Semiconductor Physics: Energy Bands - SCU · Donor and Acceptor Levels S. Saha HO #2: ELEN 251 -...
Transcript of Semiconductor Physics: Energy Bands - SCU · Donor and Acceptor Levels S. Saha HO #2: ELEN 251 -...
Semiconductor Physics: Energy Bands
HO #2: ELEN 251 - Semiconductor Physics Page 1S. Saha
• The e− in a solid exist in certain bands of allowed energy:– valence e− exist in the valence band– conduction e− or free electrons exist in the conduction band
– energy gap, Eg = Ec − Ev @ T :110810x02.7160.1)(
24
+−=
−
TTTEg (1)
Ec
Eg
Ev
Conduction Band
Valence Band
Kinetic Energy of holes
Kinetic Energy of electrons
Distance
Elec
tron
Ener
gy• When an e− in the conduction band gains energy, it moves up to an E > Ec.
• Similarly, when a hole in the valence band gains energy, it moves down to an E < Ev.
Hol
e En
ergy
Intrinsic Semiconductors
HO #2: ELEN 251 - Semiconductor Physics Page 2S. Saha
• In a pure single crystal semiconductor, all the e− in the conduction band are thermally excited from the valence band. Then at any temperature, T n = p = ni
where n = free e- (cm-3)p = free holes (cm-3)ni = intrinsic carrier (cm-3).
• Note:– T↑ ⇒ ni↑– Eg↑ ⇒ ni↓
Intrinsic Carrier Concentration
HO #2: ELEN 251 - Semiconductor Physics Page 3S. Saha
The product of the number of e− and holes is given by:
∴ ni ≅ 1.45x1010 cm-3 for silicon @ T = 300oK.
If T0 = known reference temperature (generally, 300oK), then at any temperature T, we can write:
where, Eg(T) is given by (1). Eq. (3) is used in SPICE to calculate ni at any temperature.
kTTE
i
kTTE
i
g
g
eTTn
eCTTnnp
2)(
23
16
)(32
10x9.3)( or,
)( −
−
=
==
(2)
0
023
2)(
2)(
00ii )()( kT
TEkT
TE gg
eTTTnTn
+−
⎟⎟⎠
⎞⎜⎜⎝
⎛= (3)
Extrinsic Semiconductors
HO #2: ELEN 251 - Semiconductor Physics Page 4S. Saha
• To increase and control the conductivity of semiconductors we add controlled amounts of “dopants” such as:– N type: P, As, and Sb each has five valence electrons– P type: B, Al, and Ga each has three valence electrons.
• These atoms occupy substitutional lattice sites and the extra e- or hole is very loosely bound, i.e. can easily move to the conduction or valence bands, respectively. In terms of band diagrams this can be represented as:
Ec
Ev
ED
1.1eV ≈0.05eV
Donor
Ec
Ev
EA
1.1eV ≈0.05eV
Acceptor
Donor and Acceptor Levels
HO #2: ELEN 251 - Semiconductor Physics Page 5S. Saha
Measured donor and acceptor levels for various impurities in silicon
Si
1.12
eV
Sb P As0.039 0.045 0.054
B Al Ga In
0.045 0.067 0.072 0.16
Ec
Ev
• Acceptor levels are below the gap center and measured from the top of the valance band.
• Donor levels are above the gap center and measured from the bottom of the conduction band band.
Fermi Level
HO #2: ELEN 251 - Semiconductor Physics Page 6S. Saha
Ev Ec
Den
sity
of S
tate
sZero in forbidden band
• The probability that an allowed state is occupied is described by the Fermi-Dirac distribution function:
f(E) = 1/[1 + e(E − EF)/kT] (4)EF = Fermi Level ≡ energy at which the probability of occupancy = 1/2
k = Boltzmann constant; T = temperature.
• Again, the probability of a state not being occupied is:1 − f(E) = 1/[1 + e(EF − E)/kT] (5)
= Probability a hole exists.
Fermi Level: Temperature Effect
HO #2: ELEN 251 - Semiconductor Physics Page 7S. Saha
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.0 0.5 1.0f(E)
E - E
F (e
V)
T = 0 KT = 100 KT = 200 KT = 300 KT = 400 KT = 500 KT = 600 K
At T = 0oK• all states < EF
are occupied.
• all states > EFare empty.
At T > 0oK• some e− have
E > EF.
• some holes exist E < EF.
The Boltzmann Approximation
HO #2: ELEN 251 - Semiconductor Physics Page 8S. Saha
0.0
0.5
1.0
-0.3 0.0 0.3
E - EF
f(E)
F-D functionM-B function
• Equation (5) represents Maxwell−Boltzmann function.
• This approximation is quite good for many device applications, as we will see later.
• However, for heavily doped materials Fermi function (4) must be used.
For energies > 3kT (≅ 75 meV @ room temp) above EF, f(E) can be approximated as:
f(E) ≅ e−(E − EF)/kT] (6)
Approximation
Summary of Equilibrium Carrier Distribution
HO #2: ELEN 251 - Semiconductor Physics Page 9S. Saha
• In an intrinsic or undoped semiconductor: n = p = ni = 3.9x1016T3/2e−Eg/2kT (2') EF ≅ (1/2)(Ec + Ev) ≡ Ei (7)
• Fermi level EF in doped as well as undoped materials is given by: n = nie(EF − Ei)/kT = nieq(φ − φF)/kT (8) p = nie(Ei − EF)/kT = nieq(φ
F− φ)/kT (9)
where φ = − Ei/q and φF = − EF/q.
• If both n- and p-type dopants are added in intrinsic semiconductor:n + NA = p + ND (10)
Then using np = ni2, we can show for an n-type semiconductor:
[ ][ ] ( )12 /4)2/1(
(11) 4)2/1(222
22
DiDiDn
DDiDn
NnNnNp
NNnNn
≅−+=
≅++=
Carrier Transport in Semiconductors
HO #2: ELEN 251 - Semiconductor Physics Page 10S. Saha
• In thermal equilibrium, mobile (conduction band) e- will be in random thermal motion. – avg. velocity of thermal motion, vth ≅ 107 cm/sec @ 300oK.
• In an applied ε field:
– motion of e− is in the direction opposite to ε.
– this process is called drift and causes a net current flow.
Si Crystal
− +
ε
The average e− velocity called the drift velocity is given by:
carrier of mass effective collision;between timefreemean */mobility where
(13)
====
≡
m*mq
vd
ττµ
µε
Drift - ε Field Driven Motion
HO #2: ELEN 251 - Semiconductor Physics Page 11S. Saha
• vd = µε linear relationship holds only for small fields.
• Mobility depends on parameters like:– material– doping – temperature– scattering
mechanisms.
Scattering Mechanisms
HO #2: ELEN 251 - Semiconductor Physics Page 12S. Saha
• Phonon scattering. Lattice atoms vibrate, and from Quantum Mechanics we know that discrete allowed vibrational states exist. This can be modeled as discrete particles called phonons. Transfer of energy from electrons to the lattice is called phonon scattering.
• Ionized impurity atom scattering - important at high dopant concentrations.
• Neutral impurity atom scattering - usually negligible.
• Carrier-carrier (e− - e− & e− - hole) scattering - important at high carrier concentrations.
• Surface scattering effects - important in MOSFET devices.
• Crystal defects - can be minimized by good crystals.
µn Vs. Temperature
HO #2: ELEN 251 - Semiconductor Physics Page 13S. Saha
Ionized impurity scattering is more important at low T.
Assuming that the different scattering mechanisms operate independently:
...111
++=
IL µµ
µ
Drift - ε Field Driven Motion
HO #2: ELEN 251 - Semiconductor Physics Page 14S. Saha
• At high fields, vd becomes comparable to vth which is an approximate upper bound of vd. Thus, vd saturates and is called:
vsat ≡ scattering limited velocity≅ 1 x 107 cm/sec for e− in Si≅ 8 x 106 cm/sec for holes in Si.
• The general expression for vd is: (14)ββεµ
εµ1
1⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+
=
sat
o
od
v
v
• The critical field at which vsat occurs is: ~ 2x104 V/cm = 2 V/µm.
This is an important physical limit in very small semiconductor devices like MOSFETs.
Drift - Current
HO #2: ELEN 251 - Semiconductor Physics Page 15S. Saha
The drift of carriers under an applied ε results in a current flow: I = −qAnvd (15)
where A = area; n = number of e− per unit volume.
Substituting for vd = µε and considering both e− and holes: I/A = (nqµn + pqµp)ε (16)
Usually, the majority carrier-term in (16) is important. Since for a semiconductor of length L under applied bias V, ε = V/L, then:
1/(qµnn + qµpp) = (V/L)/(I/A) = (V/I)(A/L) = R(A/L) = ρ where R = ρ(L/A) = resistance of the semiconductor:
(17)
where ρ = resistivity in ohm.cm; σ = conductivity in (ohm.cm)-1
σµµρ 11
=+
=∴pqnq pn
Resistivity Vs. Dopant Density @ T = 300°K
HO #2: ELEN 251 - Semiconductor Physics Page 16S. Saha
Diffusion - Gradient Driven Motion
HO #2: ELEN 251 - Semiconductor Physics Page 17S. Saha
Diffusion occurs from high concentration regions towards low concentration regions.
Carrierconcentration
Flux ofdiffusioncurrent
Flux ofdiffusioncurrent
t1
t2
t3
t1< t2 < t3
0 x−x The diffusion flux obeys Fick’s first law: F = −D(dn/dx) (18)
where F = flux of carriers D = diffusion constant n = carrier density.
Transport Equations
HO #2: ELEN 251 - Semiconductor Physics Page 18S. Saha
The diffusion current through a cross-sectional area is: I = qADn(dn/dx) − qADp(dp/dx) (19)
where Dn = e− diffusion constant = 38 cm2/sec in Si Dp = hole diffusion constant = 13 cm2/sec in Si
Dn and Dp are related to their respective carrier mobilities by the Einstein relationship:
Dp/µp = Dn/µn = kT/q (20)
If both drift and diffusion are important, In = qA[µnnε + Dn(dn/dx)] (21) Ip = qA[µppε − Dp(dp/dx)] (22)
We will use these basic equations in deriving device characteristics.
Generation-Recombination
HO #2: ELEN 251 - Semiconductor Physics Page 19S. Saha
• Thermal equilibrium is defined by the condition np = ni2. In
thermal equilibrium, the equations discussed earlier may be used to calculate EF, n, p etc.
• However, the equilibrium condition may be disturbed by the introduction or removal of free carriers.
φ
-+
N P
• Injection: Extra carriers are generated by optical or electrical means, and np >ni
2.
• Extraction: Here, carriers are removed electrically. And, as a result np < ni
2.
Injection of Carriers
HO #2: ELEN 251 - Semiconductor Physics Page 20S. Saha
If we shine light on the semiconductor with E = hν > Eg so that e− can be excited into the conduction band. n = ni + nL
p = ni + pL
n & p > ni
INJECTION
Eg
Ec
Ev
hν > Eg
ν = frequencyh = Planck’s constant
Low Level and High Level Injection
HO #2: ELEN 251 - Semiconductor Physics Page 21S. Saha
Now, if we shine light on an extrinsic semiconductor with:ND = 1015 cm-3.
1020
1018
1016
1014
1012
1010
1008
1006
1004
1002
1000
nno
pno
nn
pn
nnpn
ND = 1015 cm-3
ni for Si@ 300 oK
Equilibrium Lowlevel
Highlevel
In this case: n ≅ ND + nL
p ≅ ni2/ND + pL
Low level: nL << ND
High level: nL ≥ ND
In Fig., a low level injection of excess carrier = 1013 is shown.
Since the mathematics for high level injection are complex, we will consider only low level injection.
Direct (Band to Band) Recombination
HO #2: ELEN 251 - Semiconductor Physics Page 22S. Saha
If we generate excess carriers (∆n, ∆p) at a rate GL due to incident light. Then for low level injection we can show that: ∆p = ∆n = Uτ = GLτ
Where ∆p = p − po
∆n = n − no
p and n are total concentrations due to generation no and po are equilibrium concentrations U = net recombination rate = GL
τ = excess carrier lifetime = avg. time excess carriers remain free.
Thus, τ = ∆p/GL = ∆p/U = ∆n/U (23) For band to band recombination, τn = τp as the single phenomena
annihilates e− and hole simultaneously.
Ec
Ev
n electrons
p holes
Direct Recombination
HO #2: ELEN 251 - Semiconductor Physics Page 23S. Saha
• Example: Net recombination rate U = 1018 carriers cm-3 sec-1
Minority-carrier lifetime τ = 100 µsec, Then from (23), excess-carrier density:
∆p = ∆n = (1018) (100x10-6) = 1014 cm-3.
• Summary: For a semiconductor with equilibrium carrier concentrations no and
po the steady state carrier concentrations under light are: n = no + τGL nopo= ni
2
p = po + τGL np > ni2 (24)
Indirect Recombination (Trap-assisted)
HO #2: ELEN 251 - Semiconductor Physics Page 24S. Saha
In Si and Ge this is often the dominant mechanism since they are “indirect band gap semiconductors.” Physically this means that the minimum energy gap between Ec and Ev does not occur at the same point in the momentum space.
Since momentum (k) must be conserved in any energy level transition, GaAs can easily make direct transitions which makes it useful for LEDs. Ep = hν = Eg. While Si and Ge tend to recombine through intermediate trapping levels.
Egk
E
“Direct band gap Semiconductor” GaAs
Egk
E
“Indirect band gap Semiconductor” Si, Ge
Ec
Ev
Ec
Ev
Indirect Recombination
HO #2: ELEN 251 - Semiconductor Physics Page 25S. Saha
Let us consider the following example where an impurity like Au is introduced that provides a “trapping level” or a set of allowed states at energy Et.
Ec
Ev
Et
The level Et is assumed to act like an acceptor, that is, it can be neutral or negatively charged. Recombination is accomplished by trapping an e- and a hole. (The analysis can be easily extended to the case where the trap acts like a donor, i.e. + and neutral charge states).
“1” = e- capture“2” = e- emission“3” = hole capture“4” = hole emission1 + 3 or 3 +1 ≡ recombination
Ec
Ev
Et Eio
1 2
3 4EF
Indirect Recombination - SRH
HO #2: ELEN 251 - Semiconductor Physics Page 26S. Saha
• Shockley, Read, and Hall showed that for low level injection, the net recombination rate is given by:
where vth = carrier thermal velocity (≈ 107 cm/sec) σ = carrier capture cross-section (≈10-15 cm2)
• From Eq (25) we observe:– the “driving force” or the rate of recombination is ∝ np − ni
2, (that is, the deviation from the equilibrium condition).
– U = 0 when np = ni2.
– U is maximum when Et = Ei. That is, trapping levels near the mid-band are the most efficient recombination centers
( )
⎥⎦⎤
⎢⎣⎡ −
++
−=
kTEEnpn
npnNvUit
i
itth
cosh2
2σ(25)
Indirect Recombination - SRH
HO #2: ELEN 251 - Semiconductor Physics Page 27S. Saha
• Eq (25) shows that for Et → Ec, U↓ because:⇒ e− capture is more probable while hole capture is less probable.⇒ e− emission back to conduction band is highly probably.
• When Et = Ei , from (25) we get: U = vthσNt(pn - ni
2)/[n + p + 2ni] (26)
a) N-type: n >> p and n >> ni, then from (26)∴ τp = 1/(vthpσpNt) (27)
In N-type material, lots of e− are around for capture. Therefore, the limiting factor in recombination is the minority carrier lifetime τp.
b) P-type: p >> n and p >> ni
∴ τn = 1/(vthnσnNt) (28) Again, the limiting factor is the minority carrier lifetime.
Quasi-Fermi Levels
HO #2: ELEN 251 - Semiconductor Physics Page 28S. Saha
Under thermal equilibrium, we have: n = nie(EF − Ei)/kT
p = nie−(EF − Ei)/kT
In non-equilibrium conditions such as (1) injection (np > ni2) or
(2) extraction (np < ni2), we cannot use these relationships.
Here, we define two new quantities called quasi-Fermi levels such that the relationships of the type shown above hold and:
n = nie(EFn − Ei)/kT (29) p = nie−(EFp − Ei)/kT (30)
where EFn ≡ e− Quasi-Fermi level; EFp ≡ hole Quasi-Fermi level. EFn and EFp are the mathematical tools; their values are chosen
so that the “correct” carrier concentrations are given in non equilibrium situations.
In general, EFn ≠ EFp
Quasi-Fermi Levels: Example
HO #2: ELEN 251 - Semiconductor Physics Page 29S. Saha
Let us consider a p-type semiconductor with NA = 1016 cm-3, τ= 10 µsec, and we shine light on it such that GL = 1018 carriers cm-3 sec-1.
Given, po = NA = 1016 cm-3
τGL = (10x10-6 sec)(1018 carriers cm-3 sec-1) = 1013 carriers cm-3
∴ n = no + τGL = ni2/NA + τGL ≅ 1013 cm-3
p = po + τGL = (1016 + 1013) ≅ 1016 cm-3
Then from (29) and (30) we get:EFn = Ei + kTln(n/ni)
≅ 0.18 eV above Ei
EFp = Ei − kTln(p/ni)
≅ 0.36 eV below Ei
Ec
Ev
EFn
EFp
Ei
p ≅ 1016 cm-3
n ≅ 1013 cm-3
Quasi-Fermi Levels
HO #2: ELEN 251 - Semiconductor Physics Page 30S. Saha
Ec
Ev
EFn
EFp
Ei
p ≅ 1016 cm-3
n ≅ 1013 cm-3 In example, since p ≅ po, EFp is essentially unchanged while EFn moves up to reflect n >> no
enpn kTEE
iFpFn −
= 2 (31)Here,
In equilibrium: EFn = EFp = EF ⇒ pn = ni2.
However, under injection: EFn = moves up towards the CB. EFp = moves down towards the VB.
Thus, |EFn − EFp| is a measure of the degree of non-equilibrium in the semiconductor.
We will use this concept of Quasi-Fermi levels in our discussion of PN junctions.
Non-Uniform Doping Distributions
HO #2: ELEN 251 - Semiconductor Physics Page 31S. Saha
In our discussion of semiconductor statistics, we have assumed spatially uniform doping levels. Often this is not the case as we can see for the Bipolar Transistors.
xjBE xjBC
N+
N−P
Bipolar TransistorN
Depth
1 The impurity profile we have drawn represents the fixed positions of ionized donors and acceptors, i.e. ND
+, NA−.
2 The carriers (e− and holes) are mobile and attempt to diffuse away from regions of high concentrations.
Non-Uniform Doping: Built-in Electric Field
HO #2: ELEN 251 - Semiconductor Physics Page 32S. Saha
Now, consider a simpler situation where we only have N-type material (no junctions) but ND ≠ constant. Here,N
x
1018
1014
1010ND
+
1 diffusion flux is given by: F = Dn(dn/dx).
2 as e− move (drift) away, they leave behind + charged ions (ND
+) which try to pull e− back.
• Equilibrium is established when diffusion = drift. Here n(x) ≠ND(x) and a “built-in” electric field that exists is given by: Ex = − (kT/q)(1/n)[dn/dx] (32)
Similarly for holes, Ex = (kT/q)(1/p)[dp/dx] (33)
This built-in Ε−filed accelerates the minority carrier e− injected into the base from the emitter in their travel across the base or p-region.
Basic Semiconductor Equations
HO #2: ELEN 251 - Semiconductor Physics Page 33S. Saha
• To this point we have discussed:– Carrier transport − drift and diffusion– Carrier generation– Carrier recombination.
• The mechanism which ties all of these together is the condition of continuity of current.
Jp(x + ∆x)Jp(x)
∆x
U Rate of change ofnumber of holes
in ∆x
holesrecombining
Jp = hole currentU = recombination rate
From the conservation of charge,
net holesflowing
out of ∆x= +
Continuity Equations
HO #2: ELEN 251 - Semiconductor Physics Page 34S. Saha
p
oppUτ−
=
)(1)(1 xJq
xxJq
pp −∆+=
xdxdJ
qp
∆=1
From our discussion of recombination,
xpopxUp
∆−
=∆τ
∴ holes recombining =
Net holes flowing out of ∆x
Continuity Equations
HO #2: ELEN 251 - Semiconductor Physics Page 35S. Saha
Finally, the rate of change of holes within ∆x
(decrease due to recombination)xdtdp
∆−=
xppxdxdJ
qx
dtdp
p
op∆
−+∆=∆−∴
τ1
∴ dtdppp
dxdJ
q p
op−=
−+
τ1
(34)
Similarly,dtdnnn
dxdJ
q n
on−=
−+−
τ1
(35)
Eq. (34) & (35), called the continuity equations, are fundamentally important in the operation of semiconductor devices.
Drift-Diffusion Equations
HO #2: ELEN 251 - Semiconductor Physics Page 36S. Saha
The following equations that we previously derived are the transport equations (21) and (22) and are also fundamentally important in device operation.
)(dxdnDnqJ nnn += εµ
(36)
(37)
)(dxdnDpqJ ppp −= εµ
Poisson’s Equation
HO #2: ELEN 251 - Semiconductor Physics Page 37S. Saha
One final equation is needed to describe device operation and can be obtained from Poisson’s equation:
Where ε = electric field ρ = space charge density (charges/cm3) K = dielectric constant εo = permittivity of free space = 8.854x10-14 F/cm
Therefore,
If we know the charge density as f(x) we could find the electric field.
oKdxd
ερε
=
∫=oK
dxε
ρε
Poisson’s Equation
HO #2: ELEN 251 - Semiconductor Physics Page 38S. Saha
In general, the net charge density in a semiconductor is given by:
ρ = q[p+ + ND+ − n − NA]
(34) − (38) constitute a complete set of equations to describe carrier, current, and field distributions. Given appropriate boundary conditions, we can solve them for an arbitrary device structure. Generally, we will be able to simplify them based on physical approximations.
dxdV
−=ε e,Furthermor
( ) ( )[ ]DAo
NNpnKq
xdVd −+−=
ε22
that so (38)