Quantum Transport Theory with Tight-Binding Hamiltonian Stefano Sanvito Department of Physics Trinity College Dublin [email protected] September 25, 2002

Quantum Transport Theory with Tight-Binding Hamiltonian

Stefano Sanvito

Overview

• Quick Review of Landauer-B¨ uttiker formalism • Scattering Theory I: simple 1D example • Scattering Theory II: structure of the Green’s functions • Scattering Theory III: General formalism • Few Applications • Beyond the Tight-Binding: LDA Hamiltonian • Summary & Outlook

Most of the materials can be found in: S. S., “Giant Magnetoresistance and Quantum Transport in Magnetic Hybrid Nanostructures”, PhD Dissertation, University of Lancaster (UK) (1999) S. S., C.J. Lambert, J.H. Jefferson, and A.M. Bratkovsky, Phys. Rev. B 59, 11936 (1999) 1

Quantum Transport Theory with Tight-Binding Hamiltonian

Stefano Sanvito

Length-Scale System Length (L) Fermi Wavelength (λ)

λF = 2π/kF =

q

2π/ns

(1)

Elastic Mean Free path (Lm)

Lm = vF · τm

(2)

Phase relaxation length (Lϕ)

Lϕ = vF · τϕ if τϕ < τm 2 2 Lϕ = vFτmτϕ/2 if τϕ > τm

(3) (4)

Spin-flip length (Ls)

Ls = vF · τs Length λF Lm Lϕ Ls

Metals a0 10-100 ˚ A 10-100 ˚ A 10-100 nm

Semicond. (2D) 35 nm 30 µm Lm ≤ Lϕ  Lm up to 100 µm

(5) Semicond. (3D) 35 nm 100 µm Lm ≤ Lϕ  Lm up to 1 mm

Working Condition L ≥ Lϕ 2

Quantum Transport Theory with Tight-Binding Hamiltonian

Stefano Sanvito

Landauer-B¨ uttiker formalism

µ1

1

T

µ2

R

• The scattering leads electrons Basic Assumptions: • (µ1 − µ2) → 0+

inject

uncorrelated

Current I emitted from the left lead:   ∂n I = ev (µ1 − µ2) ∂E

(6)

but ∂n/∂E = ∂n/∂k · ∂k/∂E = 1/2π · 1/v¯ h and, ∆V = e∆µ

e2 I = ∆V h

(7)

If an electron has probability T to be transmitted, the conductance Γ = I/∆V becomes

Γ=

e2 hT

(T = 1 Sharvin Conductance) 3

Quantum Transport Theory with Tight-Binding Hamiltonian

Stefano Sanvito

Generalization to many scattering channels and many leads

µβ µ

γ

µ

α

µδ • All electrons injected from α-th lead have same µα • N α scattering channels for α-th lead • Tijαβ : probability to transmit i-th channel of α-th lead into j -th channel of β -th lead Then the fundamental current/voltage relation is α

I =

nX leads

Γ

αβ

β

µ

(8)

β

where 2

Γ

αβ

Nα Nβ

e X αβ e2 αβ αβ† = Tij = Tr t t h ij h

(9)

4

Quantum Transport Theory with Tight-Binding Hamiltonian

Stefano Sanvito

Scattering Theory I: simple 1D example

0 −∞ ...... i0-3

0 γ1

i0 - 2

γ1

ε0

ε0

0 α

γ2

ε0 γ2

i0-1 ... i0+1 i0+2 i0+3

......

+∞

1) Green’s functions for infinite leads

eik(E)|j−l| gjl (E) = i¯ hv(E) the crucial point here is to calculate the inverse dispersion   E −  0 −1 k = k(E) (k = cos ) 2γ

(10)

(11)

2) Green’s functions for semi-infinite leads Assume that the chain is terminated at i = i0 − 1 (no site for i ≥ i0). Use boundary conditions: g new = g old + ψ with ψ a wave-function such that ginew = 0 for l < i0 0l

e−ik(j−2i0+l) ψj (l, i0) = − i¯ hv

(12)

Note that the new Green’s functions has the same causality than the old. 5

Quantum Transport Theory with Tight-Binding Hamiltonian

Stefano Sanvito

3) Surface Green’s functions for decoupled chains It is simply ginew . Total GF’s for the two chains 0 −1,i0 −1

 g=

eik1 γ1

0 eik2 γ2

0

 (13)



4) Surface Green’s functions for coupled chains: Dyson’s equation

G=



1 γ1γ2e−i(k1+k2) − α2

e−ik2 γ2 α

α e−ik1 γ1

 (14)

5) Extract the t matrix a) First note that the most general wave-function for electrons coming from the left can be written as

ψl =

        

eik1 l v1 1/2

+

r e−ik1l 1/2 v1

l ≤ i0 − 1 (15)

t 1/2 e v2

ik2 l

l ≥ i0 + 1

this introduces reflection r and transmission t coefficients. b) Then construct the projector to map (14) onto (15). It is the same for the infinite case:

eikl glj P (j) = 1/2 v

for

l≥j

(16) 6

Quantum Transport Theory with Tight-Binding Hamiltonian

Stefano Sanvito

c) Now, use the projector to extract r and t

lim

l→(i0 −1)

Gl,i0−1P (i0 − 1) =

1 v1

1/2

+

r v1

1/2

(17)

gives

r = Gi0−1,i0−1P (i0 − 1)v1

lim l→(i0 +1)

Gl,i0−1P (i0 − 1) =

1/2

t

−1

e 1/2

ik2 l

(18)

(19)

v2

gives

t = Gi0+1,i0−1P (i0 − 1)v2

1/2 −ik2 e

(20)

Since the incoming wave-functions have unit flux

2

2

|t| + |r| = 1

(21)

7

Quantum Transport Theory with Tight-Binding Hamiltonian

Scattering wave−function

Stefano Sanvito

GF’s infinite leads Boundary Conditions GF’s semi−infinite leads

Interaction Matrix V

Projector Dyson’s Equation Surface GF’s full system

S MATRIX Landauer−Buttiker

CONDUCTANCE

8

Quantum Transport Theory with Tight-Binding Hamiltonian

Stefano Sanvito

Scattering Theory II: structure of the Green’s functions To understand the structure of GF’s for scattering problems, consider the next simplest case: 2D simple cubic

z

The Green function for such a system can be shown to be

0

0

g(z, x; z , x ) =

M X

 Nn sin

n=1

   ikzn(E)|z−z0| nπ nπ e 0 x sin x M +1 M +1 i¯ hvzn(E) (22)

with kzn(E) and vzn(E) given by the dispersion

 E = o + 2γ cos



nπ M +1



n

+ cos kz

 (23)

9

Quantum Transport Theory with Tight-Binding Hamiltonian

Stefano Sanvito

Schematically g(z, x; z 0, x0; E) is simply M X

0

n

eikz (E)|z−z | ∗ 0 0 0 βn(x ) g(z, x; z , x ; E) = βn(x) n i¯ hvz (E) n=1 The

P

(24)

is over ALL POSSIBLE CHANNELS (open or closed)

rd

ld

lm

z|

rm

Four possible scattering channels

• • • •

ld lm rd rm

left-decaying channels left-moving channels right-decaying channels right-moving channels 10

Quantum Transport Theory with Tight-Binding Hamiltonian

Stefano Sanvito

Scattering Theory III: General formalism GENERALIZATION

H1

H0

H1

H1

H0

H0

H1

z The Hamiltonian for such a system is:

      H =    

... ... ... ... ... ... ...

... H0 H−1 0 0 ... ...

... H1 H0 H−1 0 ... ...

... 0 H1 H0 H−1 ... ...

... ... 0 H1 H0 ... ...

... ... ... 0 H1 ... ...

... ... ... ... 0 ... ...

... ... ... ... ... ... ...

          

(25)

with H0† = H0 and H−1 = H1† matrices M × M and M the number of degrees of freedom in the “slice” described by H0 Note that this is identical to the linear chain case when we replace

0 −→ H0 γ −→ H1 11

Quantum Transport Theory with Tight-Binding Hamiltonian

Stefano Sanvito

GF’S FOR INFINITE LEADS

ψz is a column vector corresponding to the z -th slice. We introduce the Bloch function (vk⊥ group velocity)

ψz = √

1 ik z e ⊥ φk⊥ v k⊥

(26)

we obtain the “band structure” equation

  ik⊥ −ik⊥ H0 + H1e + H−1e − E φk⊥ = 0

(27)

First Problem: k = k(E) The “band structure” equation can be solved by mapping the (27) onto an equivalent eigenvalue problem

 H=

−H1−1(H0 I

− E)

−H1−1H−1 0

 (28)

where the eigenvalues are eik⊥z and the first half of the eigenvectors are φk⊥ . Note that H1 CAN NOT be singular. Now we calculate vk and sort out left- (φk¯ ) and right-moving (φk ) and -decaying channels

12

Quantum Transport Theory with Tight-Binding Hamiltonian

Stefano Sanvito

Green’s functions fundamental Ansatz Similarly to the 2D simple cubic case we assume the following form

gzz0 =

 P M ikl (z−z 0 ) †  φ e wk  l=1 kl   PM ¯ (z−z 0 ) †  ik  l φ e wk¯ ¯ l=1 kl

l

z ≥ z0 (29)

l

z ≤ z0

wkl and wk¯l are computed by imposing 1. The Green’s equation [(E − H)g]zz0 = δzz0 2. Continuity for z = z 0

An expression for wkl and wk¯l which involves only φ, H1 and the “dual” ˜ can be given (φ ˜† φk = φ ˜† φ¯ = δlh). vectors φ ¯ kh k h k l

l

Then, by using convenient boundary condition, right (gR) and left (gL) surface GF’s are calculated. The total surface GF for decoupled leads:

 g(E) =

gL(E) 0

0 gR(E)

 (30)

13

Quantum Transport Theory with Tight-Binding Hamiltonian

Stefano Sanvito

EFFECTIVE V MATRIX The idea is to reduces a complex V matrix to an effective coupling matrix between two surfaces. We use the decimation technique. It consists in redefining recursively the matrix V, by eliminating the degrees of freedom which do not couple the external surfaces. In V is an N × N matrix after the first iteration

(1)

Vij

= Vij +

Vi1V1j E − V11

(31)

after l iterations (l−1)

(l)

(l−1)

Vij = Vij

+

Vil

E−

(l−1)

Vlj

(32)

(l−1) Vll

finally after N − 2M steps we end up with an M × M “effective coupling matrix”

 Veff (E) =

VL∗(E) ∗ VRL (E)

∗ VLR (E) VR∗(E)

 (33)

Note that Veff (E) is energy dependent. If the interaction is short range the technique is very good for numerical optimization. Now the total Green’s function is calculated via Dyson’s equation. 14

Quantum Transport Theory with Tight-Binding Hamiltonian

Stefano Sanvito

S MATRIX AND TRANSPORT COEFFICIENTS It is a straightforward generalization of the case of the linear chain. The Bloch state ψz is normalized to unit flux (from the expression of the current)

1 ikz ψz = √ e φk vk

(34)

Note: if there is k-degeneracy the (34) does not diagonalize the current. A rotation is needed. It can be used to build up the general scattering wave-function

ψz =

      

ikl z e√ vl φkl

+

P

¯

rhl ik z √ e h φk¯h h v ¯

z≤0

h

P

thl ik z h φk √ h vh e h

(35)

z≥L

where:

• rhl = rk¯h,kl is the r coefficient for a rm electron with momentum kl scattered in a lm electron with momentum kh • thl = tkh,kl is the t coefficient for a rm electron with momentum kl transmitted in a lm electron with momentum kh Now, one defines the projector P as before, and the scattering coefficients are computed. 15

Quantum Transport Theory with Tight-Binding Hamiltonian

Stefano Sanvito

Beyond the Tight-Binding: LDA Hamiltonian We use an LSDA-LCAO code to calculate the electronic structures of heterojunctions. The final output is a tight-binding Hamiltonian, then this scattering formalism can be applied. Problems: 1. TB is non-orthogonal 2. H1 can be singular 3. Efficient treatment of large matrices is needed Solutions: 1. Introduction of overlap matrix So in the form



S−1  0 So =   ... ...

S0 S−1 0 ...

S1 S0 S−1 0

0 S1 S0 S−1

... 0 S1 S0

 ... ...   0  S1

This does not add any particular complication. For example the “band-structure” equation becomes:

h i ik⊥ −ik⊥ (H0 − ES0) + (H1 − ES1)e + (H−1 − ES−1)e φk⊥ = 0 (36) 16

Quantum Transport Theory with Tight-Binding Hamiltonian

Stefano Sanvito

THE H1 PROBLEM “Lack of Bonding”

γ ε0  H1 =

0 γ

0 0



“Excess of Bonding”

γ ε0  H1 =

γ γ

γ γ



17

Quantum Transport Theory with Tight-Binding Hamiltonian

Stefano Sanvito

Quick Fix The dispersion relation k = k(E) can be calculated by solving the generalized eigenvalue problem

 αk

0 H−1

I H0 − E



φk χk



 = βk

I 0

0 −H1



φk χk



with

e

ik

= βk /αk

Complete Fix Eliminate the singularities: DECIMATION

18

Quantum Transport Theory with Tight-Binding Hamiltonian

Stefano Sanvito

Digital Magnetic Heterostructures In Plane transport 400 Tot Ga s Ga p As s As p Mn s Mn p Mn dt Mn de

2

σ(e /h)

300

200

100

0 -5

-4

-3

-2

-5

E (eV)

-4

-3

-2

E (V)

Perpendicular to Plane transport 200 Tot Ga s Ga p As s As p Mn s Mn p Mn dt Mn de

2

σ(e /h)

150

100

50

0 -5

-4

-3

E (eV)

-2

-5

-4

-3

-2

E (V) 19