Extrinsic Semiconductors
•
•
Adding ‘correct’ impurities can lead to controlled domination of one carrier type
– n-type is dominated by electrons
– p-type if dominated by holes
Adding other impurities can degrade electrical properties
Impurities with close electronic
structure to host
isoelectronic
hydrogenic
x
x
Ge
Impurities with very different
electronic structure to host
x
x
x
P
+
deep level
-
x
x
x
x
x
x
Si
Si
Ec
Ev
©1999 E.A. Fitzgerald
Au x
Ec
ED
Ev
Ec
EDEEP
Ev
1
Hydrogenic Model
• For hydrogenic donors or acceptors, we can think of the electron or hole,
respectively, as an orbiting electron around a net fixed charge
• We can estimate the energy to free the carrier into the conduction band or
valence band by using a modified expression for the energy of an electron in
the H atom
me 4
13.6
En = 2 2 2 = − 2
8ε o h n
n
e2
(in eV)
=e
me
m *e 4 1
13.6 m* 1
εr
En = 2 2 2 ⎯⎯⎯→ 2 2 2 2 = − 2
n m ε2
8ε oh n
8ε o h n ε r
4
2
•Thus, for the ground state n=1, we can see already that since ε is on the order
of 10, the binding energy of the carrier to the center is <0.1eV
•Expect that many carriers are then thermalize at room T
•Experiment:
•B acceptor in Si: .046 eV
•P donor in Si: 0.044 eV
•As donor in Si: 0.049
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©1999 E.A. Fitzgerald
The Power of Doping
•
•
Can make the material n-type or p-type: Hydrogenic impurities are nearly fully
ionized at room temperature
– ni2 for Si: ~1020cm-3
– Add 1018cm-3 donors to Si: n~Nd
– n~1018cm-3, p~102 (ni2/Nd)
Can change conductivity drastically
– 1 part in 107 impurity in a crystal (~1022cm-3 atom density)
– 1022*1/107=1015 dopant atoms per cm-3
– n~1015, p~1020/1015~105
σ/σi~(p+n)/2ni~n/2ni~105!
Impurities at the ppm level drastically change the conductivity (5-6 orders of magnitude)
3
©1999 E.A. Fitzgerald
Expected Temperature Behavior of Doped Material (Example:n-type)
• 3 regimes
ln(n)
Eg/2kb
Eb/kb
Intrinsic
Extrinsic
Freeze-out
1/T
4
©1999 E.A. Fitzgerald
Contrasting Semiconductor and Metal Conductivity
ne 2τ
σ=
m
• Semiconductors
– changes in n(T) can dominate over τ
– as T increases, conductivity increases
• Metals
– n fixed
– as T increases, τ decreases, and conductivity decreases
5
©1999 E.A. Fitzgerald
General Interpretation of τ
• Metals and majority carriers in semiconductors
– τ is the scattering length
– Phonons (lattice vibrations), impurities, dislocations,
and grain boundaries can decrease τ
1
τ
τi =
=
1
τ phonon
li
1
=
vth vthσ i N i
liσ i N i = 1
+
1
τ impur
+
1
τ disl
+
1
τ gb
+ ...
where σ is the cross-section of the scatterer, N is the
number of scatterers per volume, and l is the average
distance before collisions
The mechanism that will tend to dominate the scattering will be the mechanism with the
shortest l (most numerous), unless there is a large difference in the cross-sections
Example: Si transistor, τphonon dominates even though τimpur gets worse with scaling
6
©1999 E.A. Fitzgerald
Example: Electron Mobility in Ge
106
μ~T-3/2 if phonon dominated
(T-1/2 from vth, T-1 from x-section σ
-3/2
T
3
Ge
(1013)
At higher doping, the
ionized donors are the
dominate scattering mechanism
µe,cm2/volt-s
105
3
(1015)
104
(1017)
3
103
10
30
100
300
T
Figure by MIT OpenCourseWare.
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©1999 E.A. Fitzgerald
Other Interpretation of τ
•
Minority carriers in semiconductors
– can think of τ as the time to recombination: recombination time
– does not affect Drude model in any way
τ, l
Ec
Recombination event: electron disappears
E
Ev
x
Imagine p-type material, so there are many more holes than electrons
holes=majority carrier
electrons=minority carrier
τ is referred to in this context as the minority carrier lifetime
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©1999 E.A. Fitzgerald
Other Recombination Pathways
• Deep levels in semiconductors act as carrier traps and/or enhanced
recombination sites
Ec
Edeep
Recombination through deep level
E
Ev
x
•Barrier to capture carrier is Eg/2
•Since the probability of the carrier transition is ~e-ΔE/kT, trapping a carrier with a deep state is
very probable
•A trapped carrier can then help attract another carrier, increasing recombination through the
deep state
1
τ
=
1
τ int rinsic
+
1
τ deep
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©1999 E.A. Fitzgerald
Recombination and Generation (Ec to Ev)
•
•
Generation
– intrinsic: photon-induced or thermally induced, G=#carriers/vol.-sec
– extrinsic:deep levels due to traps
– Go is the equilibrium generation rate Recombination
– intrinsic: across band gap, R=# carriers/vol.-sec
– extrinsic: deep levels due to traps
– Ro is the equilibrium recombination rate, which is balanced by Go
Non-equilibrium intrinsic recombination
n-type material
R=
Δp
τh
; τh =
po
Ro
Where po is the equilibrium minority carrier concentration
p-type material
R=
Δn
τe
; τe =
no
Ro
Where no is the equilibrium minority carrier concentration
©1999 E.A. Fitzgerald
Non-equilibrium extrinsic recombination
R=
Δp
τh
τh =
1
σ h vth N t
Where σh is the capture cross section for
holes and Nt is the concentration of
recombination centers
R=
Δn
τe
τe =
1
σ e vth N t
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