Simulation of transport in silicon devices at atomistic level

• Introduction• Properties of homogeneous silicon• Properties of pn junction• Properties of MOSFET

Structure, circuit symbol and I-V characteristic of an nMOSFET

Construction of Hamiltonian of Silicon devices

• Split contributions to electrostatic potential in the silicon devices into intrinsic silicon part and the part due to charge redistribution caused by applied voltage, match of Fermi energies, etc.:

SiH

• We start with studies of relatively simple structures, like homogeneous silicon, pn junction, MOS capacitors

• Parameterize the obtained

SrVeHH

VeHH

rVrVrV

rrr

Si

Si

Si

Si

)(

||

)()()(

)()()(

0

)(rVand various

• “Assemble” Hamiltonian of complex structure with above parameters:

“MOSFET” = “pure silicon”+ “pn junctions”

+ “MOS capacitors”

Properties of homogeneous silicon Band structure obtained with sp3 atomic basis

Properties of homogeneous siliconTemperature dependence of the energy band gap

With increasing temperature, interatomic distance increases, interaction of an atomic orbital with its neighbors decreases, and then band gap tends to decrease.

eVEKT

eVEKT

eVEKT

eVEKT

TTTE

g

g

g

g

g

03.1,600

06.1,500

09.1,400

12.1,300

)636/(1073.4166.1)( 24exp

eVeVE theog 12.165.1 68.0

Properties of homogeneous siliconDoping dependence of the energy band gap

The wavefunctions of the electrons bound to the impurity atoms start to overlap as the density of impurities increase, and cause the band gap to shrink.

)(10

5.22318

meVcm

NEg

Properties of homogeneous siliconConduction band structure

Experimental Results:

• The minima in <100> direction

• The minima at K=0.85(2π/a)

• Effective mass

m║=0.92

m┴=0.197

Theoretical Results:

• The minima in <100> direction

• The minima at K=0.68(2π/a)

• Effective mass

m║=0.78(1.14)

m┴=0.167(0.246)

Properties of homogeneous siliconValence band structure

Theoretical Results:• The valence bands

consist of three overlapping bands

• The maxima all at Γpoint

• Δ= 9 meV• Effective mass

mlh = 0.22(0.32)

mhh= 0.22(0.32)

msh= 0.14(0.21)

Experimental Results:• The valence bands c

onsist of three overlapping bands

• The maxima all at Γpoint

• Δ= 44 meV• Effective mass

mlh = 0.16

mhh= 0.48

msh= 0.24

Properties of pn junctionThe energy band diagram, before being joined

Properties of pn junctionThe energy band diagram at equilibrium

• A built-in potential δV(r) is established due to the charge redistribution

npbiqV

Properties of pn junctionThe charge distribution

)()()( rrr nSipSipnj

)(rpnj

)()()( rrr nSipSipnj

Properties of homogeneous siliconCharge distribution of p- and n-type silicon

)(rSi

)()( rr SinSi

)()( rr pSiSi

)(2)()( rrr SipSinSi

The distribution of charge (hole) is not uniform!

Self-consistent LDA calculation of the band bending profile over a PN junction

The profile depends on density of dopants and bias voltage

Atomic orbital can not “see” the detailed structure of the charge distribution, Hamiltonian elements

)(|)(|)( RrrVRr

change smoothly across the pn junction!

Properties of pn junctionCurrent flow in a pn junction diode

IV curve of a pn junction:Tight-binding results

The fitting curve (predicted by diffusion theory):

J = J0(exp(eV/kT)-1)

With J0 = 1.0e-18(A)And T = 420K

Properties of MOSFET

The energy band diagram at equilibrium

Properties of MOSFET The energy band diagram along the channel for various VGS

Properties of MOSFET Current flow along the channel, a lake analogy to FET

operation

Properties of MOSFET Current flow along the channel, a toy model result