Complementi di Fisica - units.it
Transcript of Complementi di Fisica - units.it
Complementi di Fisica - Lecture 24 18-11-2015
L.Lanceri - Complementi di Fisica 1
“Complementi di Fisica” Lecture 24
Livio Lanceri Università di Trieste
Trieste, 18-11-2015
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In this lecture • Contents
– Drift of electrons and holes in practice (numbers…): • conductivity σ as a function of impurity concentration • “band bending”: representation of macroscopic electric
fields – Diffusion in practice (numbers…):
• Diffusion coefficient D • Induced (“Built-in”) electric field • Einstein relation between σ and D
• Reference textbooks – D.A. Neamen, Semiconductor Physics and Devices, McGraw-
Hill, 3rd ed., 2003, p.154-188 (“5 Carrier Transport Phenomena”) – R.Pierret, Advanced Semiconductor Fundamental, Prentice
Hall, 2nd ed., p.175-215 (“6 Carrier Transport”)
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From the Boltzmann Equation… • The continuity equations for the electrical current density in
semiconductors can be obtained from the Boltzmann equation:
• Multiplying by the group velocity and integrating over the momentum space dkx dky dkz:
• One obtains the expression for current density (detailed derivation: see FELD p.187-194, MOUT p.100-104)
€
∂f∂t
= −! v g ⋅! ∇ ! r f −
! F "⋅! ∇ ! k f −
f − f0
τ
! F = −e
! E
Force on electrons
€
! v g∂f∂t∫ d3
! k = − ! v g
! v g ⋅! ∇ ! r f( )d3
! k ∫ −
! v g
! F "⋅! ∇ ! k f
'
( )
*
+ , d3! k ∫ −
! v gf − f0τ
d3! k ∫
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Drift-diffusion continuity equation
relaxation time τ is small: This new term can be neglected if frequency is not too high (few hundred MHz)
For electrons (similar for holes):
€
τ n∂! J n∂t
+! J n = qnµn
! E + 1
nkBTq! ∇ ! r n +
kB
q! ∇ ! r T
%
& '
(
) *
Drift Diffusion
Temperature gradient: also a temperature gradient can drive an electric current! Absent at thermal equilibrium
already discussed in the Drude model
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• Drift (mobility) and diffusion (diffusivity) coefficients are correlated!
• Why ? – See previous lecture – Also: no current at
equilibrium, see next
Drift and diffusion: Einstein relation
ppnn
ppnnth
qkTD
qkTD
EvEvlvD
µµ
µµ
!!"
#$$%
&=!!
"
#$$%
&=
=−=≡
drift velocity
thermal velocity
external field
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Mobility and diffusivity vs. concentration
Electrons and holes:
Low doping concentration: Lattice scattering limit
High doping concentration: Impurity scattering limit
pnpn mm µµ >⇒< ∗∗
T ~ 300 K
mobility µ (cm2 V-1 s-1) diffusivity D
(cm2 s-1)
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“Band bending”: “built-in” electrical field
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Non-uniform doping: “built-in” field
qkT
V
dxdN
NV
dxdp
pV
dxdp
pVpq
dxdp
qkT
qpq
dxdp
DqpqJ
th
A
Athth
thn
nn
nnp
≡
≈=
=##$
%&&'
(−=
=−=
=−=
11
01
0
µ
µµ
µ
Ɛ
Ɛ
Ɛ
Ɛ
p-type doping, thermal equilibrium (no “external” el.field applied!):
Ɛ is the “built-in” electric field:
“thermal voltage equivalent”
p(x)
n(x)
-N-A - n + p
Ɛ(x)
diffusion
Small “space charge” unbalance
Built-in field ⇒ drift
compensates diffusion
electrons
electrons
holes
holes
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Non-uniform doping in equilibrium • Under thermal and diffusive
equilibrium conditions the Fermi level (= total chemical potential) inside a material (or group of materials in intimate contact) is invariant as a function of position
• A non-zero (“built-in”) electric field is established in nonuniformly doped semiconductors under equilibrium conditions Ɛx = (1/ q)(dEC/dx) = = (1/ q)(dEi/dx) = = (1/ q)(dEV/dx)
0===dzdE
dydE
dxdE FFF
“band bending”, as for an external field; here the field is “built-in”
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Einstein relation - 1 Relation between electron concentration, donor concentration, Fermi level EF and intrinsic Fermi level Ei in the “quasi-neutrality” approximation (local neutrality cannot be exact, otherwise there would be no built-in field!):
€
n x( ) = nieEF −Ei x( )( ) kT ≈ ND x( )
At equilibrium, EF is constant ⇒ “band bending”:
€
EF − Ei = kT lnND x( )ni
#
$ %
&
' ( ⇒ −
dEi
dx=
kTND x( )
dND x( )dx
Induced (“built-in”) electric field:
ε x = −dVdx
= −1− q
dEi
dx= −
kTq
1ND x( )
dND x( )dx
EC
EF Ei
EV
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Einstein relation - 2 At equilibrium the total electric current (diffusion + drift) must be zero; using for the electric field εx the expression just derived one obtains: Jn,x = 0 = q nµnε x + q Dn
dndx
≈
≈ q ND x( )µnε x + q DndND x( )dx
=
= q ND x( )µn −kTq
1ND x( )
dND x( )dx
$
%&'
()+ q Dn
dND x( )dx
EC
EF Ei
EV
€
Dn
µn
=kTq
Dp
µp
=kTq
"
# $ $
%
& ' '
⇒ Einstein relation for electrons (similar for holes):
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Non-uniform doping: an example
⇒ Built-in electric field: Ɛx = (1/ q)(dEi/dx) = = (1/ q)(d(EF-Ei)/dx) = = - dΨ/dx
Electric potential V(x) = Ψ(x) = (1/ q)(EF - Ei (x))
Donor concentration:
Equilibrium: EF = constant
“band bending”:
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Example: abrupt pn junction
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pn junction
n-type p-type
Built-in electric field
At thermal and diffusive equilibrium, no external field: Net current is zero, both for electrons and holes
drift
drift
diffusion
diffusion
electrons
holes
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Einstein relation – numerical examples • In the “non-degenerate” limit:
• Typical sizes of mobility and diffusivity:
qkTD
qkTD
p
p
n
n ==µµ
s/cm26Vs/cm1000
V026.0300
22n ≈⇒=
≈⇒=
nDqkTKT
µ
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Current density equations • When an external electric field Ɛ is present in addition
to the concentration gradient: no equilibrium! – Both drift and diffusion currents will flow – The total current density is different from zero in this case!
• For electrons and holes: J = Jn + Jp
dxdp
DqpqJJJ
dxdn
DqnqJJJ
ppdiffpdriftpp
nndiffndriftnn
−=+=
+=+=
µ
µ
,,
,, Ɛ
Ɛ
drift diffusion
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High field effects
Drift velocity saturation Avalanche processes
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Drift velocity saturation Silicon: electrons and holes drift velocity - increases linearly at small fields - saturates (~ thermal velocity) at high fields
GaAs: electrons and holes drift velocity - Peculiar behavior - see next slide for an explanation
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Two-valley semiconductors
GaAs: Two-valley model of E-k band diagram: Different effective masses Different mobilities
( )( ) ( )Ev
nnnnqnnnq
mq
n
c
µ
µµµ
µµµσ
τµ
=
++≡
=+=
=∗
212211
2211
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Avalanche processes Mean free path
Ek,min > Eg
To start an avalanche: enough kinetic energy to create an e-h pair Order of magnitude estimate from energy and momentum conservation (process 1 → 2):
€
12m1vs
2 ≈ Eg + 312m1v f
2
m1vs ≈ 3m1v f
E0 =12m1vs
2 ≈1.5 Eg
GA =1qαn Jn +α p Jp( )
Band bending due to external field: Ɛx=(1/q)dEC/dx
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Ionization rates and generation rate
€
αn,α p ionization rates (cm-1)
e − h generation rate (cm-3s-1)
GA =1qαn Jn +α p Jp( )
In Silicon: E0 = 3.2 Eg (el.) E0 = 4.4 Eg (h.) Ɛ ~ 3×105 V/cm For α ~ 104 cm-1
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Lecture 24 - exercises • Exercise 1: Find the electron and hole concentrations, mobilities
and resistivities of silicon samples at 300K, for each of the following impurity concentrations: (a) 5x1015 boron atoms/cm3; (b) 2x1016 boron atoms/cm3 together with 1.5x1016 arsenic atoms/cm3; and (c) 5x1015 boron atoms/cm3, together with 1017 arsenic atoms/cm3, and 1017 gallium atoms/cm3.
• Exercise 2: For a semiconductor with a constant mobility ratio b ≡ µnµp > 1 independent of impurity concentration, find the maximum resistivity ρm in terms of the intrinsic resistivity ρi and of the mobility ratio.
• Exercise 3: A semiconductor is doped with ND (ND>>ni) and has a resistance R1. The same semiconductor is then doped with an unknown amount of acceptors NA (NA>>ND), yielding a resistance of 0.5R1. Find NA in terms of ND if the ratio of diffusivities for electrons and holes is Dn/Dp=50.
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Back-up slides
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Graded np junction AD NNN −=
N: net fixed charge
n0, p0: majority carrier concentrations
net mobile charge
net total charge
potential Ψ El.field Ɛx
electron energy-band diagram