Understanding Quantum Corrections in SCHRED and Quamc2D: Space Quantization, Slides of Computer Science

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How Quantum-Mechanical Space

Quantization is Implemented in SCHRED, Drift-diffusion

simulators (Silvaco) and Particle-Based Device Simulators

(Quamc2D)

Quantum Effects in Nanodevices

• Gate Leakage

– Device Reliability Modeling

• Quantum-Mechanical Space Quantization

– SCHRED

– Empirical Corrections to Drift-Diffusion Modeling

– Effective Potential Method for Particle-Based

Simulators

Quantum Effects in MOS Capacitors

Bacarani and Worderman  transconductance degradation (Proceedings of the IEDM, pp. 278-281, 1982)

Hartstein and Albert  estimate of the inversion layer thickness (Phys. Rev. B, Vol. 38, pp.1235-1240, 1988)

van Dort et al.  analytical model for Vth which accounts for QM effects (IEEE TED, Vol. 39, pp. 932-938, 1992)

Takagi and Toriumi  physical origins of Cinv

(IEEE TED, Vol. 42, pp. 2125-2130, 1995)

Vasileska, Schroder and Ferry  influence of many-body effects on Cinv (IEEE TED, Vol. 44, pp. 584-587, 1997)

Hareland et al.  modeling of the QM effects in the channel

(IEEE TED, Vol. 43, pp. 90-96, 1996)

Krisch et al.  poly-gate capacitance attenuation

(IEEE EDL, Vol. 17, pp. 521-524, 1996)

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=0.916m 0 , m||=mt=0.196m 0  4 -band: m=mt=0.196m 0 , m||= (ml mt)1/

 2 -band: m=ml=0.916m 0 , m||=mt=0.196m 0  4 -band: m=mt=0.196m 0 , m||= (ml mt)1/

 2 -band

 4 -band

EF

VG>





^  



z-axis 100z-axis 100

Exchange-Correlation

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=2.8x10^15 cm-3, Ns=4x10^12 cm-2, T=0 K)

Thick (thin) lines correspond to the case when the exchange-correlation corrections are included (omitted) in Distance from the interface [Å]Distance from the interface [Å] the simulations.

0.

0.

0.

0.

0.

0 20 40 60 80 100

second subband

first subband

Veff

Energy wavefunction^ Normalized

[meV]

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

00 2020 4040 6060 8080 100100

second subband

first subband

Veff

Energy wavefunction^ Normalized

[meV]

EE  E HFE HF^  E E corrcorr  E E HFHFkinkin^  E E HFHFexchangeexchange^  E E corrcorr

Total Ground State Energy of the System

Hartree-Fock Approximation for the Ground State Energy

Accounts for the error made with the Hartree-Fock Approximation

Accounts for the reduction of the Ground State Enery due to the inclusion of the Pauli Exclusion Principle

Threshold Voltage Modification

Size-Quantization in Device

Simulators

• Drift-Diffusion, Hydrodynamic
  • Quantum Moments Models
  • Quantum Correction Models
• Particle-Based Simulators
  • Effective Potential Approaches
  • Solution of the Schrödinger Equation in Slices

Quantum Moments Model

 In simple terms, the quantum moments model gives correction to the carrier temperature, in which the main task is to find the form of the quantum potential:

 The model is enabled by specifying E.QUANTUM and H.QUANTUM on the MODELS statement

 The quantum temperature is saved in the structure file by specifying T.QUANTUM parameter on the OUTPUT statement

 To achieve convergence a damping factor QFACTOR is specified in conjunction with the SOLVE statement

ln( ) 8 *

, 3

(^2 ) n m

U U k

T T q q B

q ^   



Quantum potential

Quantum-Correction Models (cont’d)

With these modifications, the energy band-gap becomes:

This results in modification of the intrinsic carrier density, which now, anywhere through the depth of the device, takes the form:

The function F ( y ) is introduced to describe smooth transition between classical and quantum description (pinch-off and inversion regions)

2 / 3

1 / 3 4

, 9

13   

  

   ^  E qk T

E E B

QMg CONVg Si

B.DORT ( MODEL )

   

i iCONV^ ^ ^ iQM

iCONV QMg CONVg B QM i n n F y F y n

n n E E k T 1 ( ) ( )

exp / 2   

 

F ( y ) 2 exp  a^2  / 1 exp 2 a^2  , a  y / yref N.DORT ( MODEL )

Quantum-Correction Models (cont’d)

The Van Dort model is activated by specifying N.DORT on the

MODEL statement.

E 0

E 1

distance

Energy n ( z )

z

 z

z^ CONV z^ QM

Classical density

Quantum-mechanical density

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  and form the auxiliary potential:

  2

(^2 2) ln

(^8) x

n m

VQ

 (^) Wigner potential or density- gradient correction

 ' ( ') 2

exp 2

2 2 x x V x a a

dx Va x  

sinh( / 2 ) ln

(^22) 2 W x a^2 a V x  a

Other Quantum Potential Formulations (cont’d)

(c) The minimization with respect to  and the minimization with respect to a^2 then give:

 Gardner and Ringhofer derived a smooth quantum potential for hydrodynamic modeling that is valid to all orders of h^2 , that involves smoothing integration of the classical potential over space and temperature:

  ( ') ( ')( ')

2 exp

( ')( ')

2 '

' ' ( , )

2 2

2

3

2

0

x ' x V x

m

m d x

d V x











    

 

 (^)  

  



  

  

  

 

  

  







(^2 21) and (^2)  0  2 2 2  0  2

coth 2

1 V x a

a x a 

    

  

  

  

   



Parameter-Free Effective Potential

The basic concept of the thermodynamic approach to effective quantum potentials is that the resulting semiclassical transport picture should yield the correct thermalized equilibrium quantum state. Using quantum potentials, one generally replaces the quantum Liouville equation

for the density matrix ( x , y ) by the classical Liouville equation

for the classical density function f ( x , k ). Here, the relation between the density matrix and the density function f is given by the Weyl quantization

 t  i  H ,  0

1  (^) t f  (^2) m * k  (^) x f   (^) xV  k f  0

f x k ( , )  W [  ]  ( x  y / 2, x  y / 2)exp( ik y dy )

Parameter-Free Effective Potential (cont’d)

The thermal equilibrium density matrix in the quantum

mechanical setting is given by eq = e- βH , where =1/kBT is the

inverse energy, and the exponential is understood as a matrix

exponential.

In the semi-classical transport picture, the thermodynamic

equilibrium density function f eq is given by the Maxwellian

distribution function.

Consequently, to obtain the quantum mechanically correct

equilibrium states in the semiclassical Liouville equation with the

effective quantum potential VQ, we set:

2 2

( , ) exp 2 * [ ] ( / 2, / 2) exp( )

k (^) Q eq H

f eq x k m V W e x y x y ik y dy

     