Numerical methods for the solution of Schrödinger … Ravaioli/Ravaioli_files... · Numerical methods for the solution of Schrödinger equation for ballistic transport Umberto Ravaioli

1

Computational Electronics Group

2002 School on Computational Material Science May 21-31, 2002

Numerical methods for the solution ofSchrödinger equation for ballistic

transport

Umberto Ravaioli

Beckman Institute andDepartment of Electrical and Computer Engineering

University of Illinois at Urbana-ChampaignUrbana, IL 61801, USA

University of Illinois

2 2002 School on Computational Material Science May 21-31, 2002

Time-dependent Schrödinger Equation• Goal of quantum mechanics is to give a quantitative

description on a microscopic scale of individual elementaryobjects (electrons, photons, atoms,…) which behave both likeparticles and waves, exhibiting diffraction and interferencephenomena.

• Propagation cannot be described by a trajectory r ( t ) as for aclassical particle, but by a field-like quantity ψ (r,t ) calledwave function. The simplest wave function is a plane wave

with mechanical and wave parameters related as

( ) ( )0 ˆ, exp = ⋅ − r t j k r tψ Ψ ω

( ) ( ); ; ;= = = =dispersion relations

p k E k E E pω ω ω


• For an electron in vacuum, the energy dispersion relationassumes the parabolic form which gives the kinetic energy

A plane wave represents a mathematical limit, since it extendsinfinitely in space and time. A realistic particle could beconstructed as a superposition of plane waves

The wave function plays the role of a probability amplitude,from which we assume the normalization for the probabilitydensity

[ ]2 2 2

310

0 09.1 10

2 2−= = = ×

p kE m kgm m

( ) ( ) ( )( ) 30 ˆ, exp = ⋅ − ∫∫∫r t k j k r k t d kψ Ψ ω

( ) 2 3, 1=∫∫∫ r t d rψ


• The evolution of the wave function is described by theSchrödinger equation

which is actually a system of two equations for the real andimaginary part of the wave function

( ) ( ) ( )2

2

0

,ˆ , ( ) ,2

∂= − ∇ +

∂r t

j r t V r r tt m

ψψ ψ

( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

22

02

2

0

ˆ

2

2

= ℜ + ℑ

∂ ℜ= − ∇ ℑ + ℑ

∂

∂ ℑ= − ∇ ℜ + ℜ

∂

j

V rt m

V rt m

ψ ψ ψ

ψψ ψ

ψψ ψ


• In operator notation the solution to Schrödinger equation is

with the system Hamiltonian

This represents a formal solution of the Schrödinger equationas a first order differential equation in time. Except for trivialcases, we cannot find an analytical form for the exponentialterm containing the Hamiltonian.

All numerical solutions must find a suitable approximation ofthe exponential on a set of discrete points.

( ) ( )ˆ

, exp ,0

= −

jr t H t rψ ψ

( )2

2

02= − ∇ +H V r

m


• Consider a discretization of the domain involving N meshpoints

The discretized form of the solution is

. . . . . . . .

1 2 3 N-1 N

( ) ( )

( ) matrix representing the discretized Hamiltonian

solution array defined on the mesh points

êxp 0

∗ =

=

+ = −

jt t H t

Ht

ψ ∆ ∆

ψ

ψ


• We need to find a numerical approximation for theexponential of the discretized operator

satisfying:

a) stability of the iteration

b) conservation of the wave function probability

c) consistency ( we really solve the intended equation )

êxp

−

j H t∆


1-D time-dependent Schrödinger equation

Let’s illustrate the properties of numerical solutions byusing finite-differences on a uniform mesh.

A discretization may require

Explicit numerical methods – if it only requires a directsubstitution of values in the formulation

Implicit methods – if it involves solution of a linear systemof equations

( ) ( ) ( )22

20

, ,ˆ ( ) ,2

x t x tj V r x t

t m xψ ψ

ψ∂ ∂

= − +∂ ∂


A. Simple (naïve) explicit method

A simple direct finite difference discretization gives

In operator notation

UNSTABLE and NON-CONSERVING

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

2

20

20

; 1 ; 1; 2 ; 1;ˆ ( ) ;2

1; 2 ; 1; ( )ˆ; 1 ; ;2

i n i n i n i n i nj V i i n

t m x

i n i n i n V ii n i n j t i nm x

y y y y y yD D

y y yy y D yD

+ - - - + += - +

Ê ˆ- - + ++ = + +Á ˜

Ë ¯

( ) ( )ˆ

1

+ = −

jn I H t nψ ∆ ψ

10

B. Three-time-levels explicit method

A centered time difference is applied

To obtain the operator algorithm, we need to eliminate theintermediate solution at iteration n

STABLE and UNITARY (CONSERVING)


( ) ( ) ( ) ( ) ( ) ( )20

1; 2 ; 1; ( )ˆ; 1 ; 1 2 ;2

i n i n i n V ii n i n j t i nm x

y y yy y D yD

Ê ˆ- - + ++ = - + +Á ˜

Ë ¯

( ) ( )( ) ( )

ˆ1 1ˆ

I jH tn n

I jH t

∆ψ ψ

∆

−⇒ + ≈ −

+

( ) ( ) ( )1 12

+ − −=

n nn

ψ ψψ

11

C. Fully implicit method

Discretize space operator at timestep ( j+1) leading to a system thatrequires a matrix inversion


STABLE but NON-UNITARY (NON-CONSERVING)


( ) ( ) ( ) ( ) ( )

( )20

1; 1 2 ; 1 1; 1 ( )ˆ; 1 ; 12

; 1

i n i n i n V ii n j t i nm x

i n

y y yy D yD

y

Ê ˆ- + - + + + ++ + + +Á ˜

Ë ¯

= -

( ) ( )1ˆ

1−

+ = +

jn I H t nψ ∆ ψ

12

D. Semi-implicit method (Crank-Nicholson, also Cayley)

The right hand side is discretized as an average of operators at thetwo timesteps


STABLE and UNITARY (CONSERVING)


( ) ( ) [ ]

( ) ( ) ( )

( ) ( ) ( )

2 20 0

2

2 20 0

2

; 1 ; 1ˆ ( 1) ( )2

4 2ˆ1; 1 ( ) 2 ; 1 1; 1

4 2ˆ1; ( ) 2 ; 1;

i n i nj RHS n RHS n

t

m x m xi n j V i i n i nt

m x m xi n j V i i n i nt

y yD

D Dy y yD

D Dy y yD

+ -= + +

Ê ˆfi - + + - - + + + +Á ˜

Ë ¯

Ê ˆ= - + + + + +Á ˜

Ë ¯

( ) ( )ˆ /(2 )1 ˆ /(2 )

−+ =

+I jH tn nI jH t

∆ψ ψ∆

13

Of the schemes analyzed, only the three time steps explicit

and the Crank Nicholson semi-implicit method

are suitable.

One can see that the numerical operators are essentiallyequivalent.


( ) ( )ˆ /(2 )1 ˆ /(2 )

−+ =

+I jH tn nI jH t

∆ψ ψ∆

( ) ( )( ) ( )

ˆ /1 1ˆ /

I jH tn n

I jH t

∆ψ ψ

∆

−+ ≈ −

+

14

The unitary finite difference schemes can be readily extended to2D. Let’s assume for simplicity a uniform mesh

The three times level scheme becomes

The numerical solution is still obtained by a simple recursion,without having to solve a matrix problem.


= =x y∆ ∆ ∆

( ) ( ) ( )

( ) ( ) ( ) ( )

2 0ˆ, ; 1 , ; 1 ( 1, ; ( , 1; )

2ˆ4 , ; 1, ; , 1; ) ( ) , ;

ti j n i j n j i j n i j nm

ti j n i j n i j n j V i i j n

Dy y y yD

Dy y y y

+ = - + - + -

- + + + + -

15

The Crank-Nicholson algorithm is

The algorithm can be rewritten for matrix implementation, bymoving all terms at timestep (n+1) to the left and all terms attimestep n to the right. Numerical solution of a 5-diagonal matrixproblem is required at each iteration.


( ) ( ) ( ) ( )

( ) ( ) ( )(

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ))

2

20

ˆ, ; 1 , ; ( , ) , ; 1 ( , ) , ;2

1, ; 1 2 , ; 1 1, ; 12

1, ; 2 , ; 1, ;

, 1; 1 2 , ; 1 , 1; 1

, 1; 2 , ; , 1; 1

ti j n i j n j V i j i j n V i j i j n

i j n i j n i j nm

i j n i j n i j n

i j n i j n i j n

i j n i j n i j n

Dy y y y

y y yD

y y yy y y

y y y

+ - = - + +ÈÎ

- - + - + + + + +

- - + + +

- + - + + + +

˘- - + + + ˚

16

To alleviate the requirements of the matrix solution, it is possibleto use an Alternate Direction Implicit (ADI) scheme. The timestep is split into two sub-steps of half duration, and twoconsecutive sets of solutions are performed:

Horizontal sweep


( ) ( )( ( )

( )( ) ( )

( ) ( )( ( )

( )( ) ( )

2

20

2

20

ˆ, ; 1/ 2 1, ; 1/ 2 2 , ; 1/ 22 2

( , )1, ; 1/ 2 , ; 1/ 22

ˆ, ; , 1; 2 , ;2 2

( , ), 1; , ;2

ti j n j i j n i j nm

V i ji j n i j n


V i ji j n i j n

Dy y yD

y y

Dy y yD

y y

È+ + - - + - +Í

ÍÎ˘+ + + + + =˙̊

È- - - -Í

ÍÎ˘+ + + ˙̊

17

Vertical sweep


( ) ( )( ( )

( )( ) ( )

( ) ( )( ( )

( )( ) ( )

2

20

2

20

ˆ, ; 1 , 1; 1 2 , ; 12 2

( , ), 1; 1 , ; 12

ˆ, ; 1/ 2 1, ; 1/ 2 2 , ; 1/ 22 2

( , )1, ; 1/ 2 , ; 1/ 22


V i ji j n i j n


V i ji j n i j n

Dy y yD

y y

Dy y yD

y y

È+ + - - + - +Í

ÍÎ˘+ + + + + =˙̊

È+ - - - + - +Í

ÍÎ˘+ + + + + ˙̊

18

The ADI procedure decouples adjacent rows and columns ofmesh points. For every sweep, a set of tridiagonal systems issolved.

Because of the decoupling, the tridiagonal solutions in eachsweep can be carried out independently, therefore the approachis suitable for supercomputing applications (vectorization,parallelization).

In the numerical scheme shown here, the contribution by thepotential is equally divided between the implicit part (left handside) and the explicit part (right hand side) of the algorithm.

Note: the intermediate solution after a horizontal sweep is stillvery biased. Only solutions obtained after two consecutivesweeps should be considered.


19

Possible discussion items:

How can one implement a self-consistent solution (Poissonequation solved at each iteration to update the potential) in athree-time-step explicit method or in a semi-implicit method,where the potential should be known for the “next” timestepalready?

What trade-offs and approximations are possible/reasonable?


20

Systems with variable effective mass

In the effective mass approximation, the potential V in theSchrödinger equation is assumed to be only the electrostaticpotential, since the effect of the periodic crystal is accounted forby the effective mass itself.

However, some of the most interesting applications of quantumsystems involve spatially varying materials and heterojunctions.The effective mass approximation can still be used with somecaution.

When the effective mass is space-dependent, the major issue isto find the Hamiltonian operator, so that it remains Hermitian.


21

If we just insert a space dependent mass in the Schrödinger as iswe know that the Hamiltonian is not Hermitian. Also, this methodwould be mathematically applicable only to slowly varying mass.A widely used Hermitian form brings the effective inside theLaplacian differential operator

This operator can be used also for transport across abruptheterojunctions, as long as the materials on the two sides havesimilar properties and band structure (e.g. AlGaAs/GaAs).

But one has to keep in mind that very close to the heterojunctionthe physical properties, described by the effective massSchrödinger equation, are not necessarily well defined.


2

*1

2 − ∇ ⋅ ∇ m

ψ

22

Let’s consider a 1-D situation with uniform mesh. We candiscretize first the outer differential operator considering themid-points of the intervals around the mesh point i, using centerdifferences

Then the derivatives on the mid-points are also evaluated withcenter differences


2 2

* * *1/ 2 1/ 2

1 1 12 2 + −

∂ ∂ ∂ ∂ − ≈ − − ∂ ∂ ∂ ∂ i ix x x x xm m mψ ψ ψ

∆

( ) ( )( )

( ) ( )( )

2

2 * *1 1

2( ) 1/ 2 1/ 2

+ − − −− −

+ −

i i i ix m i m i

ψ ψ ψ ψ

∆

i-1 i+1i

i+1/2i-1/2

23

The effective mass only needs to be known at the mid-points. Ifan abrupt heterojunction is located at point I, the abrupt changeof effective mass is treated correctly in mathematical terms andwithout ambiguity. We avoid having to assume an effective massvalue exactly at the heterojunction.

There are other possible Hamiltonians for the space-dependentmass case, which are Hermitian. The following operator isHermitian, and is obtained as linear combination of two non-Hermitian operators


22 2

* *1 1

4 − ∇ + ∇ m m

ψ ψ

24

Let’s compare the two Hamiltonians, in a 1-D formulation


2 2 2

* * 2 *

2 2 2

* 2 2 *

2 2

* 2 *

extra term

2

2 *

1 1 12 2

1 1 4

1 1 2

1 00

12

∂ ∂ ∂ ∂ ∂ − = − + ∂ ∂ ∂ ∂ ∂ ∂ ∂ − + =

∂ ∂ ∂ ∂ ∂ − + ∂ ∂ ∂

∂ + ∂

x x x xm m x m

m x x m

x xm x m

x m

ψ ψ ψ

ψ ψ

ψ ψ

ψ

25

For smoothly varying mass, the two approaches areapproximately equivalent. One can see that it is awkward todirectly discretize with finite differences the numerical operatorson the right hand side. The proper procedure is to use boxintegration, over the interval [ i-1/2; i+1/2 ]. Integration by partscan be applied


1 1 2

1 1 2

1 1 2

1 2

1 1 2

1 2

2 2

* 2 *

2

* *

1 12

1 12

+

−

+

−

+

−

∂ ∂ ∂ − + = ∂ ∂ ∂

∂ ∂ ∂ − − ∂ ∂ ∂

∫

∫

i

i

d xx x xm x m

d xx x x xm m

ψ ψ∆

ψ ψ∆

1 1 2

1 2 *1+

−

∂ ∂ + ∂ ∂ ∫i

d xx x m

ψ

same as previous result

26

Absorbing boundary conditionsNumerical solution of the time-dependent Schrödinger equationis often accomplished by assuming a gaussian wave packetdistribution describing the particle. The boundaries of thesystems are taken to be far away from the simulated region, andthe wave function there is fixed to zero.

As the simulation progresses, the wave packet spreads andintercats with potential variations, originating reflectedwavepackets. Over a sufficient number of simulation steps, thewave may reach the boundaries, where the condition of zerowave function corresponds to perfect reflection!


ψ= 0ψ= 0

27

In order to let the wave function exist the domain, we need toimplement absorbing boundary conditions.


BC1 BC2

±qVSimulation Domain

( )1 ˆ êxp( ) exp( )x xinput

A j k x B x j k xy = + -

28

Assume B(x) approximately linear near the boundary BC1


( ) ( ) ( )22

20

0, 0,ˆ (0) 0,2

t tj V t

t m xψ ψ

ψ∂ ∂

= − +∂ ∂

( )

( ) ( )

2 2 ˆ ˆ20

2 2 2 ˆ

0 0

2

ˆ,2

x x

x

j k x j k xx

xj k xx x

A e B em x

Bk kx t j em m x

ψ

−=

−= +

∂− +∂

∂+∂

…

29

The formal solution of the differential equation in time gives

At BC2 similar treatment with


( ) ( ) ( )

( )

ˆ

ˆ

0

x

j E t

jk xx

t t t eB xk e t

tm

∆ψ ∆ ψ

∆

−

−∗

= ≈ =

∂+

∂

[ ]2

2BC

m E qVk

* -=

30

Create a new discretized equation near the boundaries

In a Crank-Nicholson scheme


( ) ( ) ( )

( ) ( )max max

01

2

B x B x BBC

t xB x B x x

BCx

∆∆

∆∆

∂ −=

∂

− −=

( ) ( ) ( )12

t t tB x B x B xt x x

∆+ ∂ ∂ ∂ = + ∂ ∂ ∂

31

In general

The numerical value of the reflected momentum may differslightly from the incident momentum, during a transient,particularly in 2D and 3D situations.


ˆ ˆ( )i Rj k x j k xBC A e B x eψ −= +

32

Issue: discretization changes the relationship between(simulated) energy and momentum! Discetization creates avirtual lattice where transport is described by the Tight-BindingHamiltonian


r r x+ Dr x- DV̂

( )ˆ ˆ| 1 | | 1 || |ˆ rr

l V l l VH l ll E > < + > < -+<= +>Â

coupled pendula

33

Potential for transferring an electron from one site to a neighbor

Energy of an electron located at a lattice site

Atomic-like orbital centers at the sites (forming a regular lattice)


2

21ˆ hopping potential

2 ( )V

m x*= - =D

ˆ( ) 2rpotentialenergy

E V r V= -

| l >

r

34

In the continuum limit

For a periodic lattice


( )

( ) ( )0 0

( ) 01 ˆ| exp |

êxpl

ll

V r

k j k l x lN

E k E V j k l x

=

>= D >

= + D

Â

Â

2 2

2

finite difference approximation

ˆ ( )2

ˆ

dH H V rm d r

H

*Æ = - +

=

neighbors

35

Consider a periodic chain with constant energy parameters and mass,and uniform potential energy distribution. The discretized Hamiltonianis described by a tridiagonal matrix

Eigenvalues of the tridiagonal matrix


0

0

0

0

0

ˆ 0 0 0ˆ ˆ 0 0

ˆ ˆ0 0ˆ ˆ0 0

ˆ0 0 0

E VV E V

V E VV E V

V E

0 ˆ2 cos1n

nE E VNpÊ ˆ= + Ë ¯+

36

Dispersion relationship for the discretized Hamiltonian is not parabolic!Absorbing boundary conditions for a discretized Hamiltonian should bederived in the tight-binding formalism.


( )0

0

0

ˆˆ( ) exp( )

ˆ ˆˆ exp( ) exp( )ˆ2 cos( )

lE k E V j k l

E V j k x j k x

E V k x

= +

= + D + - D

= + D

Â

xp- D xp D0

E

k


[ ]

( )

( )

0ˆ ˆ ˆ ˆ

0

ˆ0

0

ˆ ˆ

0ˆ ˆ0 0 1 1

ˆ0 (1) ( 1)

1 0

(1)

cons

(0)( )

ˆ ˆ0 0 (0

linear at boundary( 1) (0)

tant

)

ˆ(0)

jka jka jka jka

jka

jka j

H E V

E V Ae B e Ae B e

H E V e A B

e A

BC l x a

B B BB x

B B B

B

E

V B e

ψ

ψ ψ ψ ψ

∆

∆

ψ

ψ ψ

∆

∆

− −

− −

→ = =

= + −

= + + − =

= + + + + −

= +

=

+ +

+ + +

= −

( )ˆka jkae −


[ ]

( ) ( )

( ) ( )

0( ) for TBH

ˆ

ˆ

ˆ ˆ2

ˆ ˆ

ˆˆ ˆ0 2 cos( ) 0 2 sin( )

(1) (0) (0) ( 1)

1 0

1 1 0 0

1 0 0 1

ˆ1 0 2 sin( ) 0

E k

j kaR R

j kain in

reflected

j ka j kain

j ka j kain

H E V ka j V B ka

B B B B B

B e

e

e e

e j e ka

ψ ψ ∆

∆

∆ ψ ψ

ψ ψ ψ ψ

ψ ψ ψ

ψ ψ ψ

= + −

= − = − −

= − =

= − − −

= − − −

= − −

39

Final results for absorbing boundary conditions at input

has been eliminated

An even more refined methodology for absorbing boundary conditions can be found at: L.F. Register, U. Ravaioli and K. Hess, J. Applied Physics, vol. 69, pp. 7153-7158, 1991 (+ small corrections at the Erratum section of: J. Applied Physics, vol 71, p. 1555, 1992),


[ ]0ˆ

ˆ 2

ˆ0 2 cos( ) 0

ˆ ˆˆ ˆ2 sin( ) 0 2 sin( ) 1

ˆ4 sin ( ) 0

j ka

j kain

H E V ka

j V ka j V ka e

Ve ka

y y

y y

y

= +

+ -

-

Ry

40

Time-independent transport with Schrödinger equationWe consider her an open system in steady-state conditions. Theopen system consists of a finite region connected to reservoirsof particles (contacts).Waves (particles) flow between the system and the reservoirs,with reflection and transmission properties depending on thepotential profile. Waves entering from any separate reservoircontact are studied with an independent solution.Assuming stationary states

the steady-state Schrödinger equation has the form


ˆ ˆ( , ) ( )exp( ) ( )exp( )Er t r j t r j tΨ ψ ω ψ= − = −

22

0( ) ( ) ( ) ( )

2r V r r E r

mψ ψ ψ− ∇ + =

41

Let’s consider a 1-D domain with uniform grid and mesh size ∆ x.The time independent Schrödinger equation is discretized withfinite differences

E represents the total energy, so that V(i) - E is the kineticenergy. It is often convenient to set the reference zero of thepotential V(x) at the inflow boundary, so that E corresponds tothe incident kinetic energy.

The kinetic energy varies spatially according to the potentialchanges, and the wavenumber kx(x) varies accordingly. Theknowledge of kx at the reservoir ends, allows the specification ofboundary conditions.


[ ]2

20

( 1) 2 ( ) ( 1) ( ) ( ) 02 ( )

i i i V i E im x

ψ ψ ψ ψ∆

− − + +− + − =

42

The travelling wave at the specified energy can be assumed to bea plane wave of the form

where the sign indicates the direction of propagation. Boundaryconditions are derived from this expression for both forward andbackward wave.

The exponentials are completely specified in the boundaryconditions by the knowledge of E and V(x). The prefactor A is setto a conventional reference (e.g., A=1) at the one boundary, andthe value at the other end is the outcome of the solution. Thesolution for the pre-factor A may be complex, and it contains theinformation on reflection and transmission coefficients.


( ) ( ) ( )êxp xx A x j k x xψ = ±

V = 0

E(x)

V (x)

43

In 1-D, all is needed for the solution is a recursion algorithm. Fora forward travelling wave which enters the open system at x = 0,we can set A( L ) = 1 ( L is the length of the domain) and obtainthe boundary condition

N is the index corresponding to the output boundary. Assumingthat the domain is subdivided into N mesh intervals, the index forthe left boundary is i = 0. To solve the open system problem, weneed to assume a perfect reservoir with a constant potential(ohmic contact) and no change in the wavevector kx

The knowledge of the wave function N and N+1 is sufficient tocalculate the value of ψ (N-1) using the discretized Schrödingerequation.


( ) êxp ( )xN j k N Lψ =

( ) ( )[ ]ˆ( ) ( 1) 1 exp ( )x x xk N k N N j k N L xψ ∆= + ⇒ + = +

44

The general recursion algorithm can be easily obtained byrewriting the discretized Schrödinger equation

The potential reference is V(0) = 0, so that E represents thekinetic energy

as well as the total energy at the inflow point i = 0.


( )2

01 12

2 ( )2 ( )i i im x V i E∆ψ ψ ψ− +

= + − −

2 2

0

(0)2

xkEm

=

45

Because of reflection due to the potential variations, throughoutthe device one can express the wave function as thesuperposition of an incident and a reflected wave

At the output, because of the constant potential in the reservoir,we assume that there is no reflection

At the input we can express the total wave function as


( ) ( ) ( )ˆ êxp ( ) exp ( )x xi I i j k i i x R i j k i i xψ ∆ ∆ = + −

( ) ( ) ( )0 and R N I N A N= =

( ) ( ) ( ) ( )0 0 0 0A I Rψ = = +

46

To find the amplitude of incident and reflected waves, we needan additional solution at a mesh point inside the input reservoir.With the same assumptions as for the output reservoir

By combining the above with

we obtain

Once I(0) is determined, the wave function throughout the devicecan be re-normalized to make the total probability unitary.


( ) ( ) ( )( ) ( ) ( )

; 1 1 1 1ˆ ˆ1 0 exp (0) 0 exp ( )x x

x x i I R

I j k x R j k i x

∆ ψ

ψ ∆ ∆

= − = − ⇒ − = − + −

− = − +

( ) ( ) ( ) ( )0 0 0 0A I Rψ = = +

( )( ) ( ) ( )

( )( )ˆ0 exp 0 1

0 ˆ2 sin 0x

x

j k xI

j k xψ ∆ ψ

∆

− − =

47

The procedure for inflow from the right boundary is essentiallythe same, with appropriate sign changes for the waves. Therecursion relation becomes

with the starting point

The recursions are solved for a set of energies and for inflowfrom both boundaries. In order to find the total result, one stillneeds to specify injection condition, that provide the weights foradding all the separate recursion results.


( )2

01 12

2 ( )2 ( )i i im x V i E∆ψ ψ ψ+ −

= + − −

( ) ( ) ˆ0 1 and 1 exp (0)xj k xψ ψ ∆ = − = −

48

We consider now a simple case for an n-type semiconductor,where the two reservoirs are assumed in quasi-equilibrium, witha specific Fermi level EF.

The equilibrium particle density, including the effect of spin, is

Assuming for simplicity an isotropic mass, the particle energy is

To determine the density for the injection direction, we need tointegrate over the transverse direction


( )[ ]31 1

1 exp /4x y z

F Bn dk dk dk

E E k Tπ∞−∞

=+ −∫∫ ∫

( )2 2 22 2

2 2

y zxx

k kkE E Em m

⊥ ∗ ∗

+= + = +

49

The integration is performed over the transverse momentumcomponents, in polar coordinates

The integration over the angle provides simply a factor 2π and achange of variable is performed


( )[ ]20 03

1 kk

1 exp /4x

x F Bn dk d d

E E E k Tπ

θπ

∞ ∞ ⊥⊥−∞

⊥=

+ + −∫ ∫ ∫

2k k

2k2

m EE

md dEE

∗⊥

⊥ ⊥ ⊥

∗

⊥ ⊥⊥

→ =

=

50

We obtain

The integration can be carried out exactly according to

from which


( )[ ]2 2 0 1 exp /2x

x F B

m dEn dkE E E k Tπ

∗ ∞ ∞ ⊥−∞ ⊥

=+ + −∫ ∫

[ ]log 1 exp( )1 exp( )

dx xx

= − ++∫

2 2 log 1 exp2

B F xx

B

m k T E En dkk Tπ

∗ ∞−∞

−= +

∫

51

From this result, deduce that the electron density correspondingto an incident momentum kx is

The expression above can be used as a weight to combine theincident waves which are injected from the reservoir momentumdistribution.

The Fermi-Dirac distribution in the contacts is valid close toequilibrium. Far from equilibrium, one can still use the approachto get a qualitative solution. However, in a general situation, theboundary conditions for the injection distribution should beadapted self-consistently to the bias.


2 2

2 22log 1 exp

2B F x

B

m k T E k mnk Tπ

∗ ∗ −= +

52

Let’s trace the same steps for a classical distribution of particlesfrom a Maxwellian distribution


22

22

3 22 2

3 2 22 2

0 0

2

2

2 2 12

xB B

xB B

mvmvk T k T

x y zB

mvmvk T k T

xB

mn dv dv dv e ek T

mn dv d d v v e ek T

Ev d v dEm m E

π

π

θπ

⊥

⊥

∞ − −

−∞

∞ ∞ − −

⊥ ⊥−∞

⊥⊥ ⊥ ⊥

⊥

=

=

= =

∫∫ ∫

∫ ∫ ∫

53

After changing variables


( )( )

3 2

0

3 2

0

3 2

00 exp 0

2 2 122 2

22

22

xB

xB

xB

B B

E Ek T

xB

E Ek T

xB

E Ek T

x BB

k T E k T

m En dv dE ek T m m E

mn dv dE ek T m

mn dv k T ek T m

ππ

ππ

ππ

⊥

⊥

⊥

−∞ ∞ −⊥

⊥⊥−∞

−∞ ∞ −

⊥−∞

∞−∞ −

−∞

− − +

=

=

= −

∫ ∫

∫ ∫

∫

54

Finally


2

2

2 2

3 22

1 22

1 2 1 202 2

0negative positive

22

2

2 2

x

B

x

B

x x

B B

x x

mv

k TBx

B

mv

k Tx

B

mv mv

k T k Tx x

B B

v v

m k Tn dv e

k T m

mn dv e

k T

m mn dv e dv e

k T k T

π

π

π

π π

−∞

−∞

−∞

−∞

− −∞

−∞

=

=

= +

∫

∫

∫ ∫

55

Here are the injecting distributions for the two cases


0

21 2

22 x

xB

mvk Tn dv

Bm ek Tπ

∞ − =

∫

2 2 0 log 1 exp2

B F xx

B

m k T E En dkk Tπ

∗ ∞ −= +

∫

56

Once the wave function is determined, it is possible to evaluatethe current flowing in the structure. Again using a parabolicdispersion relation, the electron current associated with aspecific momentum component is defined as

For a plane wave

from which we obtain the alternative expression


( ) kkJ v qm

ρ ψ ψ ∗∗= = −

ˆ kjψ ψ∇ =

( )J qm

ψ ψ∗∗= − ℑ ∇

57

The continuity equation for electrons has the form

where

From Schrödinger equation


2q J

t tψρ ∂∂

= − = −∇ ⋅∂ ∂

2

t t tψ ψ ψψ ψ

∗∗∂ ∂ ∂

= +∂ ∂ ∂

( )

22

2 2 2

ˆ 1 ( )ˆ2ˆ ˆ1 ( )ˆ2 2

j V rt jm

j jV rjm m

ψψ ψ ψ

ψ ψ ψ ψ ψ ψ ψ

∗∗

∗ ∗ ∗ ∗∗ ∗

∂ = ∇ + ∂

− ∇ + = ∇ − ∇

58

Then, since

we have

and from the continuity equation itself

we get yet another expression for the current density


( ) 2ψ ψ ψ ψ ψ ψ∗ ∗ ∗∇ ⋅ ∇ = ∇ + ∇ ⋅∇

( )2 ˆ

2j

t mψ

ψ ψ ψ ψ∗ ∗∗

∂= ∇ ⋅ ∇ − ∇

∂

2q J

t tψρ ∂∂

= − = −∇ ⋅∂ ∂

( )ˆ

2j qJm

ψ ψ ψ ψ∗ ∗∗= ∇ − ∇

59

Here are together all the expressions for current that we found


( )ˆ

2j qJm

ψ ψ ψ ψ∗ ∗∗= ∇ − ∇

( ) kkJ v qm

ρ ψ ψ ∗∗= = −

( )J qm

ψ ψ∗∗= − ℑ ∇

60

In practice, one cannot select all possible values of the incidentmomentum and the momentum axis is discretized. Assuming auniform discretization step ∆kx we can take the mid-value of theinterval as the average momentum for that mesh. We canexpressed the following quantities for a 1-D system

Note that these formulas are general and can be applied to bothtime-independent and time-dependent formulations


( ) ( ) ( )( ) i i ii

n x n k k x xψ ψ∗= ∆∑

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )ˆ2

ii i

i

i ii i i

i

xqJ x n k k xxm

x xqj n k k x xx xm

ψψ

ψ ψψ ψ

∗∗

∗∗

∗

∂ = − ∆ ℑ ∂

∂ ∂= ∆ − ∂ ∂

∑

∑

61


What can we do to formulate boundary conditions when quasi-equilibrium is not a good assumption?

How can we formulate a self-consistent simulation?

Video animation: Switching in a quantum nanostructure


62

1D quantum simulation using transmission line theory

Consider a single barrier system

The wave function can be written as


( ) ( ) ( )[ ]

( )( )2ˆ ˆ 2

amplitude reflecti

exp ex

on coeff

p

icient

j j m E V

x A x x

γ

ψ γ Γ γ

α β

Γ

∗= +

− −

= −

=

=

x

V2

V1

63

The system behaves like a standard transmission line


( ) ( ) ( )[ ]

( )( )

( ) ( )

( )( )

1 1 1 1

2

2 2 2

22

1 1 1

2 2

exp

ˆ 2

exp exp

ˆ 2

x A x x

j m E V

x A x

j m E V

ψ γ Γ γ

ψ γ

γ

γ ∗

∗

= − −

=

=

=

−

−

x

V2

V1

region 1 region 2

0

64

By applying continuity conditions, we obtain Γ

Then, we define the following auxiliary function


( ) ( ) ( ) ( )[ ]11 1 1exp exp1

xx x

dx Ad xj m j m

ψγ γ Γ γφ −∗ ∗= = +

( ) ( )

( ) ( )

( ) ( )( ) ( )

1 2

1 21 2

2 2 1 1

2 2 1 1

0 01 10 0d d

d x d xm m

m m

m m

ψ ψ

ψ ψ

γ γΓ

γ γ

∗ ∗

∗ ∗

∗ ∗

=

=

−⇒ =

+

65

This function is anologous to electric voltage in a transmissionline

The wave function is analogous to electric current in atransmission line

and the scaling factor behaves like characteristic impedance


( ) ( ) ( )[ ]1 1 1exp exp1 x xx Aj m

γ γ Γ γφ −∗= +

( ) ( ) ( )[ ]1 1 1 1exp expx A x xψ γ Γ γ= − −

10Zj m

γ∗=

66

The probability current flow density is analogous to time averagepower in a transmission line

At any location one can define the line impedance as

Indicate the load impedance as ZL = Z2 the input impedance of thesystem is


( ) { }1ˆ 22

Jq j m

ψ ψ ψ ψ φ ψ∗ ∗ ∗∗= ∇ − ∇ = ℜ

( ) ( )( )x

Z xx

φψ

=

2 00

0 2

cosh( ) sinh( )cosh( ) sinh( )

where

in

in

Z L Z LZ ZZ L Z L

x L

γ γγ γ

−=

−

= −

67

If a system consists of a sequence of different material layers,each layer can be treated as a transmission line with differentcharacteristic impedance, which depends on the effective mass.The reflection coefficient of the structure can be obtained withthe same algebra of transmission lines.


What are the limitations of this approach?How can we include bias and self-consistency?


Z01 Z01 Z01

Z02 Z02

68

2D Solution with iterative tight-binding Green’s functions.Let’s assume a 2-D electron gas system define by a wave guidelike pattern, and perform a uniform discretization (tight-bindinglattice). Assume a flat potential floor for simplicity. We arelooking for the structure’s reflection and transmissioncoefficients.


+ ++ +

Gji

Gii

i j

69

Scattering eigenstate in the input lead

Scattering eigenstate in the output lead


incid

reflected

ent wave

w

,

av

ˆ

s

ˆ

e

1( , ) ( )

1 ( )

m

n

j k xE m m

m

j k xnm n

nn

x y e yk

e y xk

Ψ ψ

Γ ψ

=

+ → −∞∑

tra

ˆ,

nsmitted waves

1( , ) ( )nj k xE m nm n

nnx y e y x

kΨ τ ψ= → ∞∑

70

Start with output lead as an isolated section


jq+1

hard walls

semi-infinite chain

jq+1

Gj,q+1B

Gq+1,q+1B

B

71

The Green’s function is a matrix [ n × n ] where “n” is the numberof modes in the trasverse chain (same as the number ofdiscretization nodes, 3 in the example)


1 Energy above cut-off

1 Energy below cut-off

cos ˆ2

cosh ˆ2

EV

EV

θ

α

−

−

=

=

E1

E2

E3 1

2q+1, q+13

j, q+1

0 00 00 0

GG

G

E

72

Green’s functions for the semi-infinite chain


( )

( )

( )

( )

ˆB above cut-offq+1,q+1

B below cut-offq+1,q+1

ˆ j-qB above cut-offj,q+1

j-qB below cut-offj,q+1

= ˆ

= ˆ

= ˆ

= ˆ

ji

i

j

i

i

eGV

eGV

eGV

eGV

θ

α

θ

α

−

−

73

Then, treat the next section as an isolated finite chain


qp

hard walls

finite chain

qp

Gq,pA

Gp,pA

A


Green’s function for the finite chain

( ) ( )( )[ ]

( ) ( )( )[ ]

( ) ( )( )[ ]

( ) ( )( )[ ]

( ) ( )

A above cut-offpp

A below cut-offpp

A above cut-offqp

A below cut-offqp

A A (symmetry) ; nodes along the chainpp qq

sin= ˆ sin 1

sinh= ˆ sinh 1

sin= ˆ sin 1

sinh= ˆ sinh 1

=

i

i

i

i

i i

NG

V N

NG

V N

GV N

GV N

G G N

θθ

αα

θθ

αα

+

+

+

+

=


GA is a matrix [ m × m ] where m is the number of transversemodes (number of transverse discretization nodes)

1

2

1

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

A

A

Am

Am

G

G

G

G

−

i

i

E

In wider section, moremodes are above cut-off

A B


Now, add sections A and B - Application of Dyson’s equation

jp

Gj,pA+B

GppA+B

A+B

qp

Gq,pA

Gp,pA

A jq+1

Gj,q+1B

Gq+1,q+1B

B+V̂

totalHamiltonian

Dyson's equation f

connectionsisolated regio

or the Green's functio

ns

n

0

0 0

ˆ

ˆ

H H V

G G G V G

= +

= +


jp j,q+ q1 q p+1,qÂ B B A BG G GV+ +=

qp

Gq,pA+B

A jq+1

Gj,q+1B

B

q+1,qV

pp pp pq q,q+1 q+1,pÂ B A A BAG G GG V ++ = +

qp

Gq+1,pA+B

A jq+1 B

q,q+1V

qp A

Gp,pA

+ Gp,qA


q+1,q+q+1, 1 q+1,q qpp Â B B A BG V GG + +=

qp

Gq,pA+B

A jq+1

Gq+1,q+1B

B

q+1,qV

qp qq q,q+1 1,q q pp +ˆB A A BA AG VG G G ++ = +

qp

Gq+1,pA+B

A jq+1 B

q,q+1V

qp A

Gq,pA

+Gq,q

A


Add sections A and B

( )

jp j,q+1 q+1,q qp

pp pp pq q,q+1 q+1,p

qp qp qq q,q+1 q+1,p

q+1,p q+1,q+1 q+1,q qp

qp qp qq q,q+1 q+1,q+1 q+1,q qp

qq q,q+1 q+1,q+1 q+1,q

ˆ

ˆ

ˆ

ˆ

ˆ ˆ

ˆ ˆ1

A B B A B

A B A A A B

A B A A A B

A B B A B

A B A A B A B

A B

G G V G

G G G V G

G G G V G

G G V G

G G G V G V G

G V G V G

+ +

+ +

+ +

+ +

+ +

=

= +

⇒ = +

⇒ =

= +

−

( ) 1qp qq q,q+1 q+1,q+1 q+1

qp qp

,q qpˆ ˆ1

A B A

A B A B AG G V G G

G

V+

+

−⇒ =

=

−


Get the first Green’s function for the coupled system

( )

Self-energy for the transverse chain at locat

1

jp j,q+1 q+1,q qq q,q+1 q+1,q+1 q+1,q qp

1q

jp j,q+1 q+1

p qq q,q+1 q+1,q+1 q+1,q p

q

q

,q p

ˆ ˆ ˆ1

ˆ

ˆ ˆ

1

Bq

A B B A B A

S

A B B A B

A

B

B

q

A B A

G G V G V G V G

G G V G

S

G G V G V G

−

+

−+

+ +

= −

⇓

= −

=

= ion q, due to presence of B


Get the second Green’s function for the coupled system

( )

( )

( )

1q+1,p q+1,q+1 q+1,q

pp

1pp pq q

qq qp

pp pq q,q+1 q+1,p

q qpq,q+1 q+1,q

q+1,p q+1,q+1 q+1,q qp

+1

1qp qq

q q

p

,

q

+1

ˆ 1

1

ˆ

ˆ

ˆ

1

ˆ

A B B A B A

A B

A A A B Aq

B

q

A

q

A A B

A B A B Aq

A B B A

S

B

B

G

G G G S G

G G V G S G

G G V G

V

G G V G

G G S

G V

G

−

+

−

+

+

+ +

−+

+

= −

= +

−

=

=

⇓

=

−


Here are the Green’s function results we were looking for

( )( ) 1

pp pp pq qq qp

1jp j,q+1 q+1,

.. ..

pp q

q qq qp

+1,p

1

and are diagonal matrices

and are non-diagonal matr

ˆ 1

ices

A B A A B A B Aq q

A

A B B A B

B

A A

q

B

A

B

G G G

G

S

G

G S G

NOTE

G G V G S G

G G

−+

+ +

−+

+ −

= −

=


Example of coupling matrices

1, 3 3 modes on section q+1

1, 7 7 modes on section q

n

m

= Æ

= Æ

q+1q

Here is [ 3 3 ]

BG¥

Here is [ 7 7 ]

AG¥

q,q+1

q+1,q

q,q+1 q+1,q

is [ 7 3 ]

is [ 3 7 ]

T

V

V

V V

¥

¥

= È ˘Î ˚

( ) ( ) ( )q,q+1

Overlap integral

ˆ n mnml

V V l ly y*= Â Â


Transverse Eigenvalues and Eigenfunctions

The simplest approach is to use an infinite square well model for thetransverse direction. But one could also solve a 2-D self-consistentSchrödinger/Poisson solution on the cross section and feed theeigenenergies and eigenfunctions (wave functions) to the Green’sfunction code. The tight-binding model gives as many eigevalues asnodes. Typically, 20 nodes are sufficient to resolve realistic structures.

E2

2

2 1 0 0 0 0

1 2 1 0 0 0

0 1 0 012 0 0 1 0( )

0 0 0 1 2 1

0 0 0 0 1 2

m x∆

−

− −

−

−

− −

−

i i

i i

Discretized Hamiltonian with V(x) = 0(flat potential)

L


Transverse Eigenvalues

[ ]2

2

2

2

2 2 2

2 2

1

2 ( )

1

2 ( )

2 2 2 2 21 12 2 2 2 22 2 2( ) ( )

0 0 2 cos1

01,

0

0 0 2; 1;

2 2cos1

Continuum model

( )

( 1)

n

m x

nm x

nm m mx x

na b E a bcN

c a bn N

c a b

c a a b c

nE

N

n xE

L L n N n

∆

∆

∆ ∆

πγ

γ

γ

π

π π ∆ π

= ++

⇒ ∈

= = = − =

= −+

= = =+


Transverse Eigenfunctions

Hard walls, zero wave function at the wall.

( )( )( )( )

( ) sin1

1ˆ1 exp 2 11ˆ2 2 1 exp 2 1

n n

n

n m nm

n ll AN

Aj n N NNj n N

πψ

π

π

ψ ψ δ

= +

= − +

+ ℜ − +

=


Last iteration - Add input lead (semi-infinite)

Gj,iA+B

ji

GiiA+B

A+B

ki

Gk,iA

Gi,iA

A jk+1

Gj,k+1B

Gk+1,k+1B

B+V̂


Last iteration - Add input lead (semi-infinite)

( )( ) ( )[ ]

( )( ) ( )[ ]

( )( )

( )( )

ˆ i-k-1A above cut-offi,i

i-k-1A below cut-offi,i

ˆ i-k-1A above cut-offk,i

i-k-1B below cut-offk,i

sin i-k-1= ˆ sin

sinh i-k-1= ˆ sinh

= ˆ

= ˆ

j

i

i

j

i

i

eGV

eGV

eGV

eGV

θ

α

θ

α

θθ

αα

−

−


With the application of the same algorithm obtain

Gj,iA+B

ji

GiiA+B

A+B

ABi,i

ABj,i

= [ m m ] matrix (m = nodes in input lead)

gives the reflection coefficients

= [ n n ] matrix (n = nodes in output lead)

gives the transmission coefficients

G

G

×

×


Reflection and transmission coefficients are evaluated for allmodes in the input and output leads

( )

( )

( )

ji

ii

ˆ i j, ji

elementof

ˆ 2 i,

iielementof

refle

transmission coeffic

ˆ ˆ2 sin sin

sin si

ction coeffic

n

ˆ ˆ2 sin

ie

ie ts

n s

n

t

m n

m n

jm n n m

G

jm n n m

m nm

G

j V e n G m

e

j V n G m

θ θ

θ θ

τ θ θ

Γ θ θ

θ δ

−

+

= −

= −

× +

1 for

0 for nmm n

m nδ

==

≠



How to include realistic structures?

Quasi-3D simulation

Full self-consistent 3-D simulation

Available Software: TBGreen code


Mode-matching method

The structure is partitioned into slices in which the potentialprofile is constant along the longitudinal direction.

A 1-D Schrödinger equation is solved in each section toobtain transverse eigenmodes and eigenvalues.

The continuity of the wavefunction and of its normalderivative is enforced at each interface between differentsections.

The equations are then projected onto appropriate sets oftransverse modes and solved using numerical procedures.


Mode-matching method (2-D electron gas)

xy

( ) ( ) ( )

( )

( )

( )

ˆ

2 22 2

,

1

2 sin

2 2

x

n nn

j k xn

n

y

nx y n

x y a x n

x ek

nk yn

L L

kk k Em m

ψ ϕ χ

ϕ

χ

±

∗ ∗

=

=

=

+ = =

∑



Current normalization

( )

( )( )

ˆ ˆ ˆ ˆ

2 2

ˆ2

ˆ ˆˆ2

ˆ2ˆ2

1

x x x xj k x j k x j k x j k xx x

x x

x

Jj m

C e j k C e C e j k C ej m

C j k C kj m m

m

Ck

ψ ψ ψ ψ∗ ∗∗

− −∗ ∗∗

∗ ∗

∗

= ∇ − ∇

= +

=

= to have unitary flux for each impinging mode

In units of



NM

N Reflection coefficients

M Transmission coefficients

N+M unknowns

M equations from the continuity of wave functions

N equations from the continuity of the normal derivative

M+N equations



Leftmost interface ( x = xA )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

ˆ ˆ

ˆ ˆ

ˆ ˆ

ˆ ˆ

1 1

1 1

ˆ

ˆ ˆ

i A n A

m A m A

i A n A

m A m A

j k x j k xi n n

i nn

j x j xm m m m

m mm m

j k x j k xi i n n n

nj x j x

m m m m m mm m

e y R e yk k

c e y d e y

k e y j R k e y

j c e y j d e y

ν ν

ν ν

ψ ψ

χ χν ν

ψ ψ

ν χ ν χ

−

−

= −

+ +

=

+ −

∑

∑ ∑

∑

∑ ∑



Leftmost interface (assume x = xA = 0 )wave function continuity condition

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

0 0

0

,0

1 1

1

1 1 1

L L

p i n p ni nn

L

m p mmm

L

p i p m p mi p mm

y y dy R y y dyk k

T y y dy

R T y y dyk k

ψ ψ ψ ψ

ψ χν

δ ψ χν

= −

+

⇒ = − +

∑∫ ∫

∑ ∫

∑ ∫



Leftmost interface (assume x = xA = 0 )derivative continuity condition

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( )

' '

0 0'

0'

0'

0

ˆ ˆ

ˆ

ˆ

ˆ ˆ

L L

i p n n n p nn

L

m m p mm

L

i p n

L

n n p n p pn

j k y y dy j R k y y dy

j T y y dy

j k y y dy

j R k y y dy j T

χ ψ χ ψ

ν χ χ

χ ψ

χ ψ ν

=

+

⇒ =

+

∑∫ ∫

∑ ∫

∫

∑ ∫

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Numerical methods for the solution of Schrödinger … Ravaioli/Ravaioli_files... · Numerical methods for the solution of Schrödinger equation for ballistic transport Umberto Ravaioli