Spin Interference and the Rashba Hamiltonian

Introduction Quantum Graph with Rashba Hamiltonian A Simple Spin Filter Multi-particle Spin Devices

M. Harmer

Czech Academy of Sciences, Re˘ z˘

Graph Models of Mesoscopic Systems, Newton Institute

Abstract

Scattering for a simple graph with Rashba hamiltonian and the possibility of spin ﬁltering is analysed. We discuss calculation of truth tables for multi-particle spin devices.

M. Harmer Spin Interference Introduction Quantum Graph with Rashba Hamiltonian A Simple Spin Filter Multi-particle Spin Devices Outline

1 Introduction 2 Quantum Graph with Rashba Hamiltonian Basic description Spectrum on the ring Boundary conditions Scattering matrix 3 A Simple Spin Filter The wronskian Spin ﬁlter Properties of ﬁlter 4 Multi-particle Spin Devices Description of multi-particle states Calculation of truth tables

M. Harmer Spin Interference Introduction Quantum Graph with Rashba Hamiltonian A Simple Spin Filter Multi-particle Spin Devices Quantum graph models

Offers a simple model of quantum waveguides. Finite quantum graphs solvable. Can be use to make qualitative statements/identify general features of the system. Having explicit expression for scattering properties allows optimisation of design, at least to ﬁrst order.

M. Harmer Spin Interference Introduction Quantum Graph with Rashba Hamiltonian A Simple Spin Filter Multi-particle Spin Devices Nanoelectronics with spin

Use of spin in nanoelectronic devices: Datta and Das [5] spin field effect transistor. Rashba effect and spin filtering: Kiselev and Kim [13, 14], Nitta et al [20], Splettstoesser et al [23], Streda˘ and Seba˘ [24]. 1 We consider quantum graphs with spin- 2 particles. Study simple(st) family of graphs with spin, where interaction is generated by Rashba effect. Application of this family to spin filtering. We consider multi-particle spin quantum graphs as models for (classical) gates and discuss the derivation of simplified truth tables.

M. Harmer Spin Interference Introduction Basic description Quantum Graph with Rashba Hamiltonian Spectrum on the ring A Simple Spin Filter Boundary conditions Multi-particle Spin Devices Scattering matrix Family of graphs with Rashba hamiltonian

x2 We have free hamiltonian on x1 the semi inﬁnite edges, with local coordinates x1, x2,..., xn ∈ R+. θ Assume structural inversion asymmetry on the ring so that xn the Rashba effect is present. Local coordinate on the ring is θ ∈ [0, 2π). Figure 1: Rashba ring.

The hamiltonian on the ring is

α2 H f = D2 f − f . 0 0 2 Here 1 d α D = − + σ , 0 i dθ 2 r σr = σx cos(θ) + σy sin(θ) ,

σx , σy , σz , id denote the Pauli spin matrices and α describes the strength of the Rashba spin-orbit coupling. Relative to spin axes the ring is in the x-y plane.

2 Solutions of the eigenequation H0f = k f , are

−iσz θ/2 −iσy ϕ/2 ±iσz κ±(θ−π) f±,0(θ, k) = e e e (1) q q 2 α2 1 α2 where κ± = k + 4 ± 4 + 4 and tan(ϕ) = α, π π ϕ ∈ (− 2 , 2 ). Using Pauli spin matrices we write the solution with both spin states together.

Eigenvalues 2 1 p 2 λ±,n = n − 2 ± n 1 + α − 1 (2)

are zeroes of cos(κ±π). Since λ+,n = λ−,−n we take as the spectrum [ σ (H0) = {λ+,n : n ∈ {1, 2,...}} {λ−,n : n ∈ {0, 1, 2,...}} .

Each λ±,n has multiplicity two; the eigenspace is spanned ±inθ ∓inθ by e χ↑ , e χ↓ where

cos(ϕ/2) −e−iθ sin(ϕ/2) χ = , χ = . ↑ eiθ sin(ϕ/2) ↓ cos(ϕ/2)

ϕ The twofold degeneracy of the χ↑ eigenvalues rises to a fourfold degeneracy when p 1 + α2 − 1 = m ∈ {0, 1,...} . χ↓ In this case Figure 2: Spin eigen-states on the ring. λ−,n = λ+,n+m .

The Greens function on the ring is the continuous solution of

2 2 2 H0 G θ, η; k = k G(θ, η; k ) θ=η+ ∂ 2 G(θ, η; k ) = id ∂θ θ=η− G(θ, η; k 2) = G?(η, θ; k 2) ,

for k ∈ R \ σ (H0).

Pauli matrices allow us to write both spin states together in one solution, equation (1), giving a ‘compact’ expression G θ, η; k 2 =

−iσz κ+η iσz κ−η −iσy ϕ/2 e e e iσz η/2 = f+,0(θ) − f−,0(θ) e σz cos(κ+π) cos(κ−π) 2i(κ+ + κ−) " # e−iσz θ/2 e−iσy ϕ/2 eiσz κ+(θ−η−π) e−iσz κ−(θ−η−π) = − × 2i(κ+ + κ−) cos(κ+π) cos(κ−π)

−iσy ϕ/2 iσz η/2 e e σz ; θ − η ∈ [0, 2π) . (3)

On each wire we have the ‘free’ hamiltonian

2 1 d Hj fj = Dj fj , Dj = − . i dxj

We write both spin states of the generalised eigenfunctions as ±i idkxj f±,j = e , in the z basis. Interpretation of spin measurements along x or y axes in the wires.

Assume no two wires are ψ 2 attached at the same point on ψ1 ring the ring. wire Prescribe boundary conditions ψ3 at each vertex

−1 Figure 3: Boundary β ψ1 = ψ2 = ψ3 , (4) conditions at a junction. 0 0 0 βψ1 + ψ2 + ψ3 = 0 .

Pn Continuity on the ring implies H = H0 ⊕ j=1 Hj with these boundary conditions is self-adjoint [17]. β describes the strength of the coupling between the wire and the ring. Boundary conditions are assumed to hold with the same β for each component of the spinor, ie. the coupling is spin independent. However, β may be different at each vertex. Derivation with arbitrary β. For explicit calculation of polarisation β = 1. Scattering matrix of ‘free’ junction the same as scattering matrix of T-junction in the physics literature [1].

2 Scattered waves ψi are eigenfunctions Hψi = k ψi satisfying the boundary conditions at the vertices and having the following form on the wires: ψi = f+,i + f−,i Sii ⊕j6=i f−,j Sji . (5)

Components of ψi and the scattering matrix Sji , i, j ∈ {1,..., n}, are 2 × 2 matrix valued.

n Suppose {θi }i=1 are the points where the wires are attached to the ring we deﬁne Gk = G (θ, θk ). Due to form of boundary conditions we can write the scattered wave on the ring X ψi = Gk Aki (6) k for some matrix A.

We see from (5) that on the wires

0 ψi |j = δji + Sji , ψi j = ik δji − Sji .

Likewise, deﬁning

2 Gjk = G θj , θk ; k

we have from (6)

θ=θ+ X 0 j ψi |j = Gjk Aki , ψi − = Aji θ=θj k on the ring.

Eliminating A we can write the scattering matrix

−1 S = (ikβGβ + I)(ikβGβ − I) (7)

in terms of the Greens function on the ring G and a diagonal matrix containing the coupling strengths for the n vertices β.

M. Harmer Spin Interference Introduction The wronskian Quantum Graph with Rashba Hamiltonian Spin ﬁlter A Simple Spin Filter Properties of ﬁlter Multi-particle Spin Devices The wronskian

The wronskian X W (f , g) = (hD f , gi + hf , D gi)| i i xi i

is well deﬁned on eigensolutions, H f = k 2f , H g = k 2g. Here h·, ·i is the inner product on spinors. If f , g satisfy the boundary conditions (4) then

W (f , g) = 0

the wronskian is zero.

M. Harmer Spin Interference Introduction The wronskian Quantum Graph with Rashba Hamiltonian Spin ﬁlter A Simple Spin Filter Properties of ﬁlter Multi-particle Spin Devices Symmetries

Using this and symmetries of the hamiltonian we can ﬁnd symmetries of the scattering matrix. A trivial example, the wronskian of two scattered waves gives X 0 = W ψi , ψj = hDψi , ψj i + hψi , Dψj i xk =0 k X † † = −k δik − Sik δkj + Skj + δik + Sik δkj − Skj , k

† ie. S S = I, the unitarity of the scattering matrix. Here W ψi , ψj is 2 × 2 matrix valued.

M. Harmer Spin Interference Introduction The wronskian Quantum Graph with Rashba Hamiltonian Spin ﬁlter A Simple Spin Filter Properties of ﬁlter Multi-particle Spin Devices Symmetries

Suppose the graph, and boundary conditions, are invariant with respect to a reﬂection in one of the axes on the plane

R : y ↔ −y .

Then the hamiltonian will commute with R = σy R and X 0 = W Rψi , ψj = hDRψi , ψj i + hRψi , Dψj i xk =0 k X † = k δik σy − R(Sik )σy δkj + Skj − k † δik σy + R(Sik )σy δkj − Skj .

M. Harmer Spin Interference Introduction The wronskian Quantum Graph with Rashba Hamiltonian Spin ﬁlter A Simple Spin Filter Properties of ﬁlter Multi-particle Spin Devices Symmetries

Consequently

† σˆy R(S )ˆσy S = I ⇒ S =σ ˆy R (S)σ ˆy , (8)

where σˆy is block diagonal with σy on the diagonal.

Further symmetries of S can be found from K = σy K , where K is complex conjugation and L = σz L, where L reverses the sign of α.

M. Harmer Spin Interference Introduction The wronskian Quantum Graph with Rashba Hamiltonian Spin ﬁlter A Simple Spin Filter Properties of ﬁlter Multi-particle Spin Devices Three terminal device

Consider a three terminal θ2 device with y ↔ −y symmetry. ξ The angle between the ﬁrst θ1 and second and ﬁrst and third wires is ξ ∈ (0, π). −ξ Also need the coupling θ3 constants at vertices two and three the same—for simplicity we set βi = β. Figure 4: The three terminal Rashba ring.

M. Harmer Spin Interference Introduction The wronskian Quantum Graph with Rashba Hamiltonian Spin ﬁlter A Simple Spin Filter Properties of ﬁlter Multi-particle Spin Devices Flux and polarisation

Decompose the components of the scattering matrix in terms of spin matrices X sij = sij,1 + i σαsij,α . α

Then the ﬂux Tij and polarisation in the α-axis Pij,α for waves going from wire j to wire i are

2 2 2 2 Tij = 2 sij,1 + sij,x + sij,y + sij,z (9) Pij,α = 4= sij,1s¯ij,α + sij,α−1s¯ij,α+1 . (10)

M. Harmer Spin Interference Introduction The wronskian Quantum Graph with Rashba Hamiltonian Spin ﬁlter A Simple Spin Filter Properties of ﬁlter Multi-particle Spin Devices Flux and polarisation

Now S =σ ˆy R (S)σ ˆy (8) implies

sij,1 = si0j0,1 , sij,y = si0j0,y

sij,x = −si0j0,x , sij,z = −si0j0,z

where R(sij ) = si0j0 . In particular, the polarisation satisﬁes P21,x = 4= s21,1s¯21,x + s21,z s¯21,y = −P31,x P21,y = 4= s21,1s¯21,y + s21,x s¯21,z = P31,y P21,z = 4= s21,1s¯21,z + s21,y s¯21,x = −P31,z .

For waves incident on wire 1, expectation of spin on z-axis on wires 2 and 3 opposite.

M. Harmer Spin Interference Introduction The wronskian Quantum Graph with Rashba Hamiltonian Spin ﬁlter A Simple Spin Filter Properties of ﬁlter Multi-particle Spin Devices Flux and polarisation

There are computational difﬁculties in treating a 6 × 6 matrix as a 3 × 3 matrix with Pauli spin matrix valued coefﬁcients. In this case, from (3, 7), we have  ¯   −1 b z¯1 z1 b z¯1 z1 ? ¯ USU =  z1 b z2   z1 b z2  ¯ z¯1 z¯2 b z¯1 z¯2 b Here b is a complex scalar and ! eiσz κ+(lξ−π) e−iσz κ−(lξ−π) zl = iσz κ − , cos(κ+π) cos(κ−π)

has a ‘nice form’ in the algebra generated by id, σz (not so with U).

M. Harmer Spin Interference Introduction The wronskian Quantum Graph with Rashba Hamiltonian Spin ﬁlter A Simple Spin Filter Properties of ﬁlter Multi-particle Spin Devices Flux and polarisation

This means that the matrices on the right hand side can be treated ‘almost as scalar’ to give expressions for the ﬂux

8 2 2 2 1 ¯ 2 2 T21 = |b| + |z2| |z1| − b + b z¯1 z2 + z1 z¯2 |D|2 2

and polarisation

8 cos(ϕ) 2 2 P21,z = j z¯1 z2 − z1 z¯2 . |D|2

Here the determinant D is a complex scalar and the contribution from conjugation by U just gives a multiplicative factor to the polarisation.

M. Harmer Spin Interference Introduction The wronskian Quantum Graph with Rashba Hamiltonian Spin ﬁlter A Simple Spin Filter Properties of ﬁlter Multi-particle Spin Devices Flux and polarisation

Using Maple these expressions give

8R 8 cos(ϕ) Q T (k, ξ, α) = , P (k, ξ, α) = . 21 X 2 + Y 2 21,z X 2 + Y 2 Here we need to manually cancel common factors −2 −2 cos (κ+π) cos (κ−π) which appear in the numerator and denominator to avoid numerical instability...

M. Harmer Spin Interference Introduction The wronskian Quantum Graph with Rashba Hamiltonian Spin ﬁlter A Simple Spin Filter Properties of ﬁlter Multi-particle Spin Devices Flux and polarisation

and

3 3 X = − 4κ + 3κ sin (κ+ + κ−) π + 4κ sin (κ+ + κ−)(2ξ − π) 3 − 8κ sin (κ+ + κ−)(ξ − π) 2 1 1 Y = − 6κ + 2 cos (κ+ + κ−) π − 2 cos (κ+ − κ−) π 2 2 + 2κ cos (κ+ + κ−)(2ξ − π) + 4κ cos (κ+ + κ−)(ξ − π) 3 Q = κ [cos (2κ−π) − cos (2κ+π)

+ cos (κ+2ξ + κ−2 (ξ − π)) − cos (κ+2 (ξ − π) + κ−2ξ)] 3 + 2κ [cos (κ+ (ξ − 2π) + κ−ξ) − cos (κ+ξ + κ− (ξ − 2π))] ,

M. Harmer Spin Interference Introduction The wronskian Quantum Graph with Rashba Hamiltonian Spin ﬁlter A Simple Spin Filter Properties of ﬁlter Multi-particle Spin Devices Flux and polarisation

with

2 4 1 2 R = κ + 4κ + 2 κ [cos (2κ+π) + cos (2κ−π) − cos (κ+ + κ−) ξ − cos (κ+ + κ−)(ξ − 2π) − cos (κ+ (ξ − 2π) + κ−ξ)

− cos (κ+ξ + κ− (ξ − 2π))] 4 + 2κ [cos (κ+ + κ−)(ξ − 2π) + cos (κ+ + κ−)(3ξ − 2π)] 4 + 4κ [cos (κ+ + κ−) ξ + cos (κ+ + κ−) 2 (ξ − π)] .

In these we have put β = 1.

M. Harmer Spin Interference Introduction The wronskian Quantum Graph with Rashba Hamiltonian Spin ﬁlter A Simple Spin Filter Properties of ﬁlter Multi-particle Spin Devices Vanishing of polarisation

We observe that for

κ+ = κ− + m + 1

the expression Q ≡ 0 and the polarisation is identically zero. This√ happens for values of α such that 1 + α2 − 1 = m ∈ {0, 1,...}, ie. when the eigenvalues on the ring become fourfold degenerate. Physical explanation in terms of ‘integral number of effective ﬂux quanta through the ring’.

Alternatively, we apply a gauge transformation on the ring ? VH0V ,

V (θ) = e−iσz (m+1)θ/2 eiσy ϕ/2 eiσz θ/2 .

2 − d − α 2 The hamiltonian on the ring is then dθ2 2 and all spin interaction is concentrated at the boundary conditions. √ However, for 1 + α2 − 1 = m ∈ {0, 1,...} the boundary conditions remain spin independent so that we have vanishing polarisation. A similar argument shows that the ﬂux is almost periodic in α.

M. Harmer Spin Interference Introduction The wronskian Quantum Graph with Rashba Hamiltonian Spin filter A Simple Spin Filter Properties of filter Multi-particle Spin Devices Plots of flux and polarisation

In ﬁgures 5–9 we plot the conductance T21 (upper curve) 2 and P21,z (lower curve) against the energy λ = k . We put α = 0.8—only signiﬁcant effect observed with respect to α is vanishing at points of degeneracy. Eigenvalues on the ring are indicated by

2 = λ+,n , 3 = λ−,n

Leftmost 2 and 3 are at λ+,1 and λ−,1 respectively with λ−,0 ≤ 0 off the left of the plot.

For ξ = π/2, figure 5, the polarisation is maximum at odd resonant eigenvalues and damped at even eigenvalues. A similar device is analysed, and an explanation in terms of interference effects is given, in [14]. For ξ = pπ/q flux and polarisation are periodic, repeating after 2q resonances λ±,n, with zeroes at λ±,lq. Perhaps this behaviour can also be explained by spin interference and topological phase effects [18]. ξ = 2π/3 best candidate for spin filter.

M. Harmer Spin Interference Introduction The wronskian Quantum Graph with Rashba Hamiltonian Spin ﬁlter A Simple Spin Filter Properties of ﬁlter Multi-particle Spin Devices Flux and polarisation for ξ = π/2

0.8 x u l f

d se i r 0.6 a l o p z

/ x u

l 0.4 F

0.2

0 10 20 30 40 50 lambda -0.2

-0.4

π Figure 5: T21 and P21,z for α = 0.8 and ξ = 2 .

x u l 0.8 f

d se i r a l o p z

x 0.4 u l F

0 10 20 30 40 50 lambda

-0.4

π Figure 6: T21 and P21,z for α = 0.8 and ξ = 3 .

x u

l 0.8 f

d se i r a l o p z

x 0.4 u l F

0 10 20 30 40 50 lambda

-0.4

-0.8

2π Figure 7: T21 and P21,z for α = 0.8 and ξ = 3 .

u 0.8 l f

d se i r a l o p z

x 0.4 u l F

0 10 20 30 40 50 lambda

-0.4

π Figure 8: T21 and P21,z for α = 0.8 and ξ = 4 .

x u l

f 0.8

d se i r a l o p z

x 0.4 u l F

0 10 20 30 40 50 lambda

-0.4

3π Figure 9: T21 and P21,z for α = 0.8 and ξ = 4 .

M. Harmer Spin Interference Introduction Quantum Graph with Rashba Hamiltonian Description of multi-particle states A Simple Spin Filter Calculation of truth tables Multi-particle Spin Devices Simple model of spin gate

We consider a (classical) n terminal device with p input terminals. Input is the spin state (on the z-axis) of incoming electrons on p terminals. Output is the spin expectation measured on the n − p output terminals.

In the spirit of quantum graphs we propose the simplest possible, ﬁrst order, model of such a device. The multi-particle interaction is the ‘exchange interaction’, ie. multi-particle wavefunctions on the gate given by alternating products of one particle wave functions. We need some multi-particle interaction; the sum of single particle scattering cannot produce an interesting gate. Input states modeled as scattered waves incident on the corresponding terminal/wire. All incoming waves assumed to have the same energy, temperature zero. Output calculated as sum of square norms of outward radiating coefﬁcients of multi-particle scattered wave.

For simplicity ignore spin. We consider the matrix of single particle wave functions

ikxj −ikxj ψji (x) = ψi |j = e + e Sji .

This can be thought of as a linear transformation acting on E = n C n n X X ψ(x) · ej = ψji ei = ψi (xj )ei . i=1 i=1

We ﬁx a multi-index ip = {i1 < i2 < ··· < ip}. This corresponds to a choice of the p input terminals/wires. The corresponding projection

p X P = eil iheil l=1 gives us a sum of incoming waves on the input terminals

p X P ψ(x) · ej = ψil (xj )eil , l=1 ie. those from which an input is constructed.

The multi-particle state corresponding to an input on the p input terminals is then

1 1 2 p P ψ(x ) · ej1 ∧ P ψ(x ) · ej2 ∧ · · · ∧ P ψ(x ) · ejp νp p 1 X = ψ (x1)e ∧ ψ (x2)e ∧ · · · ∧ ψ (xp)e il j1 il il j2 il il j il νp 1 1 2 2 p p p l1,l2,... = Ψ x1,..., xp e . j1 jp ip

This allows us to ‘easily’ express the desired coefﬁcients as determinants of minors of the scattering matrix.

The multi-particle spin states are indexed by the spin state of the incoming waves

σp = {σ1, . . . , σp ; σi ∈ {↑, ↓}}

Now E = C2n is an n-dimensional space of spinors and the projection p X = ih Pσp eσl ,il eσl ,il l=1 speciﬁes the choice of p input terminals/wires as well as the spin states on these terminals.

The multi-particle state corresponding to the input σp is Ψ τ , x1; ... ; τ , xp e σp 1 j1 p jp σp,ip 1 = ψ( 1) · ∧ · · · ∧ ψ( p) · . Pσp x eτ1,j1 Pσp x eτp,jp νp

Here the indices τ1, j1 denote the spinor component and the terminal/wire on which the i-th particle is being measured.

Expanding the multi-particle scattered waves (and summing over all but one particle) we get as coefﬁcients of outgoing terms

p p − 1 X · ∧ · · · Pσp S eτ1,j1 νp q − 1 τ2,j2... ∧ · ∧ · ∧ · · · ∧ · Pσp S eτq ,jq Pσp eτq+1,jq+1 Pσp eτp,jp at order Sq, q ≤ p.

Deﬁne simpliﬁed indices

αp = {(σ1, i1), (σp, ip) ...}

and denote subsets as αq ≤ αp. Then the polarisation q measured on terminal j1 is the sum over orders S where   p! X X 2 X 2  det Sβ ×α − det Sβ ×α  . ν2(q − 1)!  q q q q  p α ≤α β ,β =(↑,j ) β ,β =(↓,j ) q p q 1 1 q 1 1

M. Harmer Spin Interference Summary

Summary

1 Scattering on quantum graph with spin- 2 particles. Treating both spin states simultaneously. Demonstration of spin ﬁltering using Rashba hamiltonian on such graphs. 1 Future research: scattering on spin- 2 multi-particle graphs. Simpliﬁed derivation of truth tables.