Schroedinger-Poisson Solvers

1

Computational Electronics

Schrödinger-Poison Solvers Description andQuantum Corrections in classical simulators:

1. Schrödinger-Poisson solvers- major concerns for future integrated circuits- time-dependent Schrödinger wave equation (SWE)- discretization of the SWE for variable effective mass- time-independent Schrödinger equation

(A) Airy functions method(B) Variational approach(C) Shooting method for 1D problems – SCHRED description(D) Finding the roots of a characteristic polynomial(E) The Lanczos algorithm(F) Numerov algorithm – note on the integration of the SWE(G) 2D and 3D eigenvalue solvers

- open systems

2. The effective potential approach- Madelung and Bohm’s reformulation of quantum mechanics- other quantum potential formulations- the effective potential approach due to Ferry- simulation examples that utilize the effective potential approach

(A) Simulation of a 50 nm MOSFET device(B) Simulation of a 250 nm FIBMOS device(C) Simulation of a SOI device structure

- conclusions regarding the use of the effective potential approach


1. Schrodinger-Poisson Solvers

While current production devices are only at 0.1 µm, predictions are that they will be at 50 nm by year 2007. Issues that are currently investigated in these device structures include:

Tunnel currents in gate oxide Source to drain tunneling The role of the electron-electron interactions on carrier thermalization Statistical fluctuations: gate-length, gate-width, oxide thickness and

doping density variation:

Must go 3D instead of 2D Must model ensemble of devices instead of a single device Dopant solubility, activation and segregation at the surface

must be addressed At these sizes, we should begin to see quantum effects, as λD~3-5

nm at 300 K. The questions are: How large is the electron wave packet? How do we include space quantization effect into classical device

simulators?

1.1 Major concerns for future integrated circuits

2


Example 1: Degradation of Ctot for Different Device Technologies

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2 3 4 5 6 7 8 9 10

classical M-B, metal gates

classical F-D, metal gates

quantum, metal gates

quantum, poly-gates ND=6x1019 cm-3

quantum, poly-gates ND=1020 cm-3

quantum, poly-gates ND=2x1020 cm-3

Cto

t/Cox

Oxide thickness tox

[nm]

T=300 K, NA=1018 cm-3

Substrate

Gate

ox

oxox t

C =

polyC

invC deplC

inv

ox

poly

ox

ox

deplinv

ox

poly

ox

oxtot

CC

CC

1

C

CCC

CC

1

CC

++≈

+++

=

Conclusion: Quantization effects must be taken into account to properly describe the operation of future MOS devices.


Example 2: Heterojunction devices

• Esaki and Tsu (1969) => ionized donors can be spatially separated from the free electrons using modulation doping

• Dingle (1978) => grew the first modulation doped GaAs/AlGaAs superlattice

• The impact of the heterostructure is that the potential is abrupt, which leads to a more vigorous control of the channel charge density by the externally applied voltage.

3


5 nm GaAs (cap layer)

40 nm Si -doped AlxGa1-xAs(barrier layer)

15 nm AlxGa1-xAs (spacer layer)

0.1 µm ud -GaAs substrate

Semi - insulating GaAs substrate

-0.2

0.0

0.2

0.4

0.6

0.8

0.00 0.02 0.04 0.06 0.08 0.10

Con

duct

ion

band

[eV

]

y-axis [µm]

Fermi level EF

Conduction band profile Ec

Energy of the

ground subband E0

1 µm GaAs undoped

200 Å GaAs undoped

4x50 Å AlGaAs undoped

250 Å AlGaAs undoped

400 Å AlGaAs undoped

SI GaAs

Nd2=3x1012 cm-2 Si

Nd1=1x1012 cm-2 Si

Nd1

Nd2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 20 40 60 80 100 120 140 160

Con

duct

ion

ban

d e

dge

[eV

]

y-axis [nm]

EF

E0

EF-E0 = 12 meV

T = 4.2 K


1.2 Time-dependent Schrödinger equation

• One of the goals of the quantum mechanics is to give quantitative description on a macroscopic scale of individual particles whichbehave both like particles and waves. The simplest wavefunction is a plane-wave:

• To describe realistic situations, more complicated wavefunctionscan be constructed as superpositions of plane waves:

• The evolution of the wavefunction is described by the time-dependent Schrödinger equation:

[ ] ω==ω−⋅ψ=ψ hh Eti and ,)(exp0 k p rk

[ ] tidt )(exp)(),( 0 krkkkr ω−⋅∫∫∫ ψ=ψ

)0,(exp),(),()(),(2

),( 22

rHrrrrr ψ

−=ψ⇒ψ+ψ∇−=

∂ψ∂

ti

ttVtmt

ti

h

hh

4


• A stable and norm-preserving discretization scheme for the time-dependent SWE is the Crank-Nicholson semi-implicit scheme:

• In the actual implementation, one solves the tri-diagonal system of equations

and then calculates the value of the wavefunction at time-step (k+1) as:

[ ] KKKKKK

tHi

tHi

HHi

tψ

∆+

∆−=ψ⇒ψ+ψ−=

∆ψ−ψ ++

+

h

h

h

21

21

211

1

12

1212

21

2

221

442

1

+

−+

∆+ χ

∆+∆++

χ∆−χ∆−=ψ⇒ψ−χ=ψ

−

jj

jjkj

KK

tHi

Vti

mh

timh

ti

mh

ti

h

h

hh

h

kkk ψ−χ=ψ +1


1.3 Discretization of the SWE for variable effective mass

• Under the assumption of slowly varying material composition, one can adopt the SWE with varying effective mass. There are two ways one can make the Hamiltonian of the system to be Hermitian:

(a) Bring the effective mass inside the differential operator, i.e.

For uniform mesh size, the discretized version of Hψ is:

ψ+

ψ∇⋅∇−=ψ Vm

H*

12

2h

iii

ii

i

ii Vmm

ψ+

ψ−ψ−

ψ−ψ

∆−

−

−

+

+*

2/1

1*

2/1

12

2

2

h

5


(b) Another Hermitian operator proposed for variable mass has the form:

If one applies the box-integration method to the first two terms of this Hamiltonian, it gives the same discretized form of the equations as the fist method

ψ+

∂∂ψ+

∂∂

∂ψ∂+

∂ψ∂−=

ψ+

ψ∇+ψ∇−=ψ

Vmzmzzzm

Vmm

H

*1

21

*1

*1

2

*1

*1

4

2

2

2

22

222

h

h

2/1

2/1

2/1

2/1

2/1

2/1

2/1

2/1

2/1

2/1

2/1

2/12

2

*1

*1

*1

*1

*1

*1

+

−

+

−

+

−

+

−

+

−

+

−

∂ψ∂=

∂∂

∂ψ∂+

∂∂

∂ψ∂−

∂ψ∂=

∂∂

∂ψ∂+

∂

ψ∂

∫∫

∫∫

i

i

i

i

i

i

i

i

i

i

i

i

zmmzzmzz

zmmzzdz

zm


1.4 Time-independent Schrödinger equation • If one is considering stationary states, then:

• The time-independent SWE then reduces to:

(a) For a system confined by a potential well, the SWE is aneigenvalue problem

(b) For an open system, one has a boundary value problemwhere the energy of the wanted solution is specified and onehas to specify the spatial variation of ψ(r)

• For 1D confinement (triangular or rectangular quantum well problems) one can use either:

- analytical approaches- numerical approaches

)/exp()()exp()(),( hiEttit −ψ=ω−ψ=ψ rrr

)()()()(*2

22

rrrr ψ=ψ+ψ∇− EVmh

6


Triangular well(analytical approaches)

Airy functions method

Variational Approach

NumericalApproaches

1. Shooting method2. Finding the roots of a

characteristic polynomial3. Lanczos algorithm

1. Housholder method2. Implicitly restarted Arnoldi

method (ARPACK)

1D

2D and 3D

• Summary of approaches used to solve the SWE


(A) Airy functions method

• Suppose, we want to solve self-consistently the 1D Schrödinger-Poisson problem in a MOS structure:

• The analytical solution of the Poisson equation is of the form:

( ) ∑ ψ=+ε

−=∂∂

ψ=ψ+∂

ψ∂−

iiiA zNnnN

e

z

V

zEzzVz

zm

)(,

)()()()(

*2

22

2

2

22

h

ε−+−+−=

ε=

ψ−−

ε+

−

ε= ∫∑

avsFbulkFcd

A

d

zi

ii

A

zNeEEEEEE

Ne

EW

dzzzzzNe

Wz

zWNe

zV

2

002

0

222

)()(,2

’)’()’(2

1)(

7


• When the inversion charge density does not make significant contribution to the band-bending, The 1D SWE reduces to:

• The solutions are:

ε=ψ=ψ+

∂ψ∂−

WeNFzEzzeF

z

zm

Ass ),()(

)(*2 2

22h

3/23/12

2

43

23

*2

*2)(

+

π

≈

−=ψ

ieF

mE

eFE

zeEm

Aiz

si

s

isi

h

h


(B) Variational Approach

• When the electrons significantly affect the band bending near the interface, the Airy functions approach fails. In this case, at low temperatures one can use the variational approach due to Fang and Howard, in which the ground state wavefunction is assumed to be of the form:

• The energy of the lowest subband is:

2/203

)( bzb zez −=ψ

><+><+><+>>=<< Isdo VVVTE

*8

22

mbh

2

22 63

b

NebWNe

sc

A

sc

A

ε−

ε bNe

sc

sε16

33 2

scoxsc

oxsc beπεε+ε

ε−ε32

2

8


• When one minimizes the total energy per electron, it follows that:

sdeplsc

NNNNem

b321112

2

2

+=

ε

= *where,**

h

KTcmN

cmN

s

A

010

10212

315

==

=−

−


(C) Shooting Method for 1D problems

• To describe the shooting method, lets rewrite the time-independent SWE in the form:

• When discretized on a uniform mesh, one gets that:

• The schematics of the potential used in the discussion is:

[ ])(*2

)(,0)()(2

22

2zVE

mzkzzk

dz

d −==ψ+ψh

( ) 02 122

1 =ψ+ψ∆−−ψ −+ iiii k

9


• The solution strategy is the following one:

(1) Integrate the SWE towards larger z from zmin

(2) Integrate the SWE towards smaller z starting from zmax

(3) At the matching point zm, one matches the solutions ψ< and

ψ>

(4) The eigenvalue is then signaled by equality of the derivati-ves at zm

122

1 )2( −+ ψ−ψ∆−=ψ iiii k

122

1 )2( +− ψ−ψ∆−=ψ iiii k

[ ])()(1 ∆−ψ−∆−ψψ

=⇒ψ=ψ ><><

mmzz

zzfdz

ddz

d

mm


(C1) Description of SCHRED

• For actual device simulations, one has to solve self-consistently the 1D Schrödinger equation self-consistently with the 1D Poisson equation.

• For the case of Si, in the solution of the 1D Schrödinger equati-on one must worry about the mass anisotropy

∆2-band:m⊥=ml=0.916m0, m||=mt=0.196m0

∆4-band:m⊥=mt=0.196m0, m||= (ml mt)

1/2

∆2-band:m⊥=ml=0.916m0, m||=mt=0.196m0

∆4-band:m⊥=mt=0.196m0, m||= (ml mt)

1/2

∆2-band

∆4-band

EF

VG>0

Ε11

Ε12

Ε13

Ε14

Ε21

Ε22

z-axis [100](depth)

10


(1) Existing Features of SCHRED:

å Classical and quantum-mechanical charge descriptionFermi-Dirac and Maxwell-Boltzmann statistics

Single-valley and multiple-valley conduction bandsê Exchange and correlation corrections to the ground state

energy of the system Metal and poly-silicon gates Partial ionization of the impurity atoms

Features Recently Being Implemented in SHRED:

Á Hole quantizationÁ Non-parabolicity of the bands

Current home of the solver: Purdue Semiconductor Simulation Hub(http://www.ecn.purdue.edu/labs/punch/)

Current home of the solver: Purdue Semiconductor Simulation Hub(http://www.ecn.purdue.edu/labs/punch/)


(2) Sample input file used in SCHRED simulations:

device bulk=yesparams temp[K]=300, tox[nm]=1.5, kox=3.9body uniform=true, Nb[cm-3]=1.e19gate metal=false, Ng[cm-3]=-6.0e19ionize ionize=no, Ea[meV]=45, Ed[meV]=45voltage Vmin[V]=0, Vmax[V]=2.5, Vstep[V]=0.1charge quantum=yes, Fermi=yes, exchange=no,+ e_nsub1=4, e_nsub2=2calc CV_curve=yes, file_cv=cv.datsave charges=no, file_ch=chrg.dat,+ wavefunc=no, file_wf=wfun.datconverge toleranc=5.e-6, max_iter=2000

device bulk=yesparams temp[K]=300, tox[nm]=1.5, kox=3.9body uniform=true, Nb[cm-3]=1.e19gate metal=false, Ng[cm-3]=-6.0e19ionize ionize=no, Ea[meV]=45, Ed[meV]=45voltage Vmin[V]=0, Vmax[V]=2.5, Vstep[V]=0.1charge quantum=yes, Fermi=yes, exchange=no,+ e_nsub1=4, e_nsub2=2calc CV_curve=yes, file_cv=cv.datsave charges=no, file_ch=chrg.dat,+ wavefunc=no, file_wf=wfun.datconverge toleranc=5.e-6, max_iter=2000

11


Exchange-Correlation Correction:

Lower subband energies

Increase in the subband separation

Increase in the carrier concentration at which the Fermi level crosses into the second subband

Contracted wavefunctions

Vasileska et al., J. Vac. Sci. Technol. B 13, 1841 (1995)(Na=2.8x1015 cm-3, Ns=4x1012 cm-2, T=0 K)

Thick (thin) lines correspond to thecase when the exchange-correlationcorrections are included (omitted) inthe simulations.

Distance from the interface [Å]

0.02

0.06

0.1

0.14

0.18

0 20 40 60 80 100

second subband

first subband

Veff

Normalized wavefunctionE

nerg

y[m

eV]

E = E HF + E corr = E kinHF + E exchange

HF + E corr

Total Ground StateEnergy of the System

Hartree-Fock Approximationfor the Ground State Energy

Accounts for the error madewith the Hartree-Fock Approximation

Accounts for the reductionof the Ground State Enerydue to the inclusion of thePauli Exclusion Principle

(3) Sample of simulation results obtained with SCHRED:


0

10

20

30

40

50

60

0 5x1011

1x1012

1.5x1012

2x1012

2.5x1012

3x1012

Exp. data [Kneschaurek et al.]V

eff(z)=V

H(z)+V

im(z)+V

exc(z)

Veff

(z)=VH(z)

Veff

(z)=VH(z)+V

im(z)

Ene

rgy

E10

[meV

] T = 4.2 K, Ndepl=1011 cm-2

Ns[cm-2]

Kneschaurek et al., Phys. Rev. B 14, 1610 (1976) Infrared Optical AbsorptionExperiment:

far-ir

radiation

LED

SiO2 Al-Gate

Si-Sample

Vg

Transmission-Line Arrangement

(4) Sample of simulation results obtained with SCHRED (Cont’d):

12


(4) Sample of simulation results obtained with SCHRED (Cont’d):


(D) Finding the roots of a characteristic polynomial

• The finite-difference approximation of the 1D SWE is given by:

• The eigenvalues are the zeroes of the Nth degree characteristic polynomial of A

iiiiiiiiii

iiiii

EAAA

EV

zEzzVdz

d

ψ=ψ+ψ+ψ⇓

ψ=ψ+ψ

−+ψ

ψ=ψ+ψ

++−−

+∆∆

−∆

11,,11,

112

11

2

2

222

)()()(

)()()()(

,)()()(,)(

)()det()(

22

1,1,

22,112,221,11

1

EPAEPEAEP

AEPEAEPEAEP

EEEEP

nnnnnnn

i

N

iA

−−−

=

−−=

−−=−=⇓

−Π=−=

IA

13


• Several features help considerably in finding the roots of PA:

(1) The number of times the sequence 1, P1(E), P2(E),…,PN(E) changes sign equals the number of eigenvalues less than E

(2) To make a systematic search for all of the eigenvalues, a guidance can be the Gerschgorin’s bounds:

(3) Once the eigenvalues are determined, the eigenvectors are found by using the inverse vector iteration procedure, in which one starts with an initial guess for the eigenvector ψn

(1) associa-ted with a given eigenvalue En and refines the guess by evaluating:

;; ,,,, maxmin

+≤

−≥ ∑∑≠≠ ij

jiiii

nij

jiiii

n AAEAAE

[ ] )1(1)2( )( nnn E ψδ+−=ψ −IA


(E) The Lanczos algorithm

• This algorithm is very suitable when one is interested in many of the lowest eigenvalues.

• The strategy is to construct a set of orthonormal basis vectors ψn, in which A is explicitly tri-diagonal matrix

(1) Choose an arbitrary first vector in the basis ψ1 such that:

(2) One then forms a second vector in the basis as:

(3) Subsequent vectors in the basis are then constructed recursively:

111 =ψψT

( ) ( )[ ] 2/12111121111

111122

,

)(−

−ψψ=ψψ=

ψ−ψ=ψ

ACA

ACTT AA A

A

( ) ( )[ ] 2/121

21

1111 )(−

−+

−−++

−ψψ=

ψ−ψ−ψ=ψ

nnnnnT

nn

nnnnnnnnn

AAC

AAC

AA

A

14


• The real utility of the Lanczos method is when many, but bot all, of the eigenvalues of a large matrix are required. Suppose that one is interested in the 10 lowest eigenvalues of a matrix of dimension 1000. The procedure is then the following:

(1) Generate some number of states in a basis larger than the number of eigenvalues being sought

(2) A linear combination of these eigenvectors is then used in constructing a new basis of dimension 25

(3) The continued procedure of:-> generating a limited basis,-> diagonalizing the tridiagonal matrix-> using the normalized sum of the lowest eigenvectors as

the first vector in the next basisconverges to the required eigenvalues and eigenvectors.


(F) Numerov Algorithm – note on the integration of the SWE

• There is particularly simple and efficient method for integrating second-order differential equations called Numerov or Cowling’s method, based on the idea sketched below:

• Solving the linear system of equations for either ψi-1 or ψi+1 then provides a recursion relation for integrating either forward or backward

( ) ( ) ( ) ( )

012

1125

1212

1

2

12

2

12

1

22

2

12

1

2

21

221

22

2

2

2

2

2

2

’’’’2

’’2

11

=

∆++

∆−−

∆+

∆

+−−=−

∂∂=

∂∂

∂∂

∆+=∆

+−

−−++

+−

+−

iiiiii

iii

iiiii

kkk

kkkk

xxx

ψψψ

ψψψψψ

ψψψψψ

15


(G) 2D and 3D Eigenvalue Solvers

• When solving 2D eigenvalue problems, one can transform the matrix to tri-diagonal form by using either the Householder method, Lanczos iteration or Rayleigh quotient iteration method (S.E. Laux, and F. Stern, Appl. Phys. Lett. 49, pp. 91, 1986).

• For 3D problems, the best eigenvalue solvers of large and sparce matrices, based on Implicitly Restarted Arnoldi Method, can be found in the publicly available software package ARPACK:

-> the codes are available by anonymous ftp from

ftp.caam.rice.edu

-> or by connecting directly to the URL

http://www.caam.rice.edu/software/ARPACK


1.5 Open Systems

To calculate the conduction-band edge of the RTD structure, one needs to solve self-consistently the Poisson equation:

and the 1D Schrödinger equation:

( )nNNe

AD −−−=∇ε

ϕ2

[ ] 002

2

22

2

22

2

2

22

=ψ+∂

ψ∂⇒=ψ−+∂

ψ∂⇓

ψ=ψ+∂

ψ∂−

)()()()(

)()()(

xxkx

xxEEm

x

xExxExm

xc

c

h

h

16


• For a 1D domain limited to x∈[0,L], the SWE discretized on a uniform grid, with mesh size ∆ is:

• The traveling wave at a specified energy is assumed to be of a plane-wave type of the form:

• For a traveling wave that enters the open system at x=0, the procedure is the following one:

(1) one sets A(L)=1 at the output boundary to get that:

02 2

211 =+

∆

+− +−ii

iii k ψψψψ

[ ])(exp)()( xjxkxAx ±=ψ

( )[ ] )exp(exp

)exp(

1 ∆=∆+=

=

+ NNNN

NN

jkLjk

Ljk

ψψψ


(2) The next step is to calculate the wavefunction for i=0,1,…, N-1:

(3) For i=0, one has that:

• The procedure for inflow from the right boundary is identical:

and is started assuming

[ ] 122

1 2 +− ψ−ψ∆−=ψ iiii k

)sin(2

)exp(

)exp()exp( ;

0

1000

00001000

∆−∆

=

∆+∆−=+=

−

−

jki

jkI

jkRjkIRI

ψψψψ

( ) 12

2

12

2 −+ −

−∆+= iiii EV

m ψψψh

( ).expand1 010 ∆−== − jkψψ

17


• To find the total wavefunction from the renormalized results of the two recursions steps (from the left and from the right), oneneeds to specify the INJECTION conditions:

• Once the wavefunction is determined, one can proceed with the calculation of the current through the structure:

• The electron density is then given by:

kx

kk

x

kkkn

me

xJ jj

jj

kj

j

∆

∂

∂−

∂

∂∑−=

)(*)(

)()(*)(

2)(

ψψ

ψψ

πh

kkkknxn jjk

jj

∆∑= )()(*)()(21 ψψπ

−−+=∫=

∞+

∞− Tk

EEEm)n(kkn

dkxn

B

xCLFxx

x exp1ln)(2

)(2hππ

where


2.1 Madelung and Bohm’s Reformulation of QM

The hydrodynamic formulation is initiated by substituting the wavefunc-tion into the time-dependent SWE:

The resultant real and imaginary parts give:

ψψ === nRReiS , /h

tiV

m ∂∂=+∇− ψψψ h

h 22

2

( )

( ) qc ffQVdtd

m

mR

RmtQ

tQtVSmt

tS

S/mtRtSmt

t

+=+−∇=

ρ∇ρ−=∇−=ρ

↓

ρ++∇=∂

∂−

∇==ρ=

∇ρ⋅∇+

∂ρ∂

−

v

r

eq. Jacobi Hamilton rrr

v rr r

2/122/12

22

2

2

21

2),,(

);,,(),(21),(

)2(

;),(),(;01),(

)1(

hh

2. The effective potential approach

18


2.2 Other Quantum Potential Formulations

An alternate form of the quantum potential has been proposed by Iafrate, Grubin and Ferry, and is based on moments of the Wigner-Boltzmann equation:

Ferry and Zhou derived a form for a smooth quantum potential based on the effective classical partition function of Feynman and Kleinert. The Feynman and Kleinert idea is as follows:

(a) Calculate the smeared version of the potential V(x) as follows:

(b) Introduce a second parameter Ω and form the auxiliary potential:

( )2

22 ln

8 x

n

mVQ ∂

∂−= h Wigner potential or density-gradient correction

( ) ( ) )’(’2

1exp

2

’)( 2

22/122 xVxx

aa

dxxV

a

−−∫

π=

)’(22/

)2/sinh(ln

1),,(

~2

22

2 xVaaxWa

+Ω−Ωβ

Ωββ

=Ω


(c) The minimization with respect to Ω and the minimization with respect to a2 then give:

Gardner and Ringhofer derived a smooth quantum potential for hydrodynamic modeling that is valid to all orders of h2, that involves smoothing integration of the classical potential over space and temperature:

( ) )’()’)(’(

2exp

)’)(’(

2’

’’),(

22

23

2

0

xxx V’m

mxd

dxV

−β+ββ−βπ

β−

×∫

β+ββ−βπβ

ββ

∫ ββ=β

β

h

h

( ) ( )0202

22

22and12

coth2

1xV

axa

a∂∂=Ω

−

ΩβΩβ

Ωβ=

19


2.3 The Effective Potential Approach due to Ferry

∑ −δ∫

∑

α−

−∫∫ −δ

−δ

α−

−∫ ∑∫

∫ ∑=

ieffi

ii

ii

ii

Vd

Vdd

dVd

ndrVV

)r()rr(r~

’rrexp)’r(’r)rr(r~

)r’r(’rr

exp’r)r(r~

)r()r(

2

2

2

2

In principle, the effective role of the potential can be rewritten in terms of the non-local density as (Ferry et al.1):

Classical densitySmoothed,effective potential

Built-in potentialfor triangular po-tential approxima-tion.

Effective potential approximation

Quantizationenergy

“Set back” of charge --quantum capacitance effects

1 D. K. Ferry, Superlatt. Microstruc. 27, 59 (2000); VLSI Design, in press.


....)()(2

22 +

∂∂+=

x

VxVxVeff α

....2

)(

...)/ln(2

)()(

2

22

2

022

+∂

∂−=

+∂

∂−=

x

n

nxV

x

nnxVxVeff

βαβα

ψψ2

*

2

2

∇=

mQ

h

In semiconductors, typically the dependence of the density upon the potential is as a factor exp(-βV)

....2

)(2

1

)(2

1)(

22

22

2/2

22

2/

∫

∫∞

∞−

−

∞

∞−

−

+

∂∂+

∂∂+≅

+=

ξξξαπ

ξξαπ

αξ

αξ

dex

V

x

VxV

dexVxVeff

Within a factor of 2, the second term is now recognized as the density gradient term, but is more commonly known as the Bohmpotential

• The connection of Ferry’s Approach to the Bohm Potential is given below:

20


(1) Validation of the approach on the example of a MOS capacitor with tox = 6 nm – sheet density results:

0 100

5 1012

1 1013

1.5 1013

2 1013

1.0 2.0 3.0 4.0 5.0

1e17 - classical1e17 - SCHRED1e17 - Effective Potential1e18 - classical1e18 - SCHRED1e18 - Effective Potential

Inve

rsio

n ch

arg

e de

nsity

Ns [

cm-2

]

Gate voltage Vg [V]

(a)

The Gaussian fitting parameter a0 = 0.5 nm.


0

0.5

1

1.5

2

2.5

3

3.5

4

1.0 2.0 3.0 4.0 5.0

1e17 - classical1e17 - SCHRED1e17 - Effective Potential1e18 - classical1e18 - SCHRED1e18 - Effective Potential

Ave

rag

e d

ista

nce

[nm

]

Gate voltage Vg [V]

(b)

The Gaussian fitting parameter a0 = 0.5 nm.

(2) Validation of the approach on the example of a MOS capacitorwith tox = 6 nm – results for the average carrier displacement:

21


p-type substrate

N+ channel N+

SiO2

gate

source drain

Device specification:

• Oxide thickness = 2 nm• Channel length = 50 nm• Channel width = 0.8 µm• Junction depth = 36 nm• Substrate doping:

NA=1018 cm-3

• Doping of the source-drainregions:

ND = 1019 cm-3

2.4 Simulation examples that utilize the effective potentialapproach

(A) Simulation of a 50 nm MOSFET Device


(1) Conduction Band Profile

En

ergy

[eV

]

Dis

tanc

e [n

m]

source drain

Distance [nm]

Without EffectivePotential Veff

With the EffectivePotential Veff

VG = VD = 1 V

22


Ex – field [V/cm]D

ista

nce

[nm

]

Distance [nm]

source drain

Dis

tanc

e [n

m] Ey – field [V/cm]

Distance [nm]

source drain

With Veff Without Veff

Negative electric field responsible for charge

setback from the interface

(2) Electric field profile


1012

1013

0 50 100 150

Without the effective potentialWith the effective potential

She

et d

ensi

ty [

cm-2

]

Distance [nm]

Electron sheet density reductiondue to quantum-mechanical band-gap

widening effect

Applied bias: VG = 1 VApplied bias: VG = 1 V

0

5

10

15

20

40 50 60 70 80 90 100 110

VD=0.4 V, without V

eff

VD=0.4 V, with V

eff

VD=1 V, without V

eff

VD=1 V, with V

eff

Ave

rage

dis

tanc

e [n

m]

Distance [nm]

Additional 2-3 nm charge set-back from the interface due to quantum-

mechanical space-quantization effect

(3) Carrier density and average distance

23


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

without effective potentialwith the effective potential

Dra

in c

urr

ent

I D [m

A/µ

m]

Gate voltage VG [V]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1 1.2

VG=1.0 - without V

eff

VG=1.2 - without V

eff

VG=1.0 - with V

eff

VG=1.2 - with V

effDra

in c

urre

nt I

D [

mA

/µm

]

Drain voltage VD [V]

The observed threshold voltage shift is mainly due to the degradation of the device transconductance and is not due to mobility degradation in the channel.

The shift in the threshold voltage leads to a reduction in the on-state current.

0

5x106

1x107

1.5x107

2x107

2.5x107

0 50 100 150

without Veff

with Veff

Ve

loci

ty [

cm/s

]

Distance [nm]

VG = 1 V

VD = 1 V

(4) Transfer and output characteristics


(B) Simulation of a 250 nm FIBMOS device

Source and drain length 50 nm

Source and drain junction depth 36 nm

Gate length 250 nm

Device width P

Bulk depth 400 nm

Oxide thickness 5 nm

Source and drain doping 1019 cm-3

Substrate doping 1016 cm-3

Implant doping 1.6×1018 cm-

3

Substrate doping 1016 cm-3

Implant length 70 nm

×

24


Average electron displacement from the interface and the total channel width increase when quantum effects are included!

(1) Concentration of electrons in the channel


(2) Sheet electron density

When Quantum effects are included:

sheet density is reduced

inversion more difficult to achieve

threshold voltage is increased

25


Average energy increases and velocity decreases when quantum effects are included.

Prominent velocity overshoot evidence of non-stationary transport.

(3) Energy and velocity along the channel


With quantum effects: Drive current is lowered Threshold voltage is higher

Transconductance is degraded

(4) Device transfer and output characteristics

26


Experimental data1:

Simulate this structure using full 3D Poisson solver coupled with 2D Schrödinger solver.

Do the same calculation with effective potential

(C) Simulation of a SOI Device Structure

1 Majima, Ishikuro, Hiramoto, IEEE El. Dev. Lett. 21, 396 (2000).

A B

A

n+

n+

p-

B

Buried Oxide Substrate (400nm)

Gate oxide (34nm thick)

Channel (~10nm wide)

Thin SOI layer (7nm)

Gate

Thr

esho

ld v

olta

ge [V

]

Channel width [nm]


),,()(),,(),,(22 2

22

2

22zyxxEzyxzyxV

ymymjjj

zy

ννννν

ψ=ψ

+

∂

∂−∂

∂− hh

Approach 1:

Solve the 2D Schrödinger equation

where the electron density is given by:

Approach 2:

Use the effective potential approach to obtain the line electron density.

;),,()(2),,(23

1zyxxNzyxn j

jj

ν

=ν

ν ψ∑ ∑=

−⋅

π=

ν

−νν

Tk

ExEFTkmxN

B

FjBxj

)(2

1)( 2/1

h

(1) Proof of concept for 2D quantization

27


0

4x106

8x106

1.2x107

1.6x107

0 0.5 1 1.5 2 2.5

15 nm - Effective potential15 nm - 2D Schrodinger10 nm - Effective Potential10 nm - 2D Schrodinger7 nm - Effective Potential7 nm - 2D Schrodinger

Line

den

sity

[1/

cm]

Gate Voltage VG [V]

Excellent agreement is observed between the two approaches when using the theoretical value for the Gaussian smoothing parameter of 0.64 nm.

(2) Simulation results for a quantum wire


The effective potential approach in many cases allows for a quite accurate approximation of the quantization effects in real semiconductor devices

The numerical cost of including the effective potential is low –“more bang for the buck”

Some challenges remain, however:

Model valid within the random phase approximation.

It is still “quasi-local”, so it does not allow proper treatment phase interference effects

2.5 Conclusions regarding the use of the effective potential approach

Documents

Schroedinger-Poisson Solvers