FYS3410 Condensed matter physics
Lecture 18: Intrinsic semiconductors
Randi Haakenaasen
randi.haakenaasen@ffi.no, 63 80 73 09
UniK/UiO
Forsvarets forskningsinstitutt
12.03.2014
Outline
• Repetition: energy band structure and filling of bands
• Intrinsic semiconductors
• The Fermi level
• Carrier concentrations at equilibrium
• Semiconductor growth and characterization: HgCdTe at FFI
– FTIR – measurements of band gap
– Quantum well photoluminescence
Remember this…
•
Pauli exclusion principle: no two electrons in an interacting system can be in the
same state
•
For a crystal with N atoms, each overlapping electron level will split into N levels
which can accomodate 2N electrons
•
Material with odd number of valence electrons per unit cell -> metal
•
Even number of valence electrons per unit cell – semiconductor or insulator if filled
shell, metal or semi-metal if overlapping bands
•
In a crystal we can decribe the motion of electrons as if they were free electrons but
with an effective mass m* which includes the effect of the crystal on the electron
𝑚 ∗=
𝑑2 𝐸(𝑘)
2
ℏ
𝑑𝑘 2
−1
Filling of bands
•
The periodicity of the crystal potential introduces band
gaps in the free electron bands at the BZ boundaries
•
DOS x FD distribution gives density of occupied states
•
Highest occupied band is
–
–
–
partially filled at 0 K –> metal
completely filled at 0K –> semiconductor or insulator
Can only get current if there are empty states to go to
SC or
insulator
metal or
semimetal
metal
Semiconductors vs insulators
•
Semiconductors: band gap Eg small enough to get thermally excited carriers at
reasonable temperatures
•
Approximate: semiconductor Eg < 3eV < insulator Eg
•
Semiconductors extremely important: transistors, switches, diodes, solar cells,
detectors, LEDs…
•
Electrical conductivity varies by > 16 orders of magnitude by changes in temperature,
optical excitation or impurity content
•
Trick : vary electrical conductivity by doping extra electrons or holes
•
•
•
Group IV elemental SC: Si, Ge (diamond lattice, ie fcc lattice with a basis)
Group III-V compound SC: GaAs, InSb (zincblence, fcc w basis)
Group II-VI SC: HgCdTe (zincblende, fcc w basis)
Intrinsic semiconductors
An intrinsic semiconductor has no impurities or defects -> no carriers at 0 K
Inctrinsic σ and carr. conc. largely controlled by Eg/kT
At finite T, carriers are thermally excited from valence band VB to
conduction band CB and create electron-hole pairs EHP. The
intrinsic carrier concentrations are therefore equal:
𝑛 = 𝑝 = 𝑛𝑖
In steady state EHP recombine at same rate as they are generated
FD distribution of e- over of allowed energy levels at thermal
equilibrium = prob that available state E is occupied at T
𝑓(𝐸) =
1
1+𝑒 (𝐸−𝐸𝐹 )⁄𝑘𝑘
𝑓(𝐸𝐹 + ∆𝐸) = 1 − 𝑓(𝐸𝐹 − ∆𝐸)
symmetric about EF at all T
The Fermi level EF
•
At EF the prob of being occupied is ½:
f ( EF ) =
1
1 + e( EF − EF ) / kT
=
1
1
=
1+1 2
•
From symmetry about EF and intrinsic 𝑛 = 𝑝
– > EF close to middle of Eg
– (in middle if DOS in VB and CB are equal)
•
At room temp f(EC) and (1-f(EV)) are quite small:
10 −3
– In Si at 300K n = p = 1.5 ×10 cm
i
i
19
−3
while DOS g ( EC ), g ( EV ) ≈ 10 cm
–
As g ( EC ), g ( EV ) are large, small changes in f(E)
can result in significant changes in n and p
•
At finite T, both CB and VB are partially filled and
contibute to electrical conduction
•
In VB, electrons can hop and fill in the
empty states
– This can be treated as a pos.charged
hole moving in the opposite direction
EF in extrinsic semiconductors
•
In n-type material:
–
–
n in CB > (>>) p in VB
> f(E) must be above intrinsic position so that
f ( EC ) > f ( EV )
–
•
Thus EC − EF
gives a measure of n
In p-type material:
–
p in VB > (>>) n in CB
–
FD must be below middle of gap so [1 − f ( E )] below EV
is larger than f ( EC ) above EC
–
•
Thus EF − EV gives a measure of p
For given T, n and p can be calculated from the
position of EF if DOS of CB and VB are known
Holes
•
Electrons at the CBM are accelerated in E-field as a=(-e)E/me* , ie in the normal
direction for e- in E-field
•
Valence band almost filled except some holes around VBM.
Here a=(-e)E/(-│mh*│)=eE/│mh*│ and can be interpreted as motion of positive
carriers with positive mass –> holes. Assume mh* is positive and carriers in VB are
holes.
Carrier densities at equilibrium - qualitative
•
Concentration of e- in conduction band
1 ∞
𝑛0 = � 𝑔(𝐸)𝑓(𝐸, 𝑇)𝑑𝑑
𝑉 𝐸𝐶
•
f(E) decreases fast with energy
-> Very few e- far above EC
•
The probability of finding a hole also
decreases rapidly below EV
1 𝐸𝑉
𝑝0 = � 𝑔(𝐸) 1 − 𝑓(𝐸, 𝑇) 𝑑𝑑
𝑉 −∞
•
We get carrier densities as indicated in the
figure
•
For holes, the energy increases
downwards since E scale refers to electron
energy
Electron and hole concentrations at equilibrium
Assumptions:
1
1. 𝐸𝐶 − 𝐸𝐹 ≫ 𝑘𝑘 then 𝑓(𝐸, 𝑇) = 1+𝑒 (𝐸−𝐸𝐹)⁄𝑘𝑘 ≅ 𝑒 −(𝐸−𝐸𝐹 )⁄𝑘𝑘
in CB
kT = 0.026 eV
This means that most n in CB close to CBM and most p in VB close to VBM
1 − 𝑓(𝐸, 𝑇) = 1 −
1
≅ 𝑒 (𝐸−𝐸𝐹 )⁄𝑘𝑘
2.
For holes in VB
3.
Near CBM and VBM bands are nearly parabolic, so we approximate bands by parabolas with
(constant) effective masses 𝑚𝑒 ∗ and 𝑚ℎ ∗ given by the curvature of the band
𝑛0 =
=
1+𝑒 (𝐸−𝐸𝐹 )⁄𝑘𝑘
1 ∞ 𝑉 2𝑚𝑒 ∗ 3⁄2 1⁄2 −(𝐸−𝐸 )⁄𝑘𝑘
𝐹
𝐸 𝑒
𝑑𝑑
∫
ℏ2
𝑉 0 2𝜋2
2𝑚𝑒 ∗ 3⁄2 𝐸 ⁄𝑘𝑘 ∞ 1⁄2 −𝐸 ⁄𝑘𝑘
𝑒 𝐹
𝑑𝑑
∫0 𝐸 𝑒
2𝜋2 ℏ3
=2
=2
2𝜋𝑚𝑒 ∗ 𝑘𝑘 3⁄2 𝐸 ⁄𝑘𝑘
𝑒 𝐹
ℎ2
2𝜋𝑚𝑒 ∗ 𝑘𝑘 3⁄2 −(𝐸 −𝐸 )⁄𝑘𝑘
𝑒 𝐶 𝐹
ℎ2
= 𝑁𝑒𝑒𝑒 𝐶 𝑒 −(𝐸𝐶 −𝐸𝐹 )⁄𝑘𝑘
∞
∫0 𝑥 1⁄2 𝑒 −𝑎𝑎 𝑑𝑑 = 2𝑎
refer to bottom of CB as Ec instead of 0
𝐶
where 𝑁𝑒𝑒𝑒 = 2
2𝜋𝑚𝑒 ∗ 𝑘𝑘 3⁄2
ℎ2
𝜋
𝑎
Valid for material in thermal equilibrium, both intrinsic or doped:
𝑛0 = 𝑁𝑒𝑒𝑒 𝐶 𝑒 −(𝐸𝐶 −𝐸𝐹 )⁄𝑘𝑘
where 𝑁𝑒𝑒𝑒 𝐶 = 2
𝑝0 = 𝑁𝑒𝑒𝑒 𝑉 𝑒 −(𝐸𝐹 −𝐸𝑉 )⁄𝑘𝑘
where
3⁄2
2𝜋𝑚𝑒 ∗ 𝑘𝑘
ℎ2
𝑁𝑒𝑒𝑒 𝑉 = 2
2𝜋𝑚ℎ ∗ 𝑘𝑘 3⁄2
ℎ2
So integration over distributed electron states in CB can be represented by an effective DOS NeffC at EC
The concentration of holes in the VB can in a similar manner be represented by an effective hole DOS at EV
The product of 𝑛0 and 𝑝0 is constant at a given temperature, independent of EF (and thereby independent of doping) – law of
mass action
𝑛0 𝑝0 = (𝑁𝑒𝑒𝑒 𝐶 𝑒 −(𝐸𝐶 −𝐸𝐹 )⁄𝑘𝑘 )(𝑁𝑒𝑒𝑒 𝑉 𝑒 −(𝐸𝐹 −𝐸𝑉 )⁄𝑘𝑘 )= 𝑁𝑒𝑒𝑒 𝐶 𝑁𝑒𝑒𝑒 𝑉 𝑒 −(𝐸𝐶 −𝐸𝑉 )⁄𝑘𝑘 = 𝑁𝑒𝑒𝑒 𝐶 𝑁𝑒𝑒𝑒 𝑉 𝑒 −𝐸𝑔 ⁄𝑘𝑘 = 𝑛0 𝑝0
For intrinsic material we get:
𝑛𝑖 = 𝑝𝑖 =
𝑁𝑒𝑒𝑒 𝐶 𝑁𝑒𝑒𝑒 𝑉 𝑒 −𝐸𝑔⁄2𝑘𝑘
While for doped materials 𝑛0 𝑝0 = 𝑛𝑖 2
𝑛0 = 𝑛𝑖 𝑒 −(𝐸𝐹 −𝐸𝑖 )⁄𝑘𝑘
𝑝0 = 𝑛𝑖 𝑒 −(𝐸𝑖 −𝐸𝐹 )⁄𝑘𝑘
•
For intrinsic material we also get
•
If effective masses for electrons and holes are
𝐸𝑔 3
𝑚ℎ ∗
𝐸𝐹 =
+ 𝑘𝑘 ln
2 4
𝑚𝑒 ∗
equal, EF is in the middle of the gap
– If not, then it changes with temperature
•
For example: light electrons, heavy holes
-
•
many more holes than electrons generated as
temperature increases, unless EF moves up with
temperature
Heavy holes: bands formed from wavefunctions
with little overlap (inner or core e-) such as 4f e- in
rare earth metals
– Slow tunneling from one ion to the next is
reflected in heavy mass
Simplified view of band edge
structure of a direct band gap SC
Temperature dependence of ni
•
•
•
•
•
𝑛𝑖 𝑇 = 2
2𝜋𝑘𝑘
ℎ2
3⁄2
𝑚𝑒 ∗ 𝑚ℎ ∗
Temperature dependence in exponential, but
also in T3/2 from DOS and in EF (or Eg)
Usually ni is known for given material
Plot (neglect T dependence of ni and Eg)
ni strongly temp dep and much smaller than
those of metals
Eg
•
•
•
•
3⁄4 𝑒 −𝐸𝑔 ⁄2𝑘𝑘
GaAs
Si
Ge
HgCdTe
1.43 eV
1.11eV
0.67 eV
0 – 1.6 eV
Measurements of Eg, 𝑚𝑒 ∗ and 𝑚ℎ ∗
•
•
•
Need Eg, 𝑚𝑒 ∗ and 𝑚ℎ ∗ to calculate carrier concentrations
Get Eg from absorption experiments
Get effective masses from cyclotron resonance experiments
•
•
In a magnetic field B, electrons move in spirals around the field with
Wc=Be/me*
Strong absorption of radio frequency radiation when wr=wc
FTIR cut-on and detector cut-off λ
8
7
Apparent Q.E.
6
5
4
3
2
1
0
3
4
5
Wavelength [µm]
•
•
Fourier Transform Infrared spectroscopy or optical absorption experiments
–
Send spectrum of IR radiation onto sample, measure transmission spectrum
–
Photons with energy hν = hc /λ > Eg can create an EHP and are then absorbed
–
Photons with energy hν = hc /λ > Eg can not create an EHP and are transmitted
–
Band gap where transmission increases steeply
–
Oscillations on transmitted spectrum are interference fringes due to film thickness
Detector cut-off λ : below cut-off spectral response QE positive verdier
– Industri standard: der QE er 50% av maksimum
Some semiconductor properties
Summary
•
Semiconductors: 0 < Eg < 3 eV, get thermally excited carriers at resonable T
•
Fermi level ∼ chemical potential: f(EF)=1/2, FD symmetric about EF at all T
–
•
Position of EF -> n and p
Intrinsic semiconductors: 𝑛 = 𝑝 = 𝑛𝑖
–
Fermi level close to middle of gap
–
𝑛𝑖 = 𝑝𝑖 =
𝑁𝑒𝑒𝑒 𝐶 𝑁𝑒𝑒𝑒 𝑉 𝑒 −𝐸𝑔 ⁄2𝑘𝑘
•
For SC in thermal equilibrium and not too heavily doped: n and p can be
represented by effective DOS at Ec and Ev
•
For doped material 𝑛0 𝑝0 = 𝑛𝑖 2
•
law of mass action
At high enough temperature semiconductors become intrinsic