Therefore, (33) leads to = 0 otherwise X + cαi | · · · , nαi , · · ·i = | · · · , nαi − 1, · · ·i if nαi = 1 O = Oαβbα bβ (34) = 0 otherwise. (25) α,β + for bosons. For fermions the derivation is quite similar Hence, cαi creates an electron in state αi if state αi is as well as for V . empty and cαi anhilates an electron from state αi if αi was occupied. It is quite straightforward to check that for symmetry reasons the Fermion creation and anhilation operators have to obey the following commutation Typical Hamiltonian relations: In first quantization we consider + + + {cαi , cαj } = cαi cαj + cαj cαi = δαi,αj (26) X 2 2 X + + ~ ∂i 1 and all others ({c , c } and {c , c }) are zero. Also, H = − + U(ri) + V (ri − rj) . (35) i j i j 2m 2 c |0i = 0. The N-particle fermionic wave-function is i i6=j αi | {z } | {z } now written as H0 V Y Y ψ = S |φ (r )i = c+ |0i, (27) − αi i αi In order to write this Hamiltonian in second quantization i i we first need a basis. We chose as basis the free particle where the product of creation operators is antisymmetric basis: by construction of the {,}. Similarly, we have for bosons: 1 ikαr Y Y φα(r) = √ e , (36) + V ψ = |φαi (ri)i = bαi |0i. (28) i i Rwhich is a complete orthogonal basis with Hence, with these definitions we obtain φ∗ (r)φ (r)dr = δ . Y X Y β α α,β + + + Hence using the orthogonality relations we have, O aαi |0i = Oαβaα aβ aαi |0i (29) i α,β i 2 2 0 ~ kβ or Hαβ = δα,β +U(kβ − kα), (37) X |2{zm} | {z } + q ² O = Oαβaα aβ (30) kβ α,β where U(q) is the Fourier transform of U(r). Similarly and e2 for the Coulomb potential V (ri −rj) = we obtain, 1 |ri−rj | V = V a+a+a a , (31) αβγδ α β γ δ Z 2 e2 irj (kδ +kγ −kα−kβ ) i(ri−rj )(kγ −kα) where a+ and a are either bosonic or fermionic creation Vαβγδ = dridrje e |ri − rj| and anhilation operators. Z e2 We now show that (29) indeed follows from (14). For = δ eiqr dr kδ +kγ − kα −kβ ,0 r bosons, we have | {z } | {z } q Y X X Y 4πe2 q2 O |φαj (rj)i = Oαβ |φα(ri)ihφβ(ri)| |φαj (rj)i 2 j α,β i j 4πe X X = δkβ −kδ ,q 2 . (38) = O δ |φ (r )i · · · |φ (r )i, ··· q αβ β,αi | α1 1 {z α i } α,β i + + + Finally, this Hamiltonian in second quantized form be- bα1 ···bα ···bαN |0i comes (32) X X X 2 + + 1 4πe + + if there are p bosons in state β, one of them is replaced H = ² a a + U(q)a a + a a a a 0 . k k k k−q k 2 q2 k−q k0+q k k by state α, so that we have terms of the form: k k,q k,k0,q (39) + + p−1 + 1 + + p−1 b (b ) |0i = b ( bβb )(b ) |0i What is interesting in this formulation is that we got α β α p β β rid of the dependence in the number of particles. 1 = b+b (b+)p|0i, (33) p α β β 4

i~ TRANSPORT: THE QUANTUM APPROACH = |t|2(−ik − ik) 2m ~ One dimensional conductance = |t|22k (44) 2m Suppose that we have a perfect one-dimensional con- The total electrical current is then ductor (quantum wire) connected by two large electron Z R X kF 2 reservoirs. In real life they would be electrical con- 2 2e 2 2e 2 I = e jk = e jkdk = |t| ∆EF = |t| V, tacts. The left reservoir is fixed at chemical potential L 2π h h k kF µL = µR + eV and the right one at µR for a potential (45) 2 2 difference of V . At zero temperature the current is then ~ kF where we used that EF = 2m and we obtain the same given as expression as (40) for a transmission probability of T = |t|2. Z R EF A similar expression can also be obtained for a single I = −env = −e D(E) · V (E) dE particle HamiltonianP expressed in second quantized form. EL=ER+eV | {z } | {z } + F F Indeed, for H = Hij c cj, where ci is f.ex. expressed 1 | ∂E |−1 ∂E ij i π ∂k ~∂k in a position basis, and using 2e2 = V, (40) h ∂ i~ c = Hc where D(E) is the one dimensional density of states for ∂t i i states moving in one direction and V (E) their group ve- ∂ −i~ c+ = c+H+, (46) locity. Experimentally this conductance quantization can ∂t i i be seen in quantum point contacts, in transport through + single molecule or in break junctions. we obtain for the local density operator ρi = ci ci, We now want to generalize this result to other configurations including non-ideal ones. Using Schrödinger’s ∂ 1 ρ = (c+Hc − c+H+c ) equation: ∂t i i~ i i i i 1 X X ∂ = (c+ H c − c+H c ) i~ ψ = Hψ i~ i ij j j ji i ∂t j j ∂ 1 X −i~ ψ+ = Hψ+ (41) = (c+H c − c+H c ) ∂t i~ i ij j j ji i j X we obtain for the density operator ρ = ψ+ψ = − jij , (47) ∂ 1 j ρ = (ψ+Hψ − (Hψ+)ψ) ∂t i~ with i~ + 2 2 + = (ψ ∇ ψ − ψ∇ ψ ) i + + 2m jij = (ci Hijcj − cj Hjici), (48) i~ ~ = ∇(ψ+∇ψ − ψ∇ψ+) 2m which is the probability current density flowing from i to = −∇j, (42) j. We can now calculate the current for a Hamiltonian in where the probability current density is given by this representation by considering the expectation value of the probability current density operator for a given P + i~ solution state ψ = m ψmcm|0i. Hence, j = (ψ∇ψ+ − ψ+∇ψ). (43) 2m i ∗ ∗ This expression is only valid in a zero magnetic field but hψ|jij|ψi = (ψi Hij ψj − ψj Hjiψi). (49) for any potential V (r) since V (r) commutes with ψ+. ~ ikn For a perfect one dimensional system the solution of a For a plane wave solution ψn = te of Hi,i±1 = D and transmitted wave is of the form ψ(x) = teikx, where t is zero otherwise we obtain the transmission amplitude. In this case the probability current density is given by i j = hψ|j |ψi = D|t|2(2i sin(k)) k i,i+1 ~ i~ 2 j = (ψ∇ψ+ − ψ+∇ψ) = − D|t|2 sin(k) (50) k 2m ~ 5

This Hamiltonian corresponds to the free particle on a and deﬁning the coeﬃcients discrete one dimensional lattice, where Z T ≡ dr~ ψ∗(~r − ~r )Hψ (~r − ~r ) (58) Hψ = Eψ (51) m,n 0 n 0 m leads to we obtain the LCAO equation

N D(ψn+1 + ψn−1) = Eψn, (52) X Tm,nψm = Eψn (59) ikn which gives ψn ∼ e and E = 2D cos(k). Therefore the m=1 total electrical current can be expressed as involving only the amplitudes of the superposition Ψ and the coefficients {Tm,n}. X Z 2e 2 2 I = e jk = − |t| D sin(k)dk Nearest Neighbor Approximation ~ 2π k 2e 2 2 R L Next, we will invoke the nearest-neighbor approxima- = |t| D (cos(kF ) − cos(kF )) ~ 2π tion; that is, we will let 2e 2e2 = |t|2(ER − EL) = |t|2V (53) F F n 0 ; |m − n| 6= 1 h h T = (60) m,n ² ; m = n We now want to apply these expressions for the case of n molecular transport in the tight binding approximation. where ²n is the on-site energy of the atom at position ~rn. The coefficients {Tm,n} are the ”hopping terms” of the molecule and represent the electronic orbital overlap Tight binging approximation between neighboring atoms. We can therefore re-write eq. (59) in the form The tight-binding approximation corresponds to the X LCAO (Linear Combination of Atomic Orbitals) method. Tm,nψm = (E − ²n)ψn (61) Suppose we have molecule consisting of a linear chain of m6=n N atoms, each at a specified position ~rm and having the corresponding potential V0(~r − ~rm). The total Hamilto- which defines a system of coupled algebraic equations. nian of an electron introduced to the system is Hence, the free particle in this description is expressed as Tm,n = D for |m − n| = 1 and zero otherwise. XN H = H0 + V0(~r − ~rm) (54) m=1 MOLECULAR TRANSPORT where H0 is the free-electron Hamiltonian. We will assume that, in the N atom case, the electronic wavefunc- Infinite Chain of atoms with only one s-electron per atom: tion Ψ(~r) takes the form of a superposition of the solu- +V/2 tions ψ0(~r − ~rm) for the single-atom case and write -3 -2 -1 0 12 3 4 5 6 7 8 9 10 µ µ ε ε N s ts s ts X -V/2 Ψ(~r) = ψ ψ (~r − ~r ) (55) m 0 m ψ + ε − ψ + ψ =  ε = 3ϕ ∗ ϕ  ts p−1 ( s E) p ts p+1 0  s ∫ dr s (r)H s (r) m=1  −  ∗ ⇒ψ ∝ ikp ε = ik + ik = t = dr 3ϕ (r)Hϕ (r + a)  p e and E - s ts (e e ) 2ts cos k  s ∫ s s (Onsite and overlap energies where {ψm} are the amplitudes. The Schrödingerequa- ϕ π ε for s-orbitals: s ) e max tion therefore takes the form I(µ) = −e D(k)×v(k)× f (k)dk = −2 f (E − µ)dE ∫ { { 1FD23 h ∫ FD −π 1 1 ∂E ε N N Occ. Prob. min Chemical potential X X π h ∂k Dens. of states velocity Fermi-Dirac distribution H ψm ψ0(~r − ~rm) = E ψm ψ0(~r − ~rm) (56) m=1 m=1

ε   max   where E is the energy of the electron. We can multi- = µ − µ = = 2e 1 − 1 Itot I Left ( L ) I Right ( R ) I(V )  dE { { ∫ E−µ+eV E−µ−eV ∗ µ+ µ− h ε 2 2 ply on the left by ψ (~r − ~r ) and integrate over all of eV 2/ eV 2/ min   0 n 1+ e kT 1+ e kT  space to simplify this equation. Approximating the overlap between the single-atom solutions to be zero; that is, supposing Z ~ ∗ dr ψ0 (~r − ~rn)ψ0(~r − ~rm) = δm,n (57) 6

Left Lead Molecule Right Lead 2 2 [] 2 ik p −ik p ik p t t 4sin k sin k G(E) ψ = e L + re L ψ ∝ e R 2 tR sin kR XR LX R L ( N )1, p 3 p T E = ψ × = N-1 ( ) N +1 tX tL sin kL tRtL -3-2 -1 0 1 N+1 N+2

N  t ik ε ε ψ = XR e Rψ L tL tLX t XR tR R  N +1 N 2 N-2  tR −ik ik ik −ik ik ψ = L + L = + L + L − L ik L −1 e re 1( r)e e e ik ψ = [] −  { 14243 R G(E) ( 2it e sin k ) ψ  ψ + = e ψ + N ( N )1, LX L  0 −2isin k ε N 2 N 1  L −   ψ + ε − ψ + ψ = N 2 ψ + ε − ψ + ψ =  tL −1 ( L E) 0 tLX 1 0 tX tX t XR N ( R E) N +1 tR N +2 0 ik ⇒ −2it sin k + (t e L + ε − E )ψ + t ψ = 0 ⇒ t ψ + ( ε − E + t eik R )ψ = 0 L L L 142L 43 0 LX 1 XR N 1R23 R N +1 ik −ik ψ + ε − ψ + ψ + ψ = − −t (e L +e L )  t ( E) t t 0 − ( ik R + ik R ) L  LX 0 1 1 X 2 X 3 tR e e t  M t ik ⇒ψ = LX ik Lψ − 2 ik L sin ⇒ψ = XR e Rψ 0 e 1 ie kL  ψ + ψ + ε − ψ + ψ = N +1 N ε   t tX N −2 tX N −1 ( N E) N t XR N +1 0 t max L R 2e  1 1  ⇒ I(V ) = T (E) − dE h ∫ E−µ+eV E−µ−eV ε  2 2  min 1+ e kT 1+ e kT 

By introducing ΓL and ΓR it is possible to reexpress the transmission into the Fisher-Lee expression 2  tLX ik   e L + ε − E t t  ik 1 X X 0 L 0 ψ  it e L k  t  1  2 LX sin L  L      t ε − E t  ψ 3 X 2 0 X L 0  2   0  t N-1    =   X M O M M M ε -3-2 -1 0 1 N+1 N+2 ε      L t t t t R 2   L LX XR R  t ik  ψ   N 0 L ε − E + XR e R  N   0   N t 123 144 244 3 2 N-2  R  ψ −J 144444444444 244444444444 3 L H +Σ−E X ε N −2 tX tX

Hamiltonian of X Self-energy Energy Wavefunction Source function  2 2 2 t k t XR tLX 4sin kR sin kL []G(E) 2 2 R sin R ( N )1,  tLX ik   E 0 0   ε t  e L  L T E = ψ × =  1 X L 0  0 L 0    ( )    t  N +1 ε L  0 E L 0   t k t t t X L 0   0 0 L 0  E = L sin L R L H = 2 Σ =   X     M O 0 M O  M O 0       2   +  t ik  E  ε  XR e R  0 0 L   = ⋅Tr []Γ GΓ G −  0 0 L N   0 0 L   4 L R (Fisher Lee)  tR  (H + Σ − E)ψ = − J 2  X L  t ik  0 0 L 0   LX L ik L  e 0 L 0   − ψ = [G(E)] (−2it e sin k ) ⇒ ψ = E−H −Σ 1 J n (n )1, LX L  tL  0 0 L 0  ( X ) L Γ = Im( Σ )  144 244 3 L L  L     G ( E ) (Green’s function) Σ = Σ + Σ ;Σ = 0 0 0 ;Σ = M O L R L   R  0  ΓR = Im( Σ R ) 2 M O 0 t ik   0 0 L XR e R  3  L   t  N-1  0 0 0  R  tX ε -3-2 -1 0 1 N+1 N+2 ε L tL tLX t XR tR R N Which allows us to extend our formulation for leads in − 2 N-2 ψ = eik L p + re ik L p ψ = te ik R p p ε p more than one dimension. N −2 tX tX transmitted wave 1-dimensional lead 3 Incoming and reflected wave t N-1 − X − 2t 2tR 2 = L × − 2 j = sin k ×(t ) ε L -3-2 t -1 0 t 1 t N+1 N+2 t ε R jL sin kL 1( r ) Current probabilities R R L LX XR R  ih ∗ ∗  h h  j = [ψ∇ψ −ψ ∇ψ ] N  2m  2 N-2

2 tL sin kL 2 2 ε But current probabilities are conserved: ⇒ t = 1( − r ) = ψ + t t N −2 t sin k 123 N 1 X X R R T (E)

2 tR sin kR ⇒ T (E) = ψ + × 2 or 3 dimensional lead N 1 t sin k L L N N L 3 R t )3( N-1 )3( LX tX t XR

1 N 2 N-2

)1( )1( t t XR LX t ε 1 X t N −2 1 These expressions give us the solution for the X schrödingerequation and it was necessary to use the con- servation of the current probabilities in order to match the left and right lead solutions. The idea here is again to decompose the system in 3 7 parts, the periodic semi-infinite left lead, described by a Hamiltonian HL the finite system under consideration, corresponding to HX and the right semi-infinite lead HR. N N L 3 R t )3( N-1 )3( The total Hamiltonian then becomes: LX tX t XR 1 N 2 N-2 H = HL + HX + HR + TLX + TXR, (62) )1( )1( t tLX ε XR 1 tX N −2 1 tX where TLX and TXR describe the coupling between the

NL ≡ ≡ diﬀerent parts. In matrix form (for a tight binding hamil- ik ( L, ) p −ik ( L, ) p ⊥ ψ = (L) l + (L) l L, p,q ∑(al e bl e )sin( kl q) tonian) this corresponds to l=1 NR ( R,≡) − ( R ,≡) ⊥ ψ = (R) ik r p + (R) ik r p R, p,q ∑(ar e br e )sin( kr q)   r=1 HL TLX 0  +  t ψ + (ε − E)ψ + t ψ + t ψ + t ψ = 0 H =  T H T  , (63) L p− ,1 q L p,q L p+ ,1 q L p,q−1 L p,q+1 LX X XR  ( L,≡) + ⇒ψ ∝ eik psin(k (L, ⊥)q) and E -ε = 2t []cos( k (L,≡) ) + cos( k (L,⊥) ) 0 T H  p,q L L XR R  π π  k (L,⊥) = l ;k (R,⊥) = r l + r + which leads to the Green’s function:  N L 1 N R 1 + T (E) = 4⋅Tr [ΓLGΓRG ]  −1 E − HL −TLX 0 ΓL = Im( Σ L ) −1  +  G = (E − H) = −TLX E − HX −TXR . Γ = Im( Σ ) + R R 0 −T E − HR XR S S (64) ΣL = TXL GL TLX ;ΣR = TXR GRTRX The center element of the Green’s function GX (element (2,2)) is then N N L R )3( )3( tLX t XR G = (E − H − T + (E − H )−1T X X | LX {z L LX} ΣL )1( )1( t t XR − T (E − H )−1T + )−1 1 LX 1 | XR {zR XR } ΣR −1 = (E − HX − ΣL − ΣR) (65) L ≡ N ik ( , ) N 2 L e l L []G S = k L,⊥n k L,⊥m L n m ∑ sin( l )sin( l ) , N +1 t Using again the Fisher-Lee formula, we have n L l=1 L

(L,≡) (L,⊥) + E -ε L = 2tL [cos( kl ) + cos( kl )] T (E) = 4 · T r[ΓLGX ΓRGX ], (66)  π m  k (L,⊥) = l  l N + 1  L 1 where Γσ = =(Σσ).

N R N ik ( R ,≡) 2 R e r k R,⊥n k R,⊥m = []G S ∑ sin( r )sin( r ) R n,m n N R +1 l=1 tR

(R,≡) (R,⊥) E -ε R = 2tR [cos( kr ) + cos( kr )] m  π  k (R,⊥) = r 1  r +  N R 1 8

These eigenfunctions can then be written as:

ε   2e max  1 1  ⇒ I(V ) = T (E) − dE h ∫ E−µ+eV E−µ−eV ε  2 2  min + e kT + e kT r 1 1  2 hψ |q, mi = sin(k q) ·K sin(k m), (68) k⊥,kk N + 1 ⊥ k + | {z } (T (E) = 4⋅Tr [ΓLGΓRG ]) φq

P ∗ where φq is normalized, i.e., q φq φq = 1 and K such that

X hψk ,k |q, mihψ 0 0 |q, mi = δ 0 δ 0 . (69) ⊥ k k⊥,kk k⊥,k⊥ kk,kk q,m

2D GREEN’S FUNCTION The finite width forces the quantization of k⊥ = π n⊥ N+1 . Schrödinger’sequation of the left semi-infinite The two-dimensional Green’s function of an semi- lead is: infinite left lead, (semi-infinite in the k direction and of finite width N in the ⊥ direction) can be expressed HLhψk ,k |q, mi = 2tL cos(k⊥) + 2tL cos(kk), (70) in terms of the plane eigenfunctions with the boundary ⊥ k conditions where tL are the hopping elements of the left lead. The Green’s function corresponding to HL can then be writ- ψq,m=0 = ψq=0,m = ψq=N+1,m = 0. (67) ten as

X hp, l|ψk ,k ihψk ,k |q, mi = ⊥ k ⊥ k E − 2tL cos(k⊥) − 2tL cos(kk) k⊥,kk 2 2K X sin(k⊥q) sin(k⊥p) · sin(k m) sin(k l) = k k N + 1 E − 2tL cos(k⊥) −2tL cos(kk) k⊥,kk | {z } E⊥ X 2 2 sin(kkm) sin(kkl) = sin(k⊥q) sin(k⊥p) · K N + 1 E⊥ − 2tL cos(kk) k⊥ | {z } 1D G (E⊥) m,l µ ¶ 2 X π π π = sin(n q) sin(n p)G1D E − 2t cos(n ) , (71) N + 1 ⊥ N + 1 ⊥ N + 1 m,l L ⊥ N + 1 n⊥

which is the expression used in page 7 for the edge of the TRANSFER MATRIX APPROACH TO two-dimensional lead for elements m = 0, l = 0, where TIGHT-BINDING SYSTEMS 1D ikk G0,0(E⊥) = e /tL. A simple method to solve discrete schrödingerequa- tions is the transfer matrix technique. We start with the 9 general one dimensional tight-binding equation: two eigenvalues is always one. This gives us a simple way to obtain the bandstructure for a periodic potential. tn+1ψn+1 + tnψn−1 = (E − Vn)ψn, (72) It is then useful to define the Lyapounov exponent as which can be rewritten as   1 Yn λ = − lim ln(min |eig  Tj |) (76) µ ¶ µ ¶ µ ¶ n→∞ n ψ (E − V )/t −t /t ψ j=1 n+1 = n n+1 n n+1 n ψn 1 0 ψn−1 | {z } in terms if the eigenvalue of the product of transfer ma- T n trices. For the periodic potential case λ = 0 inside the Yn µ ¶ ψ1 band and λ > 0 inside the gap, where transmission would = Tj , (73) ψ0 be exponentially suppressed as shown in fig. 1, where we j=1 calculated the Lyapounov exponent (the dominant be- where Tj is the transfer matrix. Equation (73) allows us havior of the product of transfer matrices) for a simple to express Schrödinger’sequation in terms of the bound- periodic potential. ary condition ψ1 and ψ0. This also shows the limitations of this technique, since it only works if the initial bound- 1.6 ary conditions are either known or not important. For (V=0;1;-1) 1.4 Bands general geometries such as those discussed in the previous section this technique is often not adequate, because of 1.2 the relevance of the initial conditions. However, a prob- 1

lem where the transfer matrix technique is very useful is λ 0.8 in disordered systems. Indeed, in these systems the po- 0.6 tential is typically random, hence each conﬁguration of the potential is diﬀerent, so that what becomes important 0.4 Gap Gap are averaged quantities, such as the average transmission 0.2

0 probability, where the transfer matrix technique can be -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 E used. If we start with eq. (73) and set tn = 1 and leave only the potential to be n-dependent, it is interesting to FIG. 1: Lyapounov exponent and corresponding bandstruc- distinguish the cases where Vn is periodic or random. In ture for the periodic potential (tn = 1; V3n = 0; V3n+1 = 1; V = −1). the simplest case, where Vn = V0 the product of transfer 3n+2 matrices can be written as

n µ ¶n Y eik 0 T = T n = S S+ j 0 0 e−ik Moving towards disordered potentials j=1 µ ¶ ikn e 0 + As a next step it is interesting to consider a periodic = S −ikn S , (74) 0 e potential, where the periodicity extends over larger unit. This will lead to a much more complicated bandstruc- where E − V0 = 2 cos(k) and S the symmetric matrix, ik ture with the appearance of many bands and gaps. This which diagonalizes T0. Interestingly, the eigenvalues e −ik is illustrated in fig. 2, where we chose a potential of peri- and e are of norm 1 if |E − V0| ≤ 2 (k is real), which odicity 40 in order to calculate the Lyapounov exponent. corresponds to the allowed band and supports plane wave When we now push this much further by considering a solutions, whereas for |E − V | > 2, k is imaginary and Q 0 potential which is random the following structure of the n T diverges, which corresponds to the exponentially j=1 j Lyapounov is obtained (fig. 3). vanishing modes (or evanescent waves). For any periodic Clearly the Lyapounov exponent is now strictly pos- potential Vn the situation is very similar, where itive (λ > 0). This is known as Anderson localization,  n/a which tells us that in one dimension the transmission Yn Ya through a random potential tends exponentially to zero Tj =  Tj (75) with the size of the system. j=1 j=1 It turns out that the Lyapounov exponent defined in and a is the periodicity of Vn. Hence, the permitted eq. (76) is self-averaging for a random potential if n → bands correspond to the case where the eigenvalues of ∞. This is known as the Oseledec theorem and tells us Qa j=1 Tj are of norm one, since of tn = 1 the determinant that λ is uniquely defined in that limit. Using (73) the of Tj is one, which implies that the The product of the wave function can be written as 10

eigenvalues, one growing exponentially and the other de- creasing exponentially reﬂects the role of the boundary 0.6 condition. Indeed, the correct boundary condition would −λn 0.5 Period=40 eliminate the diverging solution, hence ψ ∼ e represents the correct typical envelope of a wavefunction in 0.4

λ a disordered potential. Since this wavefunction decays 0.3 exponentially it is called a localized wavefunction with Band Gap −1 0.2 a localization length Lc = λ . Because of this expo- nential decay the propagation of an electron (or a wave) 0.1 in a one-dimensional random potential is exponentially

0 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 suppressed, which is commonly referred to as Anderson E localization. In the next section we show this rigourously FIG. 2: Lyapounov exponent and corresponding bandstruc- with the help of the Green’s function using a perturba- ture for a periodic potential of period 40. tion calcluation.

0.7 GREEN’S FUNCTION APPROACH TO 0.6 DISORDERED SYSTEMS

0.5

0.4 Potential with 1000 random sites The idea is to calculate the localization properties of

λ the following nearest neighbor tight binding Hamiltonian: 0.3

0.2 X + + + H = Vici ci + ci+1ci + ci−1ci, (80) 0.1 i

0 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 E wherePVi is a random variable. Writing the solution as ψ = ψ c+|0i, yields FIG. 3: Lyapounov exponent for a random potential of size n n n 1000. ψn+1 + ψn−1 = (E − Vn)ψn. (81)

In matrix form the Hamiltonian is written as       V 1 0 ... 0 Yn Yn 1  1 V 1 ... 0  ψn+1 =  Tj ψ1 +  Tj ψ0. (77)  2    j=1 j=1 H =  0 1 V3 ... 0  . (82) (1,1) (1,2)  . . . . .   ......  The envelope of the wavefunction is described by e±λn. 0 0 0 ...VN Indeed, for a wavefunction of the form e−λn, the Lya- pounov exponent is simply λ using Since   E 0 0 ... 0 1  0 E 0 ... 0  λ = − lim ln |ψn|, (78)   n→∞ n   E =  0 0 E... 0  , (83)  . . . . .  which is equivalent to (76). The relationship between the  ......  Lyapounov exponent and the envelope of the wavefunc- 0 0 0 ...E tion can be understood by realizing that the product of transfer matrices gives two eigenvalues {e−λn; eλn} in the the Green’s function is simply the matrix inverse of E − n → ∞ limit, which means that in average H, i.e.,

µ ¶ G(E) = (E − H)−1 ⇒ (E − H)G(E) = 11. (84) e−λ 0 hT i = , (79) j 0 eλ From our calculation from the molecular wire, we have where h·i is the average over the disorder. This average ψ = Gˆ · J, (85) correctly gives us the expected dependence of the product of transfer matrices and also characterizes the typical where J is the source function (with J(1) 6= 0 and J(i > envelope of the wavefunction. The fact that we have two 1) = 0), Gˆ = (E − H − Σ)−1 the total Green’s function 11 including the leads, and Σ the self energy due to the 1 XN = G (E) leads. Hence, N − 1 i,i i=1 Z ψ Gˆ E XN N 1,N 0 1 0 = . (86) ⇒ λ = dE Gi,i(E ). (90) ψ1 Gˆ N − 1 1,1 −∞ i=1

If we put the coupling to the leads to zero then Gˆ = G Equation (89) is typically used to relate the density of and the Lyapounov exponent (λ) becomes states to the Lyapounov exponent and (90) will be used ¯ ¯ ¯ ¯ for the perturbative calculation of the Lyapounov expo- ¯ ¯ ¯ ¯ 1 ¯ψN ¯ 1 ¯G1,N ¯ nent. Both expressions are exact and were obtained with- λ = − ln ¯ ¯ = − ln ¯ ¯ . (87) N − 1 ψ1 N − 1 G1,1 out any approximation. We will now calculate λ using perturbation theory. From (85), we have that G1,1 is proportional to ψ1, which we can set to one, without loss of generality. Hence to calculate (87) we are left with the evaluation Small disorder perturbation of G1,N . This can be done explicitly by inverting the matrix G using the expression for the matrix inverse: In general we write the perturbation series for G as −1 i+j (Aij) = (−1) cofij(A)/ det(A) and applying it to G(1,N). Hence, G = G0 + G0VG0 + G0VG0VG0 + ... = G0 + G0V (G0 + G0VG0 + G0VG0VG0 + ...) (−1)N+1(−1)N−1 G(1,N) = , (88) = G0 + G0VG (”Dyson eq.”) det(E − H) ⇒ 11 = G0G−1 + G0V N−1 −1 0 since the cofactor of (E − H)1,N is simply (−1) . The ⇒ G = E − H − V, (91) det(E − H) is also the product of the eigenvalues, i.e., Q 0 0 det(E − H) = α(E − ²α), where ²α are the eigenvalues where H is the Hamiltonian without disorder and G of H. Therefore, its corresponding Green’s function. V is our small perturbation. Using indices the perturbation series reads: 1 X X G1,N = QN 0 0 0 0 0 0 α=1(E − ²α) Gij = Gij + GikVkGkj + GikVkGklVlGlj + ... k k,l 1 XN ⇒ λ = ln |E − ² | (92) N − 1 α α=1 We now apply this perturbation series to our expression Z of the Lyapounov expression (90), which yields = D(²) ln |E − ²|d², (89)

1 XN which is known as the Herbert-JonesP formula. Here D(²) ∂ λ = G (E) 1 E N − 1 i,i is the density of states D(²) = N−1 α δ(² − ²α), which i=1 R EF clearly satisﬁes that n = D(²)d², where n is the 1 XN X X density. We can transform λ further by calculating the = ( G0 + G0 V G0 + G0 V G0 V G0 + ...) N − 1 ii ik k ki ik k kl l li derivative, i.e., i=1 ik ikl (93) 1 XN 1 ∂ λ = E N − 1 E − ² We now take the average h·iD over the disorder, where we α=1 α assume the following properties for the disorder: hVkiD = N 2 1 X |ψ2 | 0 and hVkVliD = δk,lσ , which corresponds to an uncor- = α N − 1 E − ² related disorder of standard deviation σ. Hence, α=1 α XN PN XN XN X 1 i=1hi|ψαihψα|ii 0 0 0 0 = h Gi,i(E)iD = G + 0 + G G G hVkVliD + ... N − 1 E − ² ii ik kl li α=1 α i=1 i=1 ikl N 1 XN XN |ψ ihψ | X X = hi| α α |ii = G0 + σ2 G0 G0 G0 + ... N − 1 E − ² ii ik kk ki i=1 α=1 α i=1 ik N 1 XN 1 X X X = hi| |ii = G0 + σ2 G0 G0 G0 + ... N − 1 E − H ii kk ki ik i=1 i=1 k i 12

XN X Z E XN X 0 2 0 1 1 1 |ψαihψα| = Gii + σ Gkkhk| 0 0 |ki + ... = dE hi| |ii E − H E − H N − 1 (E − ²α − i²) i=1 k −∞ i=1 α Z N E X X X 1 1 0 2 0 = P λ − iπ dE hi|ψαihψα|iiδ(E − ²α) = Gii − σ Gkk∂Ehk| |ki + ... E − H0 −∞ N − 1 i=1 k i,α N Z X X E 1 X = G0 − σ2 G0 ∂ G0 + ... = P λ − iπ dE δ(E − ² ) ii kk E kk N − 1 α i=1 k −∞ α XN 2 = P λ − iπN(E) (98) 0 σ 0 2 = (G (E) − ∂E (G (E)) + ...) (94) ii 2 ii 1 1 i=1 where we used that x+i² = P x − iπδ(x), where P is the principle part and N(EF ) is the density. In general, P Using expression (90) we obtain corresponds to the real part of the integral, hence: ( R P N Z E 1 N X E 2 <λ = < dE Gi,i(E + i²) 1 0 0 0 σ 0 2 −∞ PN−1 i=1 hλiD = ( Gii(E )dE − (Gii(E)) + ...) D(E) = ∂ N(E) = − 1 = 1 N G (E + i²) N − 1 −∞ 2 E π N−1 i=1 i,i i=1 ( R (95) <λ = < E G+(E)dE Now we simply have to evaluate G0 (E). But since the ⇒ −∞ , (99) ii D(E) = − 1 =G+(E) 0 0 0 ikαn π eigenvalues of H are ²α = 2 cos(kα) and hψα|ni ∼ e , we have where G = Tr(G)/(N − 1). Hence, in the case without X disorder (V = 0) we have for |E| < 2 that <λ = 0 and 0 1 G (E) = 1 2 −1/2 ii 0 D(E) = (4 − E ) . E − ²α π α Z We can now verify, that the above expression for the 1 π 1 = dk density of states is the same as the standard expres- 2π E − 2 cos(k) 1 −1 −π sion: D(E) = π |∂kE| , which indeed is equivalent to 1 D(E) = 1 (4 − E2)−1/2, since E = 2 cos(k) for V = 0. It = √ . (96) π E2 − 4 is also interesting to note that up to second order in perturbation theory the density of states does not depend 0 For an energy inside the band, i.e., |E| < 2, Gii(E) = on the disorder, only <λ is aﬀected by the disorder. −i(4 − E2)−1/2 and is imaginary, hence for |E| < 2 and using (95), we obtain CONDUCTANCE QUANTIZATION σ2

FIG. 5: Conductance quantization of a quantum point con- FIG. 6: Imaging of the channels using an AFM (1 to 3 chan- tact in units of 2e2/h. As the gate voltage deﬁning the con- nels from left to right). The images were obtained by applying striction is made less negative, the width of the point contact a small negative potential on the AFM tip and then measur- increases continuously, but the number of propagating modes ing the conductance as a function of the tip scan and then at the Fermi level increases stepwise. The resulting conduc- reconstruct the 2D image from the observed change in con- tance steps are smeared out when the thermal energy becomes ductance. comparable to the energy separation of the modes.

We can also obtain this result directly using our tight In the ideal pure one dimensional case we had the con- 2 binding approach and the Green’s function approach de- ductance given by 2e /h (see eq. 40). When this is ex- scribed in the previous section. Taking for example a tended to a quasi 2-dimensional case the conductance in- width of four sites we can calculate the total transmis- creases by quantum steps (conductance quantization) of 2 sion using the Fisher-Lee formula () to obtain the energy 2e /h as seen in the experiments. This can be understood dependence shown in ﬁg. 7. simply in terms of number of channels, i.e., increasing the width of current channel increases the conductance by an integer multiple of 2e2/h. In this case the electron energy

Perfect 4 is given by channels with wide contacts ~2k2 ~2(πn/W )2 ² = x + (100) 2m∗ 2m∗ if the boundary of our narrow wire is assumed to be G[2e^2/h] sharp, since in this case the wave function has to vanish Perfect 4 channels at the boundary (like for an electron in a box of width W ). In general, if E

2 2 2 2 FIG. 7: The conductance of a 4 site wide conductor as a ~ (π/W ) ~ (2π/W ) function of energy for the case with wide leads (smooth steps) < EF < (101) 2m∗ 2m∗ or semi-inﬁnite leads of width 4 (square steps). then we recover the ideal case of a one-dimensional quantum wire, or single channel. If, however, The reason we obtain a curve which is symmetric in energy stems from the tight binding approximation. In- ~2(Nπ/W )2 ~2((N + 1)π/W )2 deed, the energy of the quasi two-dimensional system is < E < (102) given by E = 2 cos(k ) + 2 cos(k ) (the hopping term 2m∗ F 2m∗ ⊥ k is assumed to be 1). Hence, the maximum conduc- we can have n channels, where each channel contributes tance is obtained when E = 0, since then cos(kk) = equally to the total conductance. Hence in this case G = −2 cos(k⊥), which is possible for any quantized value of 2 N · 2e /h, where N is the number of channels. This k⊥ = nπ/(N + 1), where N is the width. For energies implies that a system, where we reduce the width of the closer to the band edges (E = ±4), the number of possi- conductor will exhibit jumps in the conductance of step ble values for k⊥ are reduced, which leads to the observed 2e2/n. Indeed, this is what is seen experimentally. decrease of the conductance close to the band edges. 14

INTRODUCING A MAGNETIC FIELD For the symmetric gauge (A~ = (−By/2, Bx/2, 0)) in coordinates (x, y, z) this becomes When adding a magnetic ﬁeld to the system the Hamil- tonian transverse to the magnetic ﬁeld (B~ = ∇ × A~) is written as

1 H = (−i~∇ − eA~)2. (103) 2m

µ ¶ ~2 ie ie e2 e2 H = −∂2 + By∂ − Bx∂ + y2B2 + x2B2 − ∂2 (104) 2m x ~ x ~ y 4~2 4~2 y

eBa2 φ Ba2 It is now possible to discretize this equation by defining x = la, y = ka, α = = = , φ0 = h/e the h φ0 φ0 flux quantum, a the lattice constant and φ the magnetic flux per unit cell. Using the following expressions of the derivatives:

2 2 ∂xψ = (ψl+1,k + ψl−1,k − 2ψl,k)/a and ∂xψ = (ψl+1,k − ψl−1,k)/2a (105) we obtain in the limit α ¿ 1

  ~2  ia2eBk e2a4B2k2  −ψ − ψ + 2ψ + (ψ − ψ )/2 + ψ + ··· = Eψ 2  l+1,k l−1,k l,k l+1,k l−1,k 2 l,k  l,k 2ma ~ | 4{z~ }  '0  2ma2 − e−iπαk ψ − eiπαkψ − eiπαlψ − e−iπαlψ + 4ψ  = E ψ (106) | {z } l+1,k l−1,k l,k+1 l,k−1 l,k ~2 l,k '1−iπαk

H The magnetic flux through a unit cell is then A~ = metry for two semiclassical trajectories in opposite direc- Ba2, with A~ = (−Bk/2, Bl/2, 0)a, which is consistent tions in the presence of a magnetic field. The simplest with B~ = ∇ × A~. In general, a Hamiltonian in second way to understand this is when looking at a single return quantized form and in the presence of a magnetic field trajectory. Without a magnetic field the same trajectory can therefore be written as exists in the opposing direction, hence when summing all possible trajectories one trajectory and its reverse add X up, which leads to an enhance return probability. This iφij + −iφij + H = e ci cj + e cj ci, (107) effect is destroyed by the magnetic field since time re- hiji versal symmetry is then broken and two trajectories in opposite directions will then have different phases. where hiji are the nearest neighborsP and the sum of the phases over a lattice square is φ = 2πα. Different ¤ ij Using our Green’s function technique to calculate the Gauges will lead to different forms of the phase factors, conductance for a square geometry it then possible to ob- but the sum over a unit cell is the flux per unit cell. serve this effect when introducing some disorder as shown in fig. 8. WEAK LOCALIZATION The analytical approach is obatained by a perturbative An interesting consequence of the magnetic field in the calculation on the disorder strength, which leads to the presence of disorder is the weak localization effect, which following expression due to Hikami valid in 2D (Ref: S. is an increase in conductance at non-zero magnetic field. Hikami, A. I. Larkin, and Y. Nagaoka, Prog. Theor. This effect arises from the destruction of the phase sym- Phys. 63, 707 (1980)). 15

depend on the detailed shape and given impurity con- · µ ¶ µ ¶¸ figuration. By changing slightly this configuration the e2 1 B 1 B ∆σ(B) = − Ψ + 0 − Ψ + φ (108) wavefunction will change, which in turn affects the con- πh 2 B 2 B ductance through the system. This conductance is then , characterized by fluctuations when any external param- where Ψ is the digamma function, B0 = h/(8πeDτ), eter such as the Fermi energy or the magnetic field is and Bφ = h/(8πeDτφ). The transport scattering time changed. An example of this is shown in figure 9. 2 is τ with diffusion constant D = vF τ/2 (vF the Fermi velocity) and τφ is the phase coherence time. This expression typically leads to a logarithmic increase of the resistance when lowering the temperature, following e2 ∆σ = − πh ln(τφ/τ), where τφ is temperature dependent. Weak localization (WL) also exists in 3D, where it V=0 was also observed experimentally. Moreover, this phe- V=1 nomenon is only observed when the conductance (or re- 10x100 sistance) is averaged over many disorder configurations. LR For a given configuration, WL would not be observed since the conductance in this case is dominated by universal conductance fluctuations described in the next section. The averaging over disorder is equivalent to per- forming the experiment (numerical or physical) in a system, whose size is much larger than the quantum coher- FIG. 9: Universal conductance fluctuations modeled in the ence length. Indeed, in this case the conductance of a tight binding approximation for a system of size 10*100 with system of size L is then simply the averaged conductance increasing disorder. Gφ (multiplied by a dimensional factor) of the system of size Lφ, where Lφ is the coherence length of the system, In general, the conductance fluctuations are of the or- D−2 2 i.e., G = hGφi(L/Lφ) , where h·i is the average of the der of δG ' e /h for a system where the coherence length disorder and D the dimension. Lφ exceeds the size of the system L. For a system, where L À Lφ, these fluctuationsp will be averaged out, which 2 Weal localization minimum leads to δG ' e /h· Lφ/L. Hence, in the classical limit L/Lφ → ∞ no fluctuations survive.

AHARONOV AND BOHM OSCILLATIONS AND PERSISTENT CURRENTS

Aharonov and Bohm oscillations are another interesting manifestation of the quantum coherence of the system. When the system is composed of a ring with or without leads attached to it, when applying a magnetic field through the ring, the phase of the electrons in the ring will be affected by the magnetic field (even if there is FIG. 8: Weak localization effect for different square sizes. no field in the conductor). This will lead to Aharonov and The transmission probability is plotted as a function of the Bohm oscillations oscillations as a function of the num- flux per unit cell φ for different square sizes. The dip is easily ber of quantum magnetic fluxes penetrating the ring. If seen when averaging over different disorder configurations. leads are attached to the ring the current will oscillate as a function of the magnetic field, with an oscillation given by the quantization of the magnetic flux. This can be understood by describing the ring in the tight bind- UNIVERSAL CONDUCTANCE FLUCTUATIONS ing approximation as having two branches with opposite iπα −iπα phases, te and te , where α = φ/φ0 is the magnetic Universal conductance fluctuations (UCF) is a strik- flux through the ring in units of the quantum flux. The ing example of the importance of phase coherence. In- total overlap is then simply teiπα + te−iπα = 2t cos(πα), deed, if the phase coherence of the electron is maintained which immediately shows that the transmission (∼ |t2|) over the size of the system studied, the wavefunction de- will oscillate with magnetic field with periodicity α. scribing the electron traveling through the system will Another interesting phenomena in a ring is the exis- 16 tence of persistent currents. In this case we consider For a confinement potential where the potential is infinite a closed ring of N lattice sites. In the presence of a at x=0 it is possible to show that the spectrum is given magnetic field the hopping elements are multiplied by a by the Landau levels far away enough from the edge and i2πα/N phase factor te . For an ordered ring (all hopping by En = ~ωc(2n+3/2) at the edge. Therefore, as a func- elements are equal and all onsite energies are zero) the tion of X, when approaching the edge the energy levels wave functions are simply plane-wave with a phase fac- bend up as illustrated in fig. 10. i(k−2πα/N)n tor, i.e., ψn = e . These wave functions are indeed solutions to Confining Potential i2πα/N −i2πα/N te ψn+1 + te ψn−1 = Eψn, (109) with E = 2t cos(k − 2πα/N). A closed ring penetrated by a magnetic field implies the following boundary con- EF Landau Levels i2πα dition: ψn = e ψn+N , which leads to the quantization of k = 2πl/N, where l is an integer. This implies ‹X› that each energy level (given by the quantum number FIG. 10: Landau levels in a confined potential l) will oscillate as a function of α (E = 2t cos[2π(l − α)/N]). From the associated√ normalized wavefunction It is possible to capture most of the physics of the quan- l i(2π/N)(l−α)n ψn = e / N it is possible to obtain the cur- tum Hall effect by using the tight binding approximation, rent probability expectation for state l. In the presence where the edge is naturally introduced when considering of a magnetic field the current operators have to be mod- a finite size system. In this case the magnetic field has ified in order to include the phase factor. In the contin- to be strong enough so that Landau levels develop in the uous case equ. (43) ∇ has to be replaced by ∇ − eA, system. On the other hand, when introducing the mag- where A is the vector potential and in the tight bind- netic field in the tight binding approximation we assumed ing description the hopping term t in equ. (48) has ij that amount of flux per flux quantum (α = φ/φ0) and per −iφij to be replaced by tij e , where φij is the associated unit cell is small. It turns out that the description is still gauge field induced phase factor. This leads to a current correct at large α in the case where a periodic potential expectation jl = −(2/N~)t sin(2π(l − α)/N), where the is added to the system. In most physical systems this pe- electrical current in this state is simple ejl. Alternatively, riodic potential does not exist or is irrelevant because the the expression of the current can be simply obtained from periodicity is so much smaller than the magnetic length. Hence we need to keep in mind to consider α small if we Il = −(e/h)∂αEl = −(2e/N~)t sin(2π(l − α)/N), (110) are probing Landau level physics. where El is the eigenvalue of state l. This last expres- In example of such a calculation for different fields is sion can be understood in terms of the analogy between shown in figure 11 for a two terminal configuration. the group velocity of Bloch waves (vg ∼ ∂kE) in a periodic potential applied to the ring geometry, where α plays the role of the quantum number k. 10x100 LR

V=0.1

QUANTUM HALL EFFECT

=1/8 ¡

The quantum Hall eﬀect is the quantization (due to the =1/4 ¡

Landau levels) of the Hall resistance. This effect is observed in two dimensional systems as well as in quasi-1D =1/2 ¡ systems. In order to observe a quantized Hall resistance in two dimensions only a magnetic and an edge a necessary. Hence, in order to understand its physics we need to solve the two dimensional Hamiltonian in a magnetic field and a confinement potential. Without confinement FIG. 11: Two-terminal transmission probability of a weakly the spectrum in a magnetic field is simply given by the disordered system in a strong magnetic field, showing quan- Landau levels: tum Hall plateaus. E = ~ω (n + 1/2). (111) n c In general, the quantum Hall effect is the most striking A confinement potential changes this spectrum. In when looked at in a four probe geometry, where contact general, one can introduce a quantum number X describ- contributions are minimized. In order to describe a four ing the average position perpendicular to the long edge. probe device it is necessary to introduce additional leads 17

(i = 1, 2, 3, 4). Each lead describes an electron reservoir where Vi are the voltages of the leads. If all voltages are at a given chemical potential µi. An example of such a equal this means that there cannotP be any current, hence conﬁguration is given in ﬁgure 12. From Kirchhof’s law Ii = 0. This leads to the condition j TjiVi − Tij Vj = 0, it is then possible to write the current from lead i as which implies

Z e X I = 2 dET (E)f (E) − T (E)f (E), (112) i h ji i ij j j e X I = 2 T (V − V ). (114) where fi(E) is the Fermi-Dirac distribution of lead i i h ji i j j at the chemical potential µi and Tij is the transmission probability between lead i and j. At zero temperature and assuming that Tij (E) = Tij, we then have

If we now assume without loss of generality that V = 0 e X e2 X 4 I = 2 T µ −T µ = 2 T V −T V , (113) this implies that i h ji i ij j h ji i ij j j j

      I1 T2,1 + T3,1 + T4,1 −T1,2 −T1,3 V1  I2  =  −T2,1 T1,2 + T3,2 + T4,2 −T2,3   V2  , (115) I3 −T3,1 −T3,2 T1,3 + T2,3 + T4,3 V3

which can be written in short as

20x100 −1 I~ = TˆV~ ⇒ V~ = Tˆ I.~ (116) 1 2 3 4 In a typical experimental conﬁguration the Hall resis- V=0.1 tance would be obtained by applying a current between 1 and 4 and measuring the Hall voltage between 2 and 3, which we can deﬁne as R14,23 = RH . In this case the expression for the Hall resistance reduces to

V2 − V3 −1 −1 RH = = (Tˆ )2,1 − (Tˆ )3,1, (117) I1 2 since I2 = I3 = 0 (shown in figure 12. Similarly, the FIG. 12: Four-terminal resistance in units of h/2e of a weakly diagonal resistance Rxx is obtained by flowing a current disordered system in a strong magnetic field, showing quan- between 3 and 4 and measuring the voltage drop between tum Hall plateaus. The red curve shows the Hall resistance, 1 and 2. Hence, the green curve Rxx and the blue curve the two-terminal conductance in units of 2e2/h, similar to figure 11.

V − V R = 1 2 = (Tˆ−1) − (Tˆ−1) . (118) xx I 1,3 2,3 3 ITERATIVE GREEN’S FUNCTION The transmission probabilities are simply obtained us- CALCULATION ing the Green’s function formalism, where The main difficulty in the numerical implementation of + Tij = 4tr(ΓiGΓjG ) the Green’s function method is the size of the Hamilto- X nian for 2D or 3D systems (the inversion of a large matrix G = (E − H − Σ )−1 i is the main difficulty). Realizing that the transmission i probabilities only depend on the terms of the Green’s Γ = =Σ , (119) i i function, which couple the leads it is then possible to where Σi is the self-energy due to lead i and H is the use an iterative approach to obtaining only the relevant Hamiltonian of the system without the leads. terms. For example, in a quasi 1 dimensional structure 18 it is possible to decompose the full Hamiltonian in slices Left lead Dot Right lead Hl where the full hamiltonian is ε = 0 ε = 0 ε D   t =1 t =1 . ..   tL t   R  Hl Tl,l+1  H =  +  . (120)  Tl,l+1 Hl+1  . .. FIG. 13: A typical geometry of a quantum dot The idea then is to obtain the Green’s function, which relates the elements of the two extremities H and H , 1 N of a quantum dot without interactions can be then writ- by starting with the pair of middle slices H and N/2 ten as HN/2+1. Hence, we start by obtaining the Green’s function of the middle slices (l = N/2), i.e.,   ik −1 E − e tL 0   · µ + ¶¸−1 G = tL E − ²D tR , (126) Hl+1 Tl,l+1 ik G1 = E − . (121) 0 tR E − e Tl,l+1 Hl where we used the self-energy eik due to the one- The next Green’s function relating Hl−1 and Hl+2 can dimensional lead with dispersion E = 2 cos(k)(t = 1 then be obtained by using hopping elements). In order to obtain the transmission probability we can use Fisher-Lee’s relation (). Hence, · µ ¶ ¸−1 Hl+2 0 G2 = E − − Σ1 , (122) 0 Hl−1 2 2 ΓLΓR T (E) = |2 sin(k)| |G(1, 3)| ' ¡ ¢2 , |{z} ΓL+ΓR 2 + (E − Er) where Σ is the self-energy due to the middle part given tσ ¿1 2 1 (127) by Hl, Hl+1 and Tl,l+1, which can be expressed as 2 where Γσ = |tσ sin(k)| and Γ = (ΓL +ΓR)/2 describes the 2 2 average coupling strength and Er = ²D +(tL +tR) cos(k) µ ¶ µ + ¶ Tl+1,l+2 0 Tl+1,l+2 0 is resonance energy. This Lorentzian resonance can also Σ1 = + G1 . 0 Tl−1,l 0 Tl−1,l be obtained directly from a simple double barrier geom- (123) etry. This iterative procedure can be continued until we hit An alternate approach to this quantum dot can be ob- the first and last slice H1 and HN . The last iteration tained by using the dot’s Green’s function directly. In- then reads, deed,

· µ ¶ ¸−1 2 2 ik −1 H 0 GD = (E − (tL + tR)e − ²D) , (128) G = E − N − Σ − Σ , (124) N/2 0 H N/2 L 1 since the self-energies due the leads evaluated directly on 2 ik where the dot are tσe . It is quite straightforward to show that indeed GD = G(2, 2). Separating the imaginary from the µ ¶ µ ¶ real part leads to T 0 T + 0 Σ = N−1,N G N−1,N (125) N/2 0 T + 1 0 T 1,2 1,2 (E − E ) + iΓ G = r (129) D (E − E )2 + Γ2 and ΣL is the total self energy due to the leads con- r necting H1 and HN . The number of leads is not limited Twice the imaginary part deﬁnes the spectral function to two. Indeed in the example of ﬁgure 12 a total of four (A(E) = 2=G (E)). This spectral function describes leads were used. D the density probabilityR as a function of energy and is normalized ( dEA(E) = 2π). The imaginary part of the diagonal Green’s function can also be related to the QUANTUM DOTS escape rate. Indeed, taking the Fourier transform of

A simple quantum dot can be pictured like in ﬁg. 13. iEr t −Γt =GD(E) |{z}→ e e , (130) In the tight binding approximation, the Green’s function FT 19

2 −2Γt leads to |GD(t)| = e which describes the probabil- Dephasing due to the coupling to a two level system ity of an electron to be on the dot. Hence, Γ is simply (TLS) the escape rate due to the leads of an electron placed on the dot. It is further possible to relate this escape rate Quantum mechanical dephasing is in general present to the transmission through the dot by using an expres- when our system under consideration is coupled to an sion derived by Meir and Wingreen, which relates the environment. This environment can be a lead, coupling transmission probability to the spectral function: to phonons, coupling to spins or to some other defect or system, which has some internal degrees of freedom. The 2 2 |tL| |tR| ∂E simplest example is a two level system, which is equiva- T (E) ' 2 2 A(E) lent to a spin 1/2. |tL| + |tR| ∂k Here we will look at how to describe the coupling of a ΓLΓR = ¡ ¢2 , (131) TLS to a quantum dot. Because of this additional degree ΓL+ΓR 2 2 + (E − Er) of freedom we have to use the tensor product of the two which is the same expression as obtained earlier. systems in order to describe the larger Hilbert space. The Hamiltonian of a TLS is given by µ ¶ DEPHASING IN A QUANTUM DOT Eu t HT LS = (132) t Ed The sharp resonance in a quantum dot is due to the and the quantum dot Hamiltonian with two lead sites is multiple constructive interferences which occur between the tunneling barriers. Indeed, without phase coherence   0 tL 0 one would expect the transmission probability to be sim- HQD =  tL ²D tR  . (133) ply given by T ∼ |t2 | · |t2 |. It is possible to model this L R 0 tR 0 incoherent process by introducing dephasing or decoher- ence into the quantum dot. A simple way to do this is The total Hamiltonian quantum dot plus TLS can then to add an additional lead to the dot, which is equivalent be written as to adding the self-energy due to the lead on the onsite 2 ik energy of the dot → ²D + tC e , where tC is the coupling H = HQD ⊗ I2 + I3 ⊗ HT LS + HU (134) strength to the additional lead. where In is the n×n identity matrix and HU the interaction between the TLS and the quantum dot, which could be of the form   0 0 0 µ ¶ T(E) U t H =  0 1 0  ⊗ I , (135) U t∗ 0 0 0 0 I

which corresponds to the case where there is an additional energy U when the TLS is in the Eu state and the electron in the quantum dot. There is also a modiﬁed

Increased dephasing in a quantum dot coupling t → t + tI in the TLS if the electron is in the quantum dot. With this system it is then possible to obtain the Green’s function of the total system with total energy E, which is given by

−1 G = (E − H − ΣL − ΣR) , (136) FIG. 14: The resonance transmission of a quantum dot for different values of dephasing obtained by coupling a lead di- where rectly to the dot. The dot energy was taken to be ² = 0   D 1 0 0 µ ¶ with coupling t = t = 0.1 and t increasing from 0 to 1. S S L R C   G1,1 G1,2 ΣL = 0 0 0 ⊗ S S (137) G2,1 G2,2 Similarly, when introducing a complex part to the on- 0 0 0 site energy, one effectively introduces dissipation, which and can mimic for example the effects due to electron-phonon   coupling at non-zero temperatures. The results are very 0 0 0 µ ¶ GS GS similar to introducing an additional lead and one would Σ =  0 0 0  ⊗ 1,1 1,2 . (138) R GS GS obtain a dependence similar to the one shown in fig. 14. 0 0 1 2,1 2,2 20

GS are the surface Green’s functions of the semi-inﬁnite gate voltage applied to the dot. A typical example of leads, which by analogy to the two-dimensional leads can these Coulomb blockade peaks is show in ﬁg. 16. be expressed as X S ⊥,∗ ⊥ 1D Gn,m = ψn (E⊥)ψm(E⊥)G (E − E⊥), (139)

E⊥

⊥ where ψm(E⊥) are the eigenfunctions of the TLS with 1D energy E⊥ and G the surface green’s function of the 1D leads. It is now possible to obtain the transmission probability of the quantum dot coupled to the TLS, by simply using Fisher-Lee’s relation. An example is shown in ﬁg. 15. An important point to remember is that the the energy E is the total energy of the quantum dot plus the TLS. Hence, even when the two are not coupled the trans- L. P. Kouwenhoven et al. , Z. Phys. B 85 , 367 (1991). mission probability curve has two peaks centered at the eigenvalues of the TLS (for ²D = 0 and the black curve in ﬁg. 15). FIG. 16: The experimental conductance as a function of gate voltage in the Coulomb blockade regime.

The basic physics of this can be understood in terms T(E) of simple electrostatics, where the Coulomb potential due to an additional electron in the dot acts as a potential Increased dephasing barrier, hence suppressing the current ﬂow. This phenomena is illustrated in ﬁg. 17.

T(EF ) =1 Coulomb peak

EF U H1U=[0 0 0;0 1 0;0 0 0];H2U=[0 t*exp(i*U)-t;t*exp(-i*U)-t 0];U=0 to 3.14 ε D

FIG. 15: The resonance transmission of a quantum dot for T E different values of dephasing obtained by coupling a TLS di- ( F ) 〈〈 1 rectly to the dot. The black curve corresponds to zero cou- Coulomb blockade + pling between the quantum dot and the TLS. co-tunneling U EF ε D Depending in the type of coupling and details of the TLS the dephasing effect can be non-existent or very strong. FIG. 17: The basic phenomena of Coulomb blockade, where U is the Coulomb energy and ²D the dot energy. CO-TUNNELING AND KONDO EFFECT IN QUANTUM DOTS Hence these basic features (existence of peaks in the conductance as a function of gate voltage) can be under- In most quantum dots, Coulomb interactions cannot be stood in terms of this single electron approach where the neglected and the single particle picture presented above Coulomb interaction is simply treated as an electrostatic breaks down. The most dramatic effect of Coulomb in- potential. However, there is lot of additional physics hap- teractions is the presence of Coulomb blockade, which pening when the electron interactions are treated in a occurs when the Coulomb potential necessary ti add one quantum mechanical way. Co-tunneling and the Kondo electron to the dot exceeds the Fermi energy, hence effec- resonance are prime examples, where these strong inter- tively suppressing the conductance. Experimentally this actions in quantum systems lead to novel phenomena. is see as peaks in the conductance as a function of the Co-tunneling occurs when the quantum dot is in the 21

ε +U co-tunneling D H = H + H + H + H + H , (140) 2 1 with spin-flip | D {zL R} | LD {z DR} E F H0 V ε D where the coupling terms are treated as a perturbation. The idea is then to look at processes from some initial state: |ii, which is an eigenstate of H0 (f.ex. N electrons in the dot and one electron in the left lead) to a final FIG. 18: A possible co-tunneling process, which involves a state |fi also an eigenstate of H0 (f. ex. N electron in spin flip. This is a typical second order process the dot and one electron in the right lead). Then the generalization of Fermi’s golden rule allows us to obtain the rate Γfi for going from state |ii to |fi, which can regime of Coulomb blockade, i.e., low transmission. Co- then be related to the conductance through the quantum tunneling leads to an enhancement of the tunneling prob- dot for that process. The generalized Fermi’s golden rule ability. While there exist many possible co-tunneling pro- is given by (see textbook) cesses, the main idea is illustrated in fig. 18. Essentially, 2 for an electron to get through the dot, the Coulomb Γfi = 2π|hf|T |ii| δ(²f − ²i), (141) blockade effect is circumvented by having one electron hopping out of the dot before the next one falls in again. where

This is only possible when these ”events” occur simulta- T = V + VG0V + VG0VG0V + ··· = V + VG0T (142) neously, i.e., co-tunneling, since otherwise energy conser- −1 vation is violated. These co-tunneling paths can occur in and G0 = (E − H0) is the Green’s function of H0. T many different ways and another possibility is illustrated is simply referred to the T -matrix and can be written as T = V . In principle the contribution of any possible in fig. 19, which is also a precursor for the Kondo effect. 1−G0V process can now be evaluated in this formalism and it’s usually done in a pertubative way using eq. (142). As an example we will evaluate the process depicted in fig. co-tunneling (no spin- flip) and Kondo 18, which is a second order process since it involves the resonance if summed coupling elements twice. We can write the final state as over empty states ε +U +↑ ↑ +↓ ↓ D |fi = c c c c |ii (143) 2 4 1 Empty state ²D ²L ²R ²D E F 3 which represents the process of transferring one electron ε D form the left lead to the right lead, effectively represent- ing a tunneling process. We can now use the golden rule to second order, i.e.,

¯ ¯2 FIG. 19: Another co-tunneling process, which involves no X ¯ 1 ¯ Γ = 2π |t |2|t |2δ(² −² ) ¯hf|c+↓c↓ c+↑c↑ |ii¯ . fi R L R L ¯ ²R ²D ²D ²L ¯ spin-flip. This is a fourth order process, which can lead to ²L − 2²D − U the Kondo resonance. The numbers show the order of the ²R,²L,i virtual processes. (144) Using (143) then leads to µ ¶ X 1 2 Γ = 2π |t |2|t |2δ(² − ² ) fi R L R L ² − 2² − U ² ,² L D XR L ↓ +↓ +↑ ↑ 2 Tunneling matrix formalism × |hi|c²R c²R |iihi|c²L c²L |ii| i Z µ ¶2 In order to describe quantitatively this co-tunneling 2 2 1 ∼ |tR| |tL| d²LD(²L) process it is necessary to go beyond the single particle pic- ²L − 2²D − U ture and it is important to include interactions directly. ×nF (²L − EFL )(1 − nF (²R − EFR )), (145) A powerful way to evaluate these processes is based on a where n (²) are the Fermi distributions of the leads and generalization of Fermi’s Golden rule. The starting point F D(² ) the density of state of the lead. Hence, for a dif- is to consider a quantum dot such as the one depicted in L ference in Fermi energy between the left and right lead fig. 13 and to separate the couplings t and t . Hence, in L R of eV we obtain quite general terms the total Hamiltonian can be written 2 2 as Γfi ∼ V |tR| |tL| , (146) 22 which is a typical contribution for a co-tunneling process the down spin electron in the dot and the up spin electron (linear in V and of order Γ2). in the right lead. In order to use this formalism we need This previous example was treated within a perturba- the Green’s function for H0 which is simply tive approach, where we used the second order T -matrix. −1 Similarly one could look at different processes in that G0 = (E − H0 − ΣL − ΣR) . (151) way, which is a very standard approach to co-tunneling in the perturbative regime (t small). The self energies can be found in the same way as for In the following, however, we will use a different ap- the coupling to the T LS system. The transition rate proach, which doesn’t use a perturbation on the T - Γfi = T (E) can then be obtained by using a modified matrix, but instead fixes the number of electrons in the Fisher-Lee expression: system. The approach is similar to the one of the T LS T (E) = 4 · T r[Γ T Γ T +], (152) coupled to a quantum dot described earlier. We will ap- L R ply this formalism to the fourth order process depicted where in fig. 19. The idea is to ”rewrite” the process in fig. 19 ¡ +¢ as ΓL = = L · ΣL · L ¡ +¢ Empty available state ΓR = = R · ΣR · R , (153)

ε ε D R and tR U   1 0 0 µ ¶ ε = 0 ε = 0 1 0 ε L =  0 0 0  ⊗ t =1 D t =1 0 0 0 0 0   Left lead tL DottR Right lead 0 0 0 µ ¶ 1 0 R =  0 0 0  ⊗ (154) 0 0 0 0 1 Co-tunneling (no spin-flip) also T is the T -matrix given by: FIG. 20: The 2-particle equivalent of fig. 19, which grasps most of the physics. V 1 T = = −1 . (155) 1 − G0V V − G0 This 2-particle version is equivalent to the T LS coupled to a quantum dot from eq. (134), i.e., T (E) is now the transmission probability corresponding to the co-tunneling of an up electron initially in the left H = HQD ⊗ I2 + I3 ⊗ HT LS + HU , (147) lead, co-tunneling into the right lead in the presence of a down electron in the dot with a repulsion of strength with HT LS given instead by: U in the dot. Setting ²R = 0 allows the dot electron to virtually hop onto an empty lead state before return- µ ¶ ing. With this description we can now quantitatively ²D t HT LS = (148) obtain the transmission probability corresponding to the t ²R Coulomb peak as well as the co-tunneling process. The , result is shown in fig. 21 for different values of the inter-   0 0 0 µ ¶ action parameter U. 1 0 H = U  0 1 0  ⊗ , (149) The Kondo resonance can also be obtained within this U 0 0 0 0 0 frame-work, by summing over all available empty states. These empty states are described by ²R, hence the Kondo   resonance for this process contributes as: 0 tL 0 X H =  t ² t  (150) QD L D R ∼ T²R (E), (156) 0 tR 0 ²R

and the Green’s function is given by eq. (136). However, where T²R (E) is obtained from (152) by using ²R as sum- instead of using the Green’s function we want to connect ming parameter in HT LS. Experimentally, the Kondo this formalism to the T -matrix. Hence we can define V resonance is characterized by an increase in the conduc- as all the off-diagonal components of H, which represent tance at low temperatures as seen in fig. 22. the couplings to the dots: tL and tR. A reasonable initial The temperature dependence of the Kondo resonance state |ii is the down spin electron in the dot and the up arises from the temperature dependence of available spin electron in the left lead. The final state |fi is then states ²R in the leads. Often the Kondo resonance is 23

SUPERCONDUCTIVITY

Coulomb blockade peak U = 0 U = 0 → ∞

-M (magnetization)

Type I Type II H B>O (Vortex phase) HC1 HC2

U =1000 B (internal field) B=O (Meissner) Increasing U Co-tunneling peak II

Ed=-1 I H HC1 HC2 T (Abrikosov TC vortex crystal) II FIG. 21: The 2-particle co-tunneling process as function of the I H interaction parameter. One can observe the Coulomb block- HC1 HC2 ade peak regime at low U and the emergence of a co-tunneling peak at E = ²D + 0 for large U. FIG. 23: Magnetic ﬁeld dependence in a superconductor.

The main aspects of superconductivity are

• Zero resistance (Kammerlingh-Onnes, 1911) at T < Tc: The temperature Tc is called the critical temperature. • Superconductivity can be destroyed by an external magnetic ﬁeld Hc which is also called the critical ﬁeld (Kammerlingh-Onnes, 1914). Empirically, 2 Hc(T ) = Hc(0)(1 − (T/Tc) )

(2000) • The Meissner-Ochsenfeld effect (1933). The magnetic field does not penetrate the sample, the magnetic induction is zero, B = 0. This effect distin- FIG. 22: Experimental Kondo resonance in a quantum dot, guishes two types of superconductors, type I and identified by the increase in conductance at low temperatures. type II. In Type I, no field penetrates the sample, whereas in type II the field penetrates in the form dominated by a process which involves a filled state in of vortices. the leads instead of an empty state. A down electron • Superconductors have a gap in the excitation spec- form a filled state in the lead hops to the dot, the up trum. electron from the dot hops to the right lead, then the up electron from the left lead hops onto the dot and the The main mechanism behind superconductivity is the down electron on the dot hops back into the freed lead existence of an effective attractive force between elec- state. This process can then be described by a 3-particle trons, which favors the pairing of two electrons of oppo- T -matrix and the Kondo resonance is obtained by sum- site momentum and spin. In conventional superconduc- ming over the filled lead states. tors this effective attractive force is due to the interaction with phonons. This pair of electrons has now effectively zero total momentum and zero spin. In this sense this pair behaves like a boson and will Bose-Einstein conden- sate in a coherent quantum state with the lowest possible energy. This ground sate is separated by a superconducting gap. Electrons have to jump over this gap in order to be excited. Hence when the thermal energy exceeds the gap energy the superconductor becomes normal. The origin of the effective attraction between electrons can be understood in the following way: 24

k↑ e

ξ ∼ 1000Α°

e

- k↓

FIG. 26: To take full advantage of the attractive potential

Vanadium illustrated in Fig. 25, the spatial part of the electronic pair wave function is symmetric and hence nodeless. To obey the Pauli principle, the spin part must then be antisymmetric or a singlet.

This term can be negative, hence effectively produce an attraction between two electrons exceeding the Coulomb repulsion. This effect is the strongest for k = kF and q = FIG. 24: The critical magnetic field, resistance and specific 2kF since ²(kF ) = ²(−kF ) and ω(2kF ) ' ωD (the Debye heat as a function of temperature. frequency). Hence electrons will want to form opposite momentum pairs (kF , −kF ). This will be our starting + e- point for the microscopic theory of superconductivity à ions + e- + + + + + la BCS (Bardeen, Cooper and Schrieffer). region of 8 + + + + + + v ∼ 10 cm/s F positive charge + attracts a second electron BCS theory

FIG. 25: Origin of the retarded attractive potential. Elec- To describe our pair of electrons (the Cooper pair) we trons at the Fermi surface travel with a high velocity vF . As they pass through the lattice (left), the positive ions respond will use the formalism of second quantization, which is slowly. By the time they have reached their maximum excur- a convenient way to describe a system of more than one sion, the first electron is far away, leaving behind a region of particle. positive charge which attracts a second electron. p2 H1particle = ⇒ H1pψ(x) = Eψ(x) (158) When an electron flies through the lattice, the lat- 2m tice deforms slowly with respect to the time scale of Let’s define the electron. It reaches its maximum deformation at a time τ ' 2π ' 10−13s after the electron has passed. c+(x) |0i = ψ(x) and h0|c (x) = ψ∗(x) (159) ωD 1 |{z} 1 In this time the first electron has travelled ' vF τ ' 8 cm −13 ˚ vacuum 10 s · 10 s ' 1000A. The positive charge of the lattice deformation can then attract another electron with- With these definitions, |0i is the vacuum (or ground + out feeling the Coulomb repulsion of the first electron. state), i.e., state without electrons. c1 (x)|0i corresponds Due to retardation, the electron-electron Coulomb repul- to one electron in state ψ(x) which we call 1. c+ is also sion may be neglected! called the creation operator, since it creates one elec- The net effect of the phonons is then to create an tron from vacuum. c is then the anhilation operator, i.e., + attractive interaction which tends to pair time-reversed c1c1 |0i = |0i, which corresponds to creating one electron quasiparticle states. They form an antisymmetric spin from vacuum then anhilating it again. Other properties singlet so that the spatial part of the wave function can include be symmetric and nodeless and so take advantage of the + + attractive interaction. Furthermore they tend to pair in c1|0i = 0 and c1 c1 |0i = 0 (160) a zero center of mass (cm) state so that the two electrons The first relation means that we cannot anhilate an can chase each other around the lattice. electron from vacuum and the second relation is a con- Using perturbation theory it is in fact possible to show sequence of the Pauli principle. We cannot have two that to second order (electron-phonon-electron) the effect electrons in the same state 1. of the phonons effectively leads to a potential of the form Hence,

+ + + + 2 c c = 0 ⇒ (c c )+ = 0 ⇒ c c = 0 (~ωq) 1 1 1 1 1 1 Ve−ph ∼ 2 2 (157) + + + (²(k) − ²(k − q)) − (~ω(q)) ⇒ (c1 c1)c1 |0i = c1 |0i (161) 25

+ This shows that c1 c1 acts like a number operator. It Hence the diagonalization condition for HMF fixes the counts the number of electrons in state 1. (Either 1 or angle θ of our Boguliubov transformation: 0). ga −ga We can now extend this algebra for two electrons in tan(2θ) = − ⇒ sin(2θ) = p (166) + t t2 + (ga)2 different states c1 |0i corresponds to particle 1 in state 1 + and c2 |0i to particle 2 in state 2. The rule here is that + + Hence HMF now becomes ci cj + cjci = δi,j. Finally we can write down the two particle hamiltonian p p as + + 2 2 2 2 2 HMF = (A1 A1+A2 A2) t + (ga) +(t− t + (ga) )+ga (167) + + + + It is now immediate to obtain the solutions of the H2p = t1c1 c1 + t2c2 c2 − gc1 c1c2 c2 (162) Hamiltonian, since we have the ground state |ai and we where ti is the kinetic energy of particle i and g is have a diagonal Hamiltonian in terms of Ai hence the the attraction between particle 1 and 2. We will also Ground state energy E0 is simply given by HMF |ai = suppose that t = t for the Cooper pair. The job now 1 2 E0|ai and Ai|ai = 0. The first degenerate excited states is to find the ground state of this Hamiltonian. Without + + + are Ai |ai with energy E1, where HMF Ai |ai = E1Ai |ai interactions (g = 0) we would simply have E = t1 + + + and the next energy level and state is A1 A2 |ai, with t2. The interaction term is what complicates the system + + + + energy E2 given by HMF A1 A2 |ai = E2A1 A2 |ai, hence since it leads to a quadratic term in the Hamiltonian. p 2 2 2 The idea is to simplify it by getting rid of the quadratic E2 = t + t + (ga) + ga term. This is done in the following way. From eq. (162) E = t + ga2 we have 1 p 2 2 2 E0 = t − t + (ga) + ga (168)

+ + + + H = t(c1 c1 + c2 c2) − gc1 c1c2 c2 If we take t → 0 (this corresponds to considering zero + + + + temperature and the Fermi energy as the reference en- = t(c c1 + c c2) + gc c c1c2 1 2 1 2 ergy) and define ∆ = ga we have = t(c+c + c+c ) − ga(c c − c+c+) + ga2 1 1 2 2 1 2 1 2  + + 2 + g(c1 c2 + a)(c1c2 − a) (163)  E2 = ∆ + ga | {z } 2 E1 = ga (169) '0 (Mean field approx.)  2 E0 = −∆ + ga + + + + 2 ⇒ HMF = t(c1 c1 + c2 c2) − ga(c1c2 − c1 c2 ) + ga We can now turn to what a is since, We used the mean field approximation, which re- places c1c2 by its expectation value ha|c1c2|ai = a ⇒ + + ha|c1 c2 |ai = −a, where |ai is the ground state of the a = ha|c1c2|ai Hamiltonian. The idea now is to diagonalize HMF , i.e. + P = − cos(θ) sin(θ)ha|A1A1 |ai a Hamiltonian in the form H = c+c . The trick here i i i = − cos(θ) sin(θ) ha|ai is to use the Boguliubov transformation: | {z } =1 ½ + ⇒ a = − sin(2θ)/2 (170) c1 = A1 cos(θ) + A2 sin(θ) + + c2 = −A1 sin(θ) + A2 cos(θ) Combining (166) and (170) we obtain ½ + A1 = c1 cos(θ) − c2 sin(θ) ⇒ + + (164) A2 = c1 sin(θ) + c2 cos(θ) ga 2a = p (171) + + t2 + (ga)2 It is quite straightforward to see that Ai Aj+AjAi = δij, A A + A A = 0, and A+A+ + A+A+ = 0 using the i j j i i j j i which is the famous BCS gap (∆) equation. Indeed, it properties of c . We can now rewrite H in terms of i MF has two solutions, our new operators Ai: + + + + 2 a = 0 ⇒ ha|c c |ai = 0 ⇒ Normal HMF = t(c1 c1 + c2 c2) − ga(c1c2 − c1 c2 ) + ga 1 2 + + a 6= 0 ⇒ t2 + (ga)2 = g2/4 ⇒ Superconductor(172) = t(A1 cos(θ) + A2 sin(θ))(cos(θ)A1 + sin(θ)A2 ) + ··· | {z } + + √ = (A1 A1 + A2 A2)(t cos(2θ) − ga sin(2θ)) ∆=ga= g2/4−t2 + t(1 − cos(2θ)) + ga sin(2θ) + ga2 This gives us the condition for superconductivity g ≥ + (A+A+ − A A )(t sin(2θ) + ga cos(2θ)) (165) 1 2 1 2 | {z } 2t. Hence the attraction between our two electrons has to =0 to diagonalize HMF be strong enough in order to form the superconducting 26

'=ga

Normal ½ + Ak,↑ = αck,↑ + βc−k,↓ ∗ + (177) A−k,↓+ = −β ck,↑ + αc−k,↓ Superconductor leads to the following mean ﬁeld hamiltonian: t X g/2 H = E (A+ A +A+ A )+K , (178) FIG. 27: The gap of a BCS superconductor as function of the MF |{z}k k,↑ k,↑ k,↓ k,↓ 0 kinetic energy. k,σ √ 2 2 tk+|∆k|

where ∆k is the gap. Using Ak|ai = 0, the BCS ground gap ∆. For t = 0 (zero temperature and states at the state can be written as Fermi energy, EF = 0) and using (169) this leads to a ground state energy of −∆/2 and to the ﬁrst excited Y + state of ∆/2. Hence the states at the Fermi energy have |ai = (uk + vkck,↑c−k,↓|0i, (179) disappeared in order to form a lower energy ground state k in the presence of an attractive interaction. Typically, which leads to the famous BCS result t is proportional to the temperature, hence there is a superconducting transition as a function of temperature. 2∆(T = 0) = 3.53kBTc (180)

We now want to find the expression for our supercon- −1/gd(EF ) and |∆(0)| = 2ωDe , where ωD is the phonon ducting wavefunction |ai. The most general possible form Debye frequency and d(EF ) the density of states at the is Fermi energy. How can we relate BCS theory to the observed Meiss- |ai = α|0i + β c+|0i + β c+|0i + γc+c+|0i (173) 1 1 2 2 1 2 ner effect? Including the vector potential A~ from the magnetic field B = ∇ × A into the Hamiltonian we have In addition the condition A |ai = 0 has to be verified, i p → p − e A. In second quantization the kinetic part of which leads after some algebra to c the Hamiltonian is then rewritten as Z 1 + e 2 + + T = Ψ (r)(−i~∇r − A) ψ(r)dr (181) |ai = α(1 + tan(θ)c1 c2 )|0i (174) 2m c in the presence of a magnetic field. Free electrons are This state clearly describes an electron pair, the gauge invariant, which can be expressed by Cooper pair and represents the superconducting ground ½ state of the Hamiltonian. In our derivation we only con- A → A + ∇φ , (182) sidered two electrons, but this framework can be gener- Ψ(r) → e−ieφΨ(r) alized to N electrons, where the generalized BCS Hamil- −ieφ tonian can be written as which corresponds to ci → cie in our two particle −2ieφ example. Hence ha|c1c2|ai → e ha|c1c2|ai which is X X clearly not gauge invariant in the superconducting state + + + H = t c c − g c c c c since ha|c1c2|ai = a 6= 0. This clearly shows that the BCS k k,|{z}σ k,σ q k+q,↑ −k−q,↓ k,↑ −k,↓ q superconducting state breaks the gauge symmetry. k,σ spin (175) It is further possible to show that all solutions which break gauge invariance are solution of p − e A = 0. But Here Vq is positive and represents the effective phonon c induced attraction between electrons at the Fermi level. the current density is given by j = nev, which leads to It’s maximum for q = 0. The second term describes the ne2 ~j = A.~ (183) process of one electron with momentum k and another mc electron with momentum −k which are anhilated in order to create one electron with momentum k + q and The magnetic induction is B~ = ∇~ × A~ as usual. This is another one with momentum −k − q, hence momentum in fact London’s equation for superconductivity. We can is conserved in this scattering process and a phonon with now take the rotational on both sides of (183), hence momentum q is exchanged. In mean field theory this BCS Hamiltonian become ne2 ∇~ × ~j = ∇~ × A~ X X |{z} + + + mc HMF = tkc ck,σ − (∆kc c + h.c), (176) ~ ~ 4π~ k,σ k,↑ −k,↓ ∇×B= c j k,σ k 2 ~ ~ ~ 4πne ~ P ⇒ ∇ × ∇ × B = − 2 B 0 0 mc with ∆k = k0 gkk hc−k ,↓ck,↑i. Using the generalized −x/λL Bogoliubov transformation ⇒ Bx ∼ e (184) 27 q mc2 resistance, which are the main ingredients of supercon- with λL = 2 which is the London penetration 4πne ductivity. (Reminder: Maxwell gives ∇~ × B~ = 4π~j and length. c ~ ~ 1 ∂B~ ∇ × E = − c ∂t ). A remarkable aspect of superconductivity is that one of the most fundamental symmetries is broken. Indeed, B Gauge invariance is broken because ~j ∼ A~. What hap- pens is that below Tc we have a symmetry breaking, -x/ l e L which leads to new particles, the Cooper pairs. Math- Vacuum Inside the ematically, we have superconductor U(2) = SU(2) ⊗ U(1) , (186) x | {z } | {z } | {z } Normal state SC state Gauge invariance FIG. 28: The decay of the magnetic field inside the superconductor. The decay is characterized by the London penetration where U(1) ⇔ c → ceiα (Gauge invariance), U(2) ⇔ c → length λL. Ac (A = eiφ/2a) and a+a = 1, SU(2) ⇔ φ = 0. Another important theory in superconductivity is the If we had a perfect conductor, this would imply that Ginzburg-Landau (GL) theory. It stems from the general the current would simply keep on increasing with an ex- theory of phase transitions of which superconductivity is ternal field E~ , hence an important example. It is possible to connect BCS theory to the GL theory by realizing that the gap ∆ ∼ Ψ ∂~j ne2 = E~ and the density of superconducting Cooper pairs (order 2 ∂t mc parameter) ns ∼ |Ψ| in the GL equation: ∂ ne2 ⇒ ∇~ ×~j = ∇~ × E~ ∂t mc ∂ ∂ 4πne2 2 1 2e 2 ⇒ ∇~ × ∇~ × B~ = B~ (185) [α + β|Ψ| + (−i~∇ + A) ]Ψ = 0 (187) ∂t ∂t mc2 2m c This equation is automatically verified from (184). Depending on the values of α and β we have either a Hence, (184) implies both the Meissner effect and zero type I or a type II superconductor.