schroedinger-poisson solvers
DESCRIPTION
An overview on methods to solve the classical Schroedinger-Poisson problem for application in solid state physics.TRANSCRIPT
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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
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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
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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.
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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.
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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
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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
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• 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
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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
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(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
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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
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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
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(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)(
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• 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
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(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
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• 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
==
=−
−
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(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
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• 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
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(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)
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(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/)
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(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
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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:
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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):
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(4) Sample of simulation results obtained with SCHRED (Cont’d):
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(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
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• 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
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(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
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• 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.
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(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
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(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
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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
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• 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
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(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
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• 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
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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
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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
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(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
Computational Electronics
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.
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....)()(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:
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Computational Electronics
(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.
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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:
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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
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(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
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Computational Electronics
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
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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
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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
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(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
×
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Average electron displacement from the interface and the total channel width increase when quantum effects are included!
(1) Concentration of electrons in the channel
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(2) Sheet electron density
When Quantum effects are included:
sheet density is reduced
inversion more difficult to achieve
threshold voltage is increased
25
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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
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With quantum effects: Drive current is lowered Threshold voltage is higher
Transconductance is degraded
(4) Device transfer and output characteristics
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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]
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),,()(),,(),,(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
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Computational Electronics
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
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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