Simulation of transport in silicon devices at atomistic level
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Simulation of transport in silicon devices at atomistic level
• Introduction• Properties of homogeneous silicon• Properties of pn junction• Properties of MOSFET
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Structure, circuit symbol and I-V characteristic of an nMOSFET
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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”
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Properties of homogeneous silicon Band structure obtained with sp3 atomic basis
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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
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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
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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)
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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
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Properties of pn junctionThe energy band diagram, before being joined
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Properties of pn junctionThe energy band diagram at equilibrium
• A built-in potential δV(r) is established due to the charge redistribution
npbiqV
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Properties of pn junctionThe charge distribution
)()()( rrr nSipSipnj
)(rpnj
)()()( rrr nSipSipnj
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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!
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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!
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Properties of pn junctionCurrent flow in a pn junction diode
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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
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Properties of MOSFET
The energy band diagram at equilibrium
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Properties of MOSFET The energy band diagram along the channel for various VGS
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Properties of MOSFET Current flow along the channel, a lake analogy to FET
operation
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Properties of MOSFET Current flow along the channel, a toy model result