Metal-Oxide-Semiconductor Field-Effect Transistors (MOSFETs)
• Introduction.
The figure below illustrates schematically the MOSFET structure (an n-channel MOSFET – or nFET - is
shown. p-channel devices – or pFETs – are doped in a complementary manner): Heavily n-doped source and
drain regions are separated by a p-type region. An insulating layer (the ‘oxide’, SiO)2 in the most common
implementation of the device) separates the p-type substrate from the third electrode, the ‘gate’, typically either
a metal or heavily-doped n-type polycrystalline Si.
When the gate is grounded or negatively biased, the p-substrate is accumulated. The application of a voltage
between the source (taken as grounded) and the drain (under a positive bias VDS ) contacts cannot result in
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any current, because of the built-in potential (possibly enhanced by a negative gate bias). If, however, we apply
a positive gate bias, VG , the effect is that of a forward bias applied to the source-substrate n-p junction. In
other words, as the p-type substrate is being inverted by the positive gate bias, a conductive ‘channel’ is formed
at the substrate-oxide interface and current flows from the source to the drain. A higher postive gate bias will
increase the electron density in the inversion layer, thus enhancing the source-drain current.
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It is this ability to control the conductivity of the channel with a third terminal, the gate, which renders this
device a resistor whose resistivity is modulated by the gate. As an analog device, it is an amplifyier, since a
small signal on the gate can be amplified by applying a large source-to-drain bias. As a digital device, it can
be toggled between conduction and no-conduction, thus performing the function of a switch (”0” or ”1” logical
states). The figure below shows the result of a Monte Carlo simulation: Electrons – indicated by little spheres
colored according to their kinetic energy – pool up in the source (at the right) and spill over into the channel,
dropping to the drain. The surface on which they move is the potential energy as seen from the ‘top’. The
potential barrier of the SiO2 insulator has been deleted from the figure.
• Drain current: Drift-diffusion analytic model.
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The figure above shows the potential energy for electrons in an n-channel MOSFET. The drain current is due to
electrons wich exit the source region and drift along the channel towards the drain at the right. Let’s consider
the x axis along the channel and the y axis pointing downward, the interface being located at y = 0. Let’s
start by viewing the channel as a resistor, and let V (x) be the potential at y = 0 at the point x along the
channel. The current density along the interface will be:
jnx (x, y = 0) = e µn n(x, y = 0) Fx = −eµn n(x, y = 0)
dV (x)
.
dx
The total inversion charge in the channel at position x along the channel will be
∞
Qinv (x) = − e
n(x, y) dy .
(532)
(533)
0
Thus, from these equations, the total drain current will be
∞
dV (x)
ID = −W
Jnx (y) dy = −W Qinv (x) µn
.
dx
0
(534)
This expression can be viewed as Ohm’s law: In a small element of the channel of length dx we have
dV (x) = ID dR(x) , with resistance dR(x) =
−dx
.
W µnQinv (x)
(535)
In order to estimate how the total inversion charge Qinv varies along the channel, let’s recall that V (x) = 0
near the source, while V (x) = VD , the drain voltage, near the drain. When the semicondcutor surface is in
strong inversion, we have ψs = 2ψB near the source and ψs = 2ψB + VD near the drain. Generalizing this
along the entire channel,
ψs (x) = 2 ψB + V (x) .
(536)
Now, since the charge in the gate at x will be the sum (with opposite sign) of the inversion and depletion
charges in the sustrate:
−QG (x) = Qd (x) + Qinv (x) ,
(537)
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and since the total voltage drop in the oxide is given by
Vox (x) = VG − [VF B + ψs (x)] = VG − [VF B + 2ψB + V (x)] =
QG (x)
,
Cox
(538)
we have from Eqns. (537) and (538):
Qinv (x) = −Cox {VG − [VF B + 2ψB + V (x)]} − Qd (x) .
(539)
Since the surface potential varies along the channel as described by Eq. (536) above, the width of the depletion
region also varies and so also the depletion charge (denoted here as WD to avoid confusion with the width of
the device, denoted by W ):
Qd (x) = −NA WD (x) = −{2es NA [2ψB + V (x)]}1/2 .
(540)
Integrating now Eq. (534) from source to drain:
L
0
ID dx = ID L = −W
V
D
dV (x)
Qinv (x) µn
Qinv (x) dV , (541)
= − W µn
dx
channel
VS
where L is the channel length. Using Eqns. (538) and (540) to express the inversion charge and recalling that
ID is constant along the channel:
V
D
W
ID = µn Cox
{VG − [VF B + 2ψB + V ] − (1/Cox ) [2es NA (2ψB + V )]1/2 } dV . (542)
L VS
Setting γ = (2es NA )1/2 /Cox and VDS = VD − VS we have:
W
VDS
2
ID = µnCox
VG − VF B − 2ψB − VS −
VDS − γ[(2ψB + VD )3/2 − (2ψB + VS )3/2
L
2
3
(543)
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10
VFB = 0, VG = 5.0 V,
tox = 10 nm, NA=1016 cm–3, µn = 200 cm2/Vs
8
ID (10–4 A)
W=L
saturation
6
ID,sat
4
2
VD,sat
0
0
2
4
6
VDS (V)
8
10
This equation describes the drain current correctly up to the drain bias at which ID reaches a maximum, that
is, dID /dVD = 0. This voltage,
2
VD,sat = VG − VF B − 2ψB +
2 1/2
γ
γ
− γ VG − VF B +
2
4
,
(544)
is called ‘saturation voltage’. When this happens, one can see that Qinv (x = L) = 0, that is, the inversion
layer is not formed at the drain-end of the channel. The analysis leading to Eq. (543) is not valid any longer: The
drain current does not decrease, as predicted by that equation, but remains pinned at its value at VD = VD,sat ,
value called ‘saturation current’ and indicated by ID,sat . At any drain bias above VD,sat the channel is said
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to be ‘pinched-off’ and the portion of the channel in which there is no inversion layer is called the ‘pinched-off
region’.
Note that even in pinch-off the channel remains conductive. A simple argument can be given why the current
must saturate: The voltage-drop across the pinched-off region of the channel is VD − VD,sat , while in the
rest of the channel the voltage drop is always VD,sat , irrespective of the drain bias applied. Therefore, the
conduction properties of the channel do not change and the current saturates.
The threshold voltage will be the total voltage (drop in the oxide + drop in the substrate) needed to set up strong
inversion, ψs = 2ψB . Since VG = ψs + Vox = ψs + QG /Cox , noticing that at the onset of strong inversion
most of the charge in the semiconductor will be due to ionized impurities, eNA WD,max = [2es NA 2ψB ]1/2 ,
and that this must be equal (and opposite) to QG , we have, also accounting for the shift VF B :
(4NAes ψB )1/2
VT 0 = VF B + 2ψB +
= VF B + 2ψB + γ(2ψB )1/2 ,
Cox
(545)
where the subscript ‘0‘ indicates that this is the threshold voltage when the source is grounded. When the
source is not grounded, we must shift this expression by VS (since it is the threshold at the source-end of the
channel which controls the flux of carries towards the drain):
VT S ≈ VF B + 2ψB + VS + γ [VS + 2ψB ]1/2 ,
(546)
• Drain current: Simplified model. The model developed so far can be simplified by linearizing the dependence
of the depletion region on the local surface potential V (x):
Qd (x)
[2s eNA (2 ψB + V (x))]1/2
−
=
≈ γ(2ψB )1/2 + δ V (x) ,
Cox
Cox
(547)
where δ is the linearization constant (equal to γ/[2(2ψB )1/2 ]). Now, from Eqns. (545) and (546):
VT S = VT 0 + VS + γ[(2ψB + VS )
1/2
− (2ψB )
1/2
] ≈ VT 0 + VS + δVS ,
(548)
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while from Eqns. (539) and (540) the inversion charge becomes:
Qinv (x) ≈ −Cox {VG − [VT 0 + (1 + δ)V (x)]} .
(549)
Let’s now define the ‘body factor’ n = 1 + δ , so that the expression above becomes:
Qinv (x) ≈ −Cox {VG − [VT 0 + nV (x)]} .
(550)
In order to extract the drain current in this approximation, we proceed as before, integrating the current from
source to drain:
L
V
D
ID dy = −W µn
Qinv (x) dV (x) ,
(551)
0
VS
so that, since ID does not depend on x by current continuity:
V
D
W
ID =
[VG − (VT 0 + nV )] dV .
µn Cox
L
VS
Re-writing the linearized version of Eq. (548) as VT S ≈ VT 0 + n VS , we have:
W
1
2
ID ≈
µn Cox (VG − VT S )VDS − nVDS .
L
2
(552)
(553)
Once more, this equation describes a parabolic dependence of ID on VDS , but it is physically meaningfull
only untill the channel is pinched-off. This occurs when VDS = VD,sat , where the saturation drain voltage –
already seen in Eq. (544) above – is now approximated as:
VD,sat ≈
1
(VG − VT S ) + VS .
n
(554)
The region for which Eq. (543) holds true (i.e., for VD ≤ VD,sat ) is called the ‘linear region’. The bias-region
VD > VD,sat is called the ‘saturation region’. Let’s consider separately these two regions.
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2
– In the linear region, let’s consider Eq. (543) for small VD (so that we can neglect terms of order VD
).
Setting VS = 0 so that VDS = VD we can write the drain current in the following form (see Eq. (553)
which is fully equivalent after having introduced the linearization parameter δ ):
W
1
(s eNA /ψB )1/2
2
VD ,
ID = µn Cox
(555)
(VG − VT 0 ) VD −
+
L
2
4Cox
or, ignoring terms non-linear in VD :
ID ≈ µn Cox
W
(VG − VT 0 ) VD ,
L
(556)
for VD << VG − VT 0 . The ‘threshold voltage’, VT = VF B + 2ψB + (2s e NA )1/2 /Cox , is also called
the ‘linear threshold voltage’, VT,lin ). It can be measured by plotting ID as a function of VG for small VD
and extrapolating the linear dependence at the VG -axis.
Two important parameters to estimate the performance of a MOSFET are the ‘channel (or output) conductance’
gD and the ‘transconductance’ gm :
gD =
W
∂ID
≈ µn Cox
(VG − VT 0 ) ,
∂VD
L
(557)
W
∂ID
≈ µn Cox
VD .
∂VG
L
– In the saturation region, substituting the value for VD,sat from Eq. (554) into Eq. (553) we get:
gm =
ID,sat ≈ µn Cox
W 1
(VG − VT,sat )2 .
L 2n
(558)
(559)
In principle, for low doping, the ‘saturation threshold voltage‘ VT,sat coincides with VT 0 = VT,lin. However,
at higher doping VT,sat becomes dependent on VG . We shall ignore these corrections. The output
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conductance and transconductance are now given by:
gD,sat ≈ 0 ,
and
gm,sat ≈ µnCox
(560)
W 1
(VG − VT s ) .
L n
(561)
• Velocity Saturation. We have to consider two corrections to the model developed so far: The first correction
deal with quantization effects in inversion layers and we shall deal with it below. The second correction
concerns electron-heating effects. We know that the electron mobility decreases as electrons become ‘hotter’
(see Eq. (296), page 86):
µn ≈
µn,th
(1 + ceµn,th F 2 /T )1/2
=
µn,th
(1 + F 2 /Fc2 )1/2
,
(562)
where we have defined a ‘critical field’ Fc2 = T 2 /(ecµn,th ). Therefore, in the saturated region, when the
large VDS applied will cause the carrier velocity to saturate, the drain current will be reduced with respect
to the value we have estimated ignoring velocity saturation. To see how this happens, note that in Eq. (562)
F = Fx = −dV (x)/dx. Thus, we can multiply both sides of Eq. (541) by (1 + F 2 /Fc2 )1/2 . In so doing
the righ-hand side will remain unchanged, but the left-hand side will become
L
0
ID
1 +
1/2
F 2
dx = Csat (VDS ) ID > ID ,
Fc
(563)
where Csat(VDS ) – a function of the applied bias – is always larger than 1, approaching unity only when VDS
approaches zero. Thus, under velocity saturation, the drain current will be reduced by a factor Csat(VDS ).
Besides this reduction of ID , velocity-saturation also implies a linear dependence of ID on VG , unlike the
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quadratic dependence predicted by Eq. (559). This dependence is of the form
ID,sat ≈ Cox W (VG − VT ) vsat .
(564)
Physically, this can be obtained from Eq. (556): When velocity saturation occurs we must replace µnVD /L in
that equation with the saturated velocity vsat . However, note that this is not a derivation of Eq. (564), since
Eq. (556) is valid only for small VDS . A ‘proof’ of Eq. (564) relies on more sophisticated arguments. In this
limit the transconductance becomes:
gm,sat =
∂ID
≈ Cox W vsat ,
∂VG
(565)
expression which replaces Eq. (561). Note that the transconductance per unit width divided by the oxide
capacitance, gm,sat /(Cox W ), has dimension of a velocity and can be interpreted as some average carriervelocity in the channel.
• Operation in sub-threshold.
So far we have considered the device above threshold, ignoring what happens for a gate bias which leaves the
interfacial region depleted (so, when the channel hasn’t been yet formed). Yet, the operation of the device under
these conditions is quite interesting for low-power applications because it is associated with the way the device
switches on or off. This is called the ‘subthreshold region’.
In subthreshold the electrons diffuse over the source/channel barrier, so that the drain current will be given by:
ID = eADn
n(0) − n(L)
,
L
(566)
where A is the cross-section across which the electrons flow,
n(0) = np0 eβψs
(567)
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and
n(L) = np0 eβψs −βVD ,
(568)
where, as usual, β = e/(kB T ). We can estimated the cross-section A by noticing that the electron
concentration drops exponentially away from the interface. Let’s assume that the concentration is a constant
up to a distance ykT away from the interface at which the potential has dropped by kB T /e, so that
ykT = kB T /(eFs ), where the surface field is Fs = eNA /CD . Then, recalling Einstein’s relation
Dn = µn (kB T /e), setting A = W ykT and recalling also that np0 = n2i /NA :
W
ID = e
L
kB T
e
2
n2i eβψs
βV
µn
[1 − e D ] .
NA Fs
(569)
Since ψs depends linearly on the gate bias VG in substhreshold, the drain current increases exponentially
with gate bias. It is customary to express this exponential behavior by the ‘inverse subthreshold slope’ (or
‘subthreshold swing’ or simply ’subthreshold slope’) S : By definition this is
dVG
ln(10)
=
.
d(ln ID )
d log ID
(570)
d(ln ID )
1 dID dψs
=
,
dVG
ID dψs dVG
(571)
S =
dVG
Since
and, from Eq. (569):
we have
dID
d
dψs −1
dψs
= ID β −
,
dψs
dψs
dy
dy


dψs
d
d ln ID
dψs
dψs
dy

= β −
.
dψs
dVG
dVG
(572)
(573)
dy
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Now recall that the depletion capacitance CD is just CD = dQd /dψs = s (dFs /dψs ), so that, since
Fs = dψs /dy :
d
CD
dψs
.
(574)
= −
dψs
dy
s
Also, from this equation and the fact that Fs = eNA /CD (by Gauss law) we have:
d
dψs
dψs
dy
dψs
dy
=
2
CD
=
eNA s
1
.
2ψs
(575)
Finally, inserting this equation into Eq. (573) and neglecting 1/(2ψs ) compared to β = 2/(kB T ), we have
S =
ln(10)
d(ln ID )
dVG
≈ ln(10)
kB T dψs
.
e dVG
(576)
Since VG = Vox + VF B + ψs = Qd /Cox + VF B + ψs , we have
dψs
1
=
,
CD
dVG
1+ C
(577)
ox
so that
kB T
S =
ln(10)
e
CD
1 +
Cox
= n
kB T
ln(10) ,
e
(578)
where n is the body factor. Note that, from the definition of the subthreshold slope S , Eq. (570), we have
d ln ID = dVG /S , so that
qVG
ID ∝ exp
(579)
,
nkB T
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where we recall that n = 1 + CD /Cox .
If interface states are present, then the associated capacitance Cit is in parallel with the depletion capacitance
CD and the subthreshold slope takes the form:
kB T
CD + Cit
S =
(580)
ln(10) 1 +
.
e
Cox
At room temperature S ≈ 60 meV/decade. This is the minimum inverse swing we can expect. Deviations
from ideality only worsen (that is, increase) this number, as shown by the effect of interface traps in Eq. (580).
Clearly, smaller inverse slopes (and so higher slopes 1/S and so a faster dependence of ID on VG ) are preferable
since they imply that the device will turn on and off more abruptly with gate drive.
• Surface mobility and transport in inversion layers.
We saw before that quantization effects in inversion layers modify the transport properties (see Lecture Notes,
Part 2, pp. 160-167). In this section we shall be a little more quantitative.
– Carrier mobility in inversion layers.
The Boltzmann Transport Equation (BTE) has been derived assuming that all quantities change slowly on
the length-scale of the electron wavelength. In inversion layers this assumption is clearly violated along
the direction normal to the interface (which we’ll assume is the z -direction). Indeed, this is the origin of
quantization effects in inversion layers. However, on the plane of the interface the electrostatic potential still
varies slowly (at least in sufficiently long devices). Therefore, having accounted for the confinement along the
z -axis, we can still describe transport on the (x, y) plane of the interface with a two-dimensional BTE.
The carrier mobility can be computed by linearizing the BTE for small electric fields, obtaining the KuboGreenwood expression we have seen before (see Lecture Notes, Part 1, pp. 79-80). There are two major
modifications: First, several subbands (or even ladders of subbands) will be populated, so that we must
consider separately the mobility µα,j for each subband j in ladder α. The total mobility will then be expressed
as the average of the mobilities in each subband weighted by the fractional occupation nα,j /ns of each
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subband:
µ =
nα,j
µα,j ,
ns
(581)
α,j
where nα,j is the carrier population in the subband (α, j) and ns is the total carrier sheet density
ns =
α,j nα,j . The second modification is caused by the two-dimensional nature of transport. The
mobility µα,j will be given by (assuming for simplicity the diagonal element (xx) of the mobility tensor):
e
µα,j =
mx,α,j kB T nα,j
∞
Eα,j
(p,x)
dE (E − Eα,j ) ρα,j (E) τα,j (E) fα,j (E)[1 − fα,j (E)] , (582)
where fα,j (E) = {1 + exp[(E − Eα,j − EF )/(kB T )]}−1 is the Fermi function for the subband j in
(d)
(d)
ladder α, Eα,j is the bottom of the subband, ρα,j (E) = mα,j /(πh̄2 ) is the density of states, mα,j the
(p,x)
DOS mass in the subband, and τα,j (E) is the relaxation time for the x-component of the momentum.
This expression is fully analogous to the 3D Kubo-Greenwood expression, Eq. (273) on page 81 of the Lecture
Notes, Part 1. Like in the bulk, 3D case, complications related to the anisotropic nature of the dispersion (that
is, the presence of different transverse and longitudinal effective masses on the (x, y) plane for the unprimed
valleyes in Si) have been ignored.
– Scattering processes.
Dealing with carrier mobility in bulk semiconductors we considered two major scattering processes: Scattering
with phonons (both acoustic and optical) and Colomb scattering with charged impurities (ionized dopants).
The same scatterers should be considered in inversion layers, with the addition of another scattering process
due to the presence of the Si-SiO2 interface, namely scattering with interfacial roughness (often called ‘surface
roughness’, SR).
The scattering rate for a carrier in subband j in ladder α (let’s use the Greek letters µ, ν , etc. to indicate
the pair of indices (α, j) and let’s also employ upper case symbols the 2D electron wavector K, scattering
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wavevector Q, etc.) can be evaluated with the 2D Fermi Golden Rule:
1
2π 2
|Hµν (Q)| δ[Eµ (K) − Eν (K + Q) ± ∆E(Q)] ,
=
τµν (K)
h̄
(583)
Q
where ∆E(Q) is the energy exchanged in the collision (phonon energy for phonon scattering, zero for elastic
processes such as Coulomb or SR scattering). The term
Hµν (Q) =
∞
0
∗
dz ψµ
(z) VQ (z) ψν (z)
(584)
is the ‘matrix element’ of the scattering potential V (r) = V (R, z) (where
1
VQ (z) =
(2π)2
e−iQ·R V (R, z)
(585)
is the 2D Fourier transform of the potential V ) between the initial state ψν (z)e−iK·R/A (where A is the
normalization area) and the final state ψµ (z)e−i(K+Q)·R/A. The momentum relaxation rate (along the x
axis, for example) will be:
2π Qx
2
|Hµν (Q)|
≈
δ[Eµ (K) − Eν (K + Q) ± ∆E(Q)] ,
(p,x)
h̄
K
x
τµν (K)
Q
1
(586)
which is just the expression for the scattering rate with the extra factor Qx /Kx = Kx − Kx /Kx =
1 − K cos φ/K , the fractional change of momentum along the x axis. In the case of scattering with
phonons an additional integration over the z -component of the phonon wavevector qz must be accounted for.
Phonon scattering.
Let’s consider in detail the case of scattering with acoutic phonons since the matrix element exhibits an
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interesting dependence on the strength of the quantum confinement. In this case the matrix element has the
form:
(ac)
Hµν
(q ) =
1/2
∞
h̄
∗
∆ac |q|
dR
dz ψµ
(z) eiK ·R e−iQ·R e−iqz z ψν (z) e−iK·R =
2ρωq
0
A
=
1/2
∞
h̄
∗
−iqz z
∆ac |q|δ(K − K − Q)
dz ψµ(z) e
ψν (z) .
2ρωq
0
(587)
Therefore, squaring and integrating over q as demanded by Eq. (584) or (586) (recall that the relaxation rate
equals the scattering rate for isotropic and elastic processes):
dQ
∞
∞
∆2ac kB T
(ac)
2
dqz |Hµν (q)| ≈
dqz
2ρc2s
−∞
−∞
∞
∗
−iqz z
dz ψµ (z) e
ψν (z)
0
2
, (588)
having used the elastic, equipartition approximation (nq ≈ kB T /(h̄ωq, the phonon energy ignored in the
energy-conserving delta-function) and having approximated h̄ωq as h̄cs q . Now let’s consider the integral in
Eq. (588). Let’s write it as:
∞
−∞
=
∞
dqz
0
∞
∞
0
∗
−iqz z
dz ψµ (z) e
ψν (z)
dz
0
∞
0
iq z ∗ dz ψµ (z ) e z ψν (z ) =
∗ ∗
dz ψν (z )ψν (z)ψµ (z)ψµ(z )
∞
−∞
−iqz (z−z )
dqz e
.
∞
Since −∞ dqz exp[−iqz (z − z )] = 2πδ(z − z ), the term above becomes:
∞
Fµν = 2π
dz |ψν (z)|2 |ψµ (z)|2 .
0
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(590)
186
Note that for a quantum well of width L this term behaves as L−2 . In inversion layers we can have an idea
of how this term behaves by using the Stern-Howard ground-state ‘variational wavefunction’
ψ0 (z) ≈
3
b
2
1/2
−bz/2
ze
.
(591)
The parameter b can be obtained by minimizing the expectation value of the energy over this wavefunction:
b=
3
1/3
∝ ns ,
z0
(592)
where z0 is the ‘centroid’ of ψ0 . Using this approximation, the momentum relaxation time for scattering with
acoustic phonons behaves like:
3mz b ∆2ac
−1
−1/3
≈
→
µ
∝
b
∝
n
.
ph
s
τ (p)
64h̄3 ρc2s
1
(593)
Similar expressions hold in the case of scattering with optical (inter-valley) phonons. Note that both in
the case of a quantum-well (Fµν ∝ 1/L2 ) as well as in the case of inversion layers (Fµν ∝ 1/z0 ), the
scattering (or momentum relaxation) rate increases with increasing confinement. In particular, in inversion
layers the phonon-limited component of the mobility, µph , behaves as:
−1/3
µph ∝ ns
.
(594)
Coulomb scattering.
The squared matrix element for Coulomb scattering with NC charged centers per unit area (ionized dopants
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in the semiconductor, interface traps, and oxide charges) has the form:
1/2
e2 NC
(C)
G̃µν (Q) ,
Hµν (Q) ∝
2
(595)
Q
where G̃µν (Q) is a more complicated matrix element involving the initial and final wavefunctions (of course)
and the Green’s function for Poisson equation in the semiconductor/insulator geometry. Dielectric screening,
somewhat more complicated than in bulk semiconductors, may be accounted for in a perhaps oversimplified
2
way by replacing Q2 in the denominator of Eq. (595) with Q2 + β2D
, where β2 is the 2D screening parameter.
Note that the component µC of the mobility limited by Coulomb scattering behaves as:
4/3
ns
µC ∝
.
NC
(596)
Surface roughness.
In the ’70s Ando proposed a simple model for the effect of surface roughness on the mobility. He assumed the
presence of steps at the interface, caused by the existence of ‘terraces’ in the semiconductor surface (see the
figure illustrating the qualitative atomic configuration of the Si surface with terraces).
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(100)–surface [110]–section
(1x2) TERRACE
(2x1) TERRACE
As seen in the figure below, each steps causes a ‘shift’ of the wavefunction. Mathematically this corresponds
to a change in the boundary conditions ψ(z = 0) = 0 → ψ̃[z = ∆(R)] = 0, where ∆(R) is the height
of the step occurring at the position R on the plane of the interface. The scattering rate is proportional to
the matrix element of the free-electron Hamiltonian H between the shifted and the ‘unshifted’ wavefunctions:
∞
(SR)
∗
Hµν
=
dr ψ̃µ(r) H ψν (r) .
(597)
0
It can be shown that – in the case of inversion layers –
(SR)
Hµν
≈
†
h̄2 dψµ dψν dR −i(K−K)·R
∆(R)
e
2π
2m dz dz ,
(598)
z=0
so that, introducing the 2D Fourier transform of the roughness, S(Q):
(SR)
Hµν
(Q) = S(Q)
†
h̄2 dψµ dψν 2m dz dz .
(599)
z=0
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The form of S(Q) to be employed can only be known either from experiments or from fitting procedures
(theoretical calculations of the structure of the interface are only now beginning to yield tentative results).
Ando assumed the only possible ‘reasonable’ form based on a ‘random’ correlation, a Gaussian auto-correlation
spectrum:
|∆(R)|2 =
dQ −iQ·R
|S(Q)|2 ,
e
2π
(600)
where:
|S(Q)|
2
2
2
= πΛ ∆ exp
2
2
Λ Q
4
.
(601)
The parameter Λ is the correlation length among steps (a sort of ‘average distance’ between adjacent steps),
while ∆ is root-mean-square (rms) height of the steps.
Note that carriers at the Fermi energy contribute mostly to the mobility. Therefore, SR-scattering will have
its most significant effect when the Fermi wavevector of the 2D carriers will approach the peak of Eq. (601),
roughly of the order of Λ−1 . Since Λ is usually short (of the order of a few nm), SR-scattering dominates at
high carrier densities. Indeed the SR-limited mobility decreases as the carrier density (and so the confinement)
grows:
µSR ∝ ∆
−2 −2 −2
Λ ns .
(602)
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∆(R)
SiO2
∆(R)
Si
There are other components of the SR scattering Hamiltonian which have been considered by Ando: They
arise from dipoles present at the steps and from their image charges. In addition, dielectric screening affects
the perturbing potential. Nevertheless, while important from a quantitative point of view, these Coulomb
terms do not alter the qualitative features of the SR scattering processes, which are well captured by Eq. (599).
Thus, we shall not discuss these additional terms.
Using Matthiessen’s rule and the qualitative bahavior of the phonon-limited, Eq. (594), Coulomb-scatteringlimited mobility, Eq. (596), and SR-limited mobility, Eq. (602), we obtain the qualitative behavior of the total
mobility (for electrons) shown in the figure below.
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3
2
µeff ( cm2 / V s )
1049
8
7
6
5
4
µSR
µC
(nD+ns)–2
ns1.3
3
2
ns–0.3
µph
1039
8
7
6
5
µtot
4
1011
1012
ns ( cm–2 )
1013
We see that the electron mobility depends on the electron density ns , and so both on the confining field Fs
as well as on the substrate doping (since this affects both the surface field, Fs , as well as the shape of the
potential in the depletion region).
About 20 years ago it observed that if one expresses the mobility as a function of what’s been called the
‘effective field’
Q + η|Qinv |
Fef f = d
,
(603)
s
(where η = 1/2 for Si n channels (electrons) and η = 1/3 for p channels (holes)), the mobility measured
in channels with various (uniform) substrate doping follows a ‘universal’ curve, shown in the figure below, as
long as Coulomb scattering with the dopants is not dominant. The ‘universal mobility curve’ is now widely
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used in the device-modeling community. But we should keep in mind that this picture lacks any rigorous
physical justification and deviations from the ’universal’ behavior can be seen in the case of susbstrates
with non-uniform doping (case always true in today’s VLSI technology demanding ‘retrograde doping’ for
short-channel devices, as we shall see).
1039
1039
4
4
3
3
8
7
6
5
µ (cm2/Vs)
µ (cm2/Vs)
8
7
6
5
2
1029
8
7
6
5
2
1029
8
7
6
5
4
4
3
3
2
2
101
1011
10
12
10
ns (cm–2)
13
10
14
NA = 1015, 3x1015, 1016,..., 3x1018 cm–3
101
104
105
106
Feff (V/cm)
107
– Effective mobility.
From the discussion above regarding the electron mobility in inversion layer, we see that a complication arises:
If we look back at Eq. (541), we see that we performed the integral over the channel by taking µn as a
constant. However, we have just seen that the electron mobility varies along the channel, since the confining
field Fs also varies along the channel, usually decreasing from source to drain. Thus, the integration performed
in Eq. (541) becomes impossible (unless one does it numerically). An approximated way to deal with this
problem analytically is to approximate the carrier mobility with empirical expressions, function of the ‘vertical
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field’ Fs , of the density, of the effective field Fef f , of the doping density, or more complicated expressions
functions of all of these variables. As an example, let’s consider the simplest possible case of a mobility
modulated by the effective field:
µn0
µn0
=
,
1 + KFef f (x)
1 + (K/s )[Qd (x) + |Qinv (x)|/2]
µn (x) =
(604)
where K is some fitting parameter with dimension of inverse field. We know that Qd + Qinv =
−Cox (VG − VF B − ψs ), so that |Qinv | = Cox (VG − VF B − ψs ) − Qd . Also, Qd = NA W =
1/2
(2es NA ψs )1/2 = γψs
µn(x) =
(see Eqns. (539) and (540) replacing 2ψB + V (x) with ψs here), so that
µn0
1 + [KCox /(2s )][γCox ψs (x)1/2 + VG − VF B − ψs (x)]
.
(605)
Let’s now go back to our starting point, the integration in Eq. (541). Before carrying out the integration, we
have in each infinitesimal element of length dx along the channel:
ID dx = W µn(x)Qinv (x) dV (x)
(606)
From Eq. (605) we have:
ID dx =
W Qinv (x)µn0
1 + [KCox/(2s )][γCox ψs (x)1/2 + VG − VF B − ψs (x)]
dV (x) .
(607)
Let’s rewrite it as:
ID {1+[KCox/(2s )][γCox ψs (x)
1/2
+VG −VF B −ψs (x)]} dx = W Qinv (x)µn0 dV (x) . (608)
Let’s define an ‘effective mobility’, constant along the channel, such that
ID dx = W Qinv (x)µef f dV (x) ,
(609)
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so that, inserting this expression for ID into Eq. (608) we get:
W Qinv (x)µef f
dV (x)
{1 + [KCox/(2s )][γCoxψs (x)1/2 + VG − VF B − ψs (x)]}dx =
dx
= W Qinv (x)µn0 dV (x) ,
(610)
which can be written as:
{1 + [KCox/(2s )][γCox ψs (x)
1/2
+ VG − VF B − ψs (x)]}dx =
µn0
dx .
µef f
(611)
Let’s now integrate along the channel:
L
0
{1 + [KCox /(2s )][γCox ψs (x)1/2 + VG − VF B − ψs (x)]}dx = L
µn0
.
µef f
(612)
Let’s recall that ψs (x) = 2ψB + V (x) and let’s assume a linear voltage drop along the channel, so that
dV /dx ≈ (VD − VS )/L. Then, integrating the left-hand side with a change of variable x → V we get:
2
− VS2
VD
2 γ 3/2
3/
(2ψB + VD )
(VG − VF B − 2ψB ) −
+
− (2ψB + VS )
2VDS
3 VDS
(613)
Let’s now assume that the source is grounded (so, VS = 0) and consider only the case of small VD , which
is appropriate when considering the low-field concept of ‘mobility’. Thus, we can ignore the third term in the
equation above, notice that the last term is approximately γ(2ψB )1/2 and we get, finally:
µn0
KCox
= 1+
µef f
2s
µef f ≈
µn0
,
ox [V − V
1/2 ]
1 + KC
+
2γ(2ψ
)
G
T0
B
2s
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195
recalling that VT 0 = VF B − 2ψB − γ(2ψB )1/2 . This shows that the effective mobility is depressed at
large gate bias, mainly because of the physical effects described above: The growth of the electron-phonon
matrix element with increasing confinement and the effect of SR-scattering.
• Short-channel effects.
Devices have been made smaller and smaller since the dawn of ICs in the ’70s because this ‘scaling’ affords
increased perfomances and reduced costs: The availability of smaller FETs mean one can pack more functionality
per unit area of Si wafers. Since the cost of Si ‘real estate’ has remained constant, this translates into reduced
cost per function. The availability of smaller devices also means improved performance of a single device, thus
increased speed of operation and, so, reduced cost per operation performed. However, as devices are scaled to
maller dimensions, several problems arise. We shall now discuss problems related to the degraded electrostatic
behavior of small devices (the proper ‘short-channel’ effects, namely VT -shift, channel-length modulation, and
the associated increase of the subthreshold current) as well as transport issues: Non equilibrium transport (which
renders concepts like µn and vsat meaningless, so undermining the derivation of the ID − VD characteristics
we have discussed so far) and hot-electron degradation effects. These have been a dominant concern in the last
decade, but their importance is waning (albeit not completely), as the supply voltage is being reduced in the
present VLSI technology. We shall later discuss how to circumvent (or, at least, minimize at the best of our
ability) these short-channel effects by following proper ‘scaling rules’ or, if this is not a technologically feasible
option, how to modify the structure of MOSFETs to achieve our goals.
– Electrostatic short-channel effects.
The figure in the net page illustrates the electrostatic problems we must face when shrinking the device:
At left we see a MOSFET at VDS = 0. The depletion regions due to the source, drain, and gate are
schematically sketched. Note that the area of the depletion region controlled by the gate has a trapezoidal
shape. For very long devices, one may assume that the ‘top’ and ‘bottom’ length of the depletion region, L
and L1 respectively, are approximately equal. But as L is reduced to dimensions approaching the width of
the source/drain depletion widths, we see that the area of the trapezoidal depletion region is reduced. Simple
geometrical arguments show that the ‘underhang’ x is given by
x = rj [(1 + 2WD,max /rj )1/2 − 1] ,
(615)
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where rj is the S/D junction depth. Thus, the total depletion charge controlled by the gate Qd will be
reduced from the long-channel value Qd0 = WD,max NA by the amount:
Qd
1
=
Qd0
2
rj
x
rj
L1
1+
L
= 1−
rj
L
[(1 + 2WD,max /rj )1/2 − 1] .
L
L
GATE
GATE
OXIDE
yS
WD,max
OXIDE
(616)
yD
WD,max
rj+WD,max
WS
L1
WD
L1
Since the threshold voltage depends on Qd (see Eq. (545) and discussion preceeding it), this reduction of
depletion charge will result in a reduction of VT 0 :
VT 0 = VF B + 2ψB +
eWD,max NA
Cox
{1 −
rj
L
[(1 + 2WD,max /rj )1/2 − 1]} .
(617)
This is a serious issue regarding yield: The ‘drawn channel length’ (that is, the length of the channel as
specified in the lithographic mask set) will always be translated into a ‘real’ channel length subject to statistical
processing errors (mask misalignement, variations in etching rates, fluctuations of doping profiles, etc.). The
best tolerance control thus puts a limit to the minimum L one can emply without sacrificing too many devices
with VT 0 outside the specification.
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The figure on the previous page illustrates another problem: The modulation of the channel length caused by
the drain bias: Comparing the drawing at left (VDS = 0) with the drawing at right (VDS = 0), we see that
as VD increases, so does the width of the depletion region at the drain, WD = [2es (2ψB + VD )/NA ]1/2 .
This, in turn, modifies both the threshold voltage (the quantity VT,sat mentioned above) as well as the
channel which, at the surface, shrinks by an amount ∆L(VD ) ≈ yS + yD . From any of the expressions
for the drain current considered above (such as Eq. (543) or Eq. (553)), replacing L with L + ∆L(VD ) we
see that the current will increase. In saturation (see Eq. (559) or Eq. (564)), this dependence of ID,sat on
∆L(VD ) will result in a nonzero output conductance gD . In most cases the ID − VD characteristics of
the device in saturation can be approximated by a linear saturation behavior, characteristics at different VG
converging to ID = 0 at a common voltage VA known as the ‘Early’ voltage (as shown in the figure below
in which we compare the ideal long-channel behavior - left – with the short-channel behavior – right). In this
case the output conductance in saturation will be approximately gD,sat = ID,sat /VA .
10
10
VFB = 0,
VFB = 0,
NA=1016 cm–3,
tox = 10 nm,
8
µn = 200 cm /Vs
6
VG = 5.0
4
µn = 200 cm /Vs
6
VG = 5.0
4
VG = 4.0
2
VG = 3.0
2
4
6
VDS (V)
VA
VG = 3.0
0
0
NA=1016 cm–3,
2
VG = 4.0
2
W=L
tox = 10 nm,
8
2
ID (10–4 A)
ID (10–4 A)
W=L
8
10
0
–10
–5
0
VDS (V)
5
10
An alternative way to look at the problem is to consider the effect of the drain bias on the height of the barrier
at the source-channel junction. This barrier controls the flow of electrons injected into the channel from the
source. Ideally, in an electrostatically long-channel device, only VG controls the height of this barrier. But
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as the width of the depletion region near the drain grows and approaches the source-region, ‘drain-induced
barrier lowering’ (DIBL) may occur. As VD grows, the barrier-height shrinks. This results in the undesired
non-zero output conductance of electrostatically short-channel devices.
When the channel becomes sufficiently small, the ‘effective channel length’ L − yS − yD may approach zero
at large VD . In this case the gate does not have any more control of the charge (and so, of the conductivity)
of the channel: Carriers may flow directly from the source depletion-region to the drain depletion-region
regardless of gate bias. This situation – in which the source and dran depletion-regions merge – is called
‘punch-through’. It is a catastrophic failure of the device, since it cannot be turned-off any more by the gate
bias.
Finally, as seen in the figure on page 196, the VD -induced modulation of the channel-length will also cause an
increase of the subthreshold current. Indeed, when the width yDS of the depletion region at the body-drain
becomes comparable to the channel-length L, Eq. (569) becomes:
W
ID = e
L − yS − yD
kB T
e
2
n2i eβψs
µn
[1 − eβVD ] .
NA Fs
(618)
Simply put, coupling this observation with the onset of punch-through, when the channel becomes too short
the device does not turn-off as well.
The three phenomena we have just discussed – VT shift, channel-length modulation (and punch-through),
and degradation of the subthreshold current – are purely electrostatic effects. As we shall see below discussing
scaling laws, they have to do mainly with the design of the device: A device may be made short while mantaining
‘long-channel electrostatic behavior’ if scaled properly. Conversely, relatively long devices may exhibit bad
short-channel problems if poorly designed. Thus, we may talk of ‘electrostatic long-channel or short-channel
behavior’. Dimitry Antoniadis of MIT has coined the term ‘well tempered MOSFET’ (paraphrasing J. S. Bach’s
‘well tempered clavier’ discovery in musical theory) to describe well-designed devices exhibiting electrostatic
long-channel behavior.
– Hot electron effects.
A second set of short-channel-related problems is related to transport issues. The first of such problem is
related to the length of the channel relative to the mean-free-path λ of the carriers in the channel itself. In
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this section we have made ample use of the drift-diffusion equatins (DDE) to derive approximate expressions
for the drain current ID . Let’s recall that we assumed implicitly that the (drift) velocity is a function of
the local electric field Fx = −dV /dx via vdrif t = µn Fx . In using the concept of mobility we have also
assumed that the electric fields are small enough to keep carriers at kinetic energies near the thermal value,
so that µn is a well-defined concept. (An exception to this approximation was discussed when dealing with
velocity saturation). All of these assumptions are valid when carriers scatter many times along the channel,
so that these many collisions keep them ‘cool’ and in local equilibrium with the field. Thus, we must require
L >> λ. When, on the contrary, the channel-length shrinks to such an extent that L ∼ λ, several problems
arise:
1. Off-equilibrium effects. With only few collisions occurring as the carriers transit along the channel, the
‘equilibrium’ concept of mobility ceases to be valid. Carrier entering the high-field region of the channel at
the source end, past the source/body barrier, will be accelerated to velocities larger than vsat in a time
shorter than the scattering time. Only later will they scatter, but their velocity will be given neither by
the mobility-field product nor by vsat . This ‘velocity overshoot effect’ had resulted in ‘effective velocities’
gm /(W Cox ) larger than vsat at low temperature (77 K) in 0.1 µm channel-length devices in experiments
performed at IBM and MIT as early as 1986. The results of these experiments is shown in the figure below.
Note how the transconductance in saturation increases as the channel length is reduced, in sharp contrast
with the predictions of Eq. (565). Only considering higher moments of the BTE or, better yet, obtaining
exact solutions of the BTE via Monte Carlo methods (used to obtain the ‘simulation’ result in the figure
above) one can account for these strong non-equilibrium effects.
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TRANSCONDUCTANCE (µS/µm )
1200
Experiments
(Sai–Halasz et al.)
Simulation
1000
800
77 K
600
400
300 K
200
0.00
0.10
0.20
0.30
METALLURGICAL CHANNEL LENGTH ( µm )
2. Electron heating. As a result of the small number of collisions in the channel, the carrier energy will increase
above the thermal value. In addition to invalidating the assumptions behind the DDE, the excess carrier
energy may cause real (as opposite to ‘theory-related’) problems: Electrons (which tend to get hotter more
easily than holes) can impact-ionize as they gain enough energy approaching the drain-end of the channel.
The generated holes can damage the Si-SiO2 interface. Similar damage can be caused by hot-electrons
hitting the interface. Or, for VD large enough (¿ 3.2 eV), electrons may be injected into the SiO2 conduction
band. The net result will be the generation of interface traps and/or oxide charge due to electron trapped
in the insulator. These charges will cause VT shifts (via VF B shifts we have considered before (see Lecture
Notes, Part 2, pp. 156-159) and/or degradation of the sub-thresold slope S (see Eq. (580). These effects
will change with time, as the devices operate, and will depend on the history of each device. Eventually, the
circuits will stop operating correctly, as a number of devices will fail to turn on or off at the specified gate
bias.
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Experimentally the presence of hot electrons in the channel can be monitored by measuring the substrate
current (due to holes generated by impact ionization and diffusing to substrate contact) and the gate current
(due to electrons injected into SiO2 and collected by the gate contact. The letter at first grows with
increasing VD at a given VG . But as VD grows larger, along an increasingly larger fraction of the channel
the surface electric field Fs decreases, thus preventing electron injection into the insulator. Usually, the
maximum gate current is observed for VD ≈ VG /2.
– Scaling laws.
So, having seen that short channel-length pose all sort of problems, how do we circumvent these difficulties in
our attempts to scale devices? The severity of hot-electron-degradation problems are clearly reduced as the
applied bias is reduced. For many years IC and system manufacturers had been reluctant to move away from
the ‘standard’ supply voltage of VDD = 5V. This depended manily on the practical considerations of retaining
the same power supplies without wasting precious Si-wafer area populated by voltage-reduction circuits. But
in the late-’80s the electric fields present in devices became untolerably high with serious-to-catastrophic hot
electron issues (even felt in Wall Street!). Thus VDD was reduced to 3.3 V. Once the standard of 5 V had
been abandoned, it was only a matter of time to see it dropping even more. Clearly, it will be driven as low as
noise and tolerance margins will allow (probably down to 0.6 V at 300 K, maybe less if low-T operation will
be considered).
Off-equilibrium effects are a problem for those trying to understand device operations, but do not hamper
device operation.
Electrostatic problems can be solved (to some extent, we shall discuss these limits) by reducing the linear
dimensions of the device while at the same time mainting the same aspect-ratio and reducing the junction
depth rj and the width of the depletion regions. Two basic strategies can be considered, requiring slighlty
different scaling criteria: Constant-voltage and constant-field scaling.
Constant-voltage scaling, as the name suggests, was of interest when the ‘holy value’ VDD = 5 V was
employed. We shall not discuss it, although historically it has dominated the scene for a couple of decades.
Constant-field scaling requires shrinking the device while at the same time reducing the applied bias by the
same factor, so that the electric fields inside the device remain roughly unchanged. Let’s call this scaling
factor λ−1 (called 1/κ by Fritz Gaensslen and Bob Dennard in their pioneering 1974 paper). So, for example,
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for λ = 21/2 , we want to reduce the channel-length by 40% and so the device area by a factor of 2, thus
increasing device-density on chip also by a factor of 2 (which, according to Moore’s law, happens every 18
months). The first two columns of the table below are taken from the literature (and the Colinge-Colinge text)
and tell us how to increase doping, reduce oxide thickness and junction depth in order to maintain long-channel
electrostatic behavior. They also tell us how the performance will be affected in the DDE-context, but it
should be taken with a grain of salt, given the approximate (to say the least!) validity of the DDEs for
small devices: For example, according to the DDEs, the current is expected to scale as λ, according to the
expression ID,sat ≈ (W/nL)µCox(VG − VT )2 . But we know that this is not necessarily true (at least
for channel-length longer than about 50 nm): From the figure on page 200 (experiments in which Cox was
kept constant rather than being scaled as λ) we expect ID /W ∼ gm /W ∝ λ, when accounting for an
increasing Cox ∝ λ and a compensating reduction of VDD , thanks to the reduction of the channel length
L, so that ID ∝ constant, as W is also reduced. Thus, in the third column we see what would happen if
the drain current remained constant as we scale the channel length. The gate capacitance is W LCox , so it
is reduced by a factor λ−1 . The power consumption goes as VDD ID . The gate delay is ∝ VDD Cox/ID .
The scaling of the doping concentration is a little tricky: A reduction of the depletion regions by a factor of
λ requires an increase of doping concentration by λ2 . But since built-in potentials do not scale and VT is
reduced with VDD , the issue is more subtle and a factor of λ shows how we attempt to reduce the width of
the depletion regions.
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Parameter
Scaling factor (DDE)
Scaling factor (BTE)
Dimensions L, W , tox
Applied voltage VDD
Threshold voltage VT
Current ID
Doping concentration NA , ND
Oxide capacitance Cox
Gate capacitance W LCox
Power density ID VDD /(W L)
Current density ID /(W L)
Power consumption ID VDD
Gate delay W LCox VDD /ID
2
Power-delay product W LCox VDD
Integration density (transistors/unit-area)
λ−1
λ−1
λ−1
λ−1
λ
λ
λ−1
1
λ
λ−2
λ−1
λ−3
λ2
λ−1
λ−1
λ−1
1
λ
λ
λ−1
λ
λ2
λ−1
λ−2
λ−3
λ−2
– Scaling limits.
If we could follow the recepies of the table above to the letter we would be able to scale MOSFETs endlessly.
Clearly, something is going to give at some length scale. The search for the ‘ultimate limits’ of device scaling
has always fascinated researchers worldwide. Often, papers have appeared in scientific journals forecasting
‘doomsday’ scenarios: The one-micrometer barrier will put a halt to scaling, then it was the 0.25 µm barrier,
then the 0.1µm. Now (having changed units) we argue about whether 80 or 40 nm will mark the end.
What limits scaling? Several factors. We list here only the most obvious ones:
1. Doping cannot be increased without limits: For every dopant impurity there is a ‘solid solubility’ limit for
the concentration. Above this limit no more impurities can go substitutional and so be electrically active.
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2.
3.
4.
5.
6.
7.
For n-type Si, this limit is around 1021 impurities per cm3 . Thus, depletion regions will not shrink forever.
Processing requires several high-temperature treatments. The total thermal history (usually referred to as
‘thermal budget’) causes dopant diffusion. This, in turn, results in less and less sharp junctions, so in
excessively wide depletion layers.
The energy gap and the thermal energy kB T cannot be scaled as VDD is reduced. This has a wide range
of effects. For example: The built-in potential between source and channel must be overcome, so we cannot
reduce endlessly VDD ; noise and tolerances issues put a realistic limit of about VDD ≈ 0.6 V for Si at
300 K.
As the doping increases, Zener tunneling at the drain-body junction will cause untolerable leakage.
The oxide thickness cannot be reduced below 1 nm (maybe even slightly more): The quantum-mechanical
tunneling current across the insulator causes unwanted power dissipation (and also loss of drain current, in
the limit of extremely thin oxides, as electron travel from the source to the gate instead of ending up in the
drain).
Fluctuations: Reducing the area W L of the device implies a reduction of the total number of dopants.
For L = 10 nm, W = 100 nm and NA Wdepl,max = 1013 cm−2 , we have only 100 impurities in the
√
channel. Thus we expect an rms fluctuation of 1/ 100 ≈ 10% on VT , which is unacceptably large. Also,
as the oxide thickness shrinks, thickness fluctuations of only a single atomic layer become relatively huge,
causing again VT fluctuations and mobility reduction.
Ultimately, the barrier present between source and drain below threshold could be penetrated by quantummechanical tunneling for small-enough L. In this case the device will not turn off (or it will do so at
unreasonable high VT ).
Each of these issues has been receiving considerable attention and there ideas about how to bypass them, at
least in part:
1. Novel device structures may reduce electrostatic short-channel effects: Silicon-on-insulator (SOI) and the
use of retrograde doping put a limit to the depth of the depletion region controlled by the gate, so that the
short-channel effects of on page 190 are reduced or eliminated. Double-gate devices constitute the ultimate
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2.
3.
4.
5.
6.
7.
implementation of these ideas and have been proven (theroretically) to be scalable at least down to L ∼ 30
nm, possibly even smaller.
Ion implantation (more precise and controllable then the diffusion processes originally employed) is now
standard, replacing gas-phase diffusion. Larger dopant ions (As instead of P for n doping, In instead B for
p doping) are now employed, to make use of their reduced diffusion constants. Rapid thermal annealing (or
‘spike’ anneals) are employed to reduce the thermal budget as much as possible.
Low-T operation can provide functionality at reduced VDD . Lower-gap materials will also exhibit smaller
built-in potentials (however, this is bad from the point of view of issue 4 below).
MOSFET structures with undoped channels (fully-depleted SOI, double-gate FETs) will reduce this problem
by reducing Vbi at the junctions.
Alternative insulators with higher dielectric constant (‘high-κ’ or ’high-k’ insulators) are being investigated as
possible substitutes for SiO2 : If ox is increased, we can increase the capacitance Cox = ox /tox without
reducing tox excessively. This will reduce significantly the tunneling-induced gate-leakage problem. Metal
gates (replacing poly-silicon gates, employed to simplify VT problems for pMOSFETs and nMOSFETs in the
CMOS technology) ) are being also considered to avoid the ‘poly-depletion’ problem: In inversion, the gate
will be depleted. This depletion effectively increases the oxide thickness and reduces the gate capacitance.
Double-gate and fully-depleted SOI FETs can be designed with undoped channels. Thus, dopant fluctuations
will be minimized. There is no solution for thickness fluctuations, though.
There is no known solution to this problem, which may be the ultimate stumbling block.
We could keep going discussing these issues, since each ‘solution’ presents its own drawbacks. We shall discuss
below in slightly more detail the alternatives to ‘conventional scaling’ which are being investigated at present.
• Terminal capacitances and other parasitic elements.
A note on resitivity and effective channel length: For an isotropic conductor, we define the conductivity σ such that j = σ F,
where j is the current density and F the electric field. The ‘resistivity ρ (measured in Ωcm) is defined as 1/σ . For a conductor (or
semiconductor) with mobility µ and carrier density n σ = enµ. If the conductor of length L has a cross section A = W d, then the
resistance of the conductor is R = ρL/A = (ρ/d)(L/W ) = Rsh L/W , where Rsh = ρ/d is called the ‘sheet resistance’ of the
conductor, often measured in Ω/square (since for a square L = W the resistance is independent of the size of the square).
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The channel length L we have used so far is a vaguely defined quantity: We can talk of the gate length LG , of the ‘metallurgical
channel length Lmet , or of the actual (‘effective’) channel length Lef f = L − yS − yD . A correct way to define it would be to
see how long is the region along the channel over which the electron (for n channels) or hole (for p channels) concentration deviates
significantly from the equilibrium unbiased situation.
The characteristics of an ideal devices do not tell us the whole story about the performance of a device. In
general, devices have to charge ‘loads’ (capacitive, resistive, inductive, or mixed). Typically the load is dominated
by ‘wires’ (that is, interconnects among devices on the chip) or, more often by another gate. Thus, as devices
switch, the speed of operation is controlled by the amount of charges which must be moved to fill the channel,
charge the gate, modify the width of depletion regions. Thus it is important to know the capacitances associated
with various junctions in a MOSFET. In addition, the source and drain regions will be connected to the S/D
contacts via a ‘series resistance’ due to the finite resistivity of the source and drain regions themselves.
Regarding the devices capacitances, the figure above, left frame, illustrates the various components:
– Cj , the ‘junction (or diffusion) capacitance’, exists at the S/D junctions when the transistor is on. The
depletion capacitance CD can be thought as part of Cj , but its contribution is small. Cj is given by the
usual expression:
s
es NA
Cj =
=
(619)
,
WDj
2(Vbi + Vj )
where Vbi is the built-in potential of the junction, WDj is the width of the depletion region at the junction
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and Vj is the voltage applied across the junction (so, VS or VD ). The total capacitance CJ will be
CJ = W dCj ,
(620)
where d is the diffusion width.
– Cov is the ‘overlap capacitance’. The right frame of the figure above shows its components: 1. A ‘direct
overlap’ component
Cdo = W lov Cox = ox W lov /tox ,
(621)
where lov is an equivalent overlap length; 2. An ‘outer fringe’ component
2ox W
Cof =
ln
π
1 +
tgate
;
tox
(622)
3. An ‘inner fringe component
2s W
Cif =
ln
π
1 +
xj
tox
,
(623)
which exists only below threshold (VG < VT ) since when the channel is formed the 2DEG screens the gate
field.
To analyze the series resistance, let’s consider the source side with reference to the figure below.
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Looking at the path of the current from the source contact to the channel, the resistance can be subdivided in
the following form:
Rsource = Rac + Rsp + Rsh + Rco ,
(624)
where:
– Rac is the ‘accumulation resistance’, that is, the resistance of the accumulation layer in the overlap region. It
depends on VG and, thus, it is usually lumped into the channel resistance.
– Rsp is the ‘spreading resistance’ given by:
2ρj
xj
Rsp ≈
ln 0.75
(625)
,
πWS
xc
where ρj = 1/(enS µS ) is the resistivity of the source region with depletion depth WS characterized by
a (bulk) electron density nS = ND with mobility µS . Typically xj /xc ≈ 40 and Rsp ≈ 2ρj /WS .
Since the doping concentration is never uniform, this equation is just a rough approximation. The carries will
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follow a spreading path such that Racc + Rsp is minimized. Only 2D transport simulation can accurately
estimate the values of these resistances in a 2D highly-inhomogeneous situation. For abrupt junctions, the
injection point is close to the metallurgical end of the channel, so that the ‘effective channel length’ Lef f is
approximately Lmet . For the more realistic graded horizontal profiles, the injection moves ‘to the left’, so
that Lef f > Lmet . These are the dominant components of the total series resistance.
– Rsh is the ‘sheet resistance of the source/drain diffusions, given by
Rsh ≈ ρSD
S
,
WS
(626)
where ρSD is the sheet resistivity of the source/drain regions. Rsh is typically negligible compared to the
channel resistance, provided the length S of the source diffusion is not excessively large.
– Rco the ‘contact resistance’ approximately given by:
(ρSD ρc )1/2
ρSD 1/2
,
Rco ≈
coth lc
(627)
WS
ρc
where lc is the width of the contact window and ρc is the interfacial contact resistivity. Note that for a short
contact (lc << (ρc /ρSD )1/2 ) we have:
(ρSD ρc )1/2 −1
Rco ≈
lc
WS
ρc
ρSD −1/2
=
.
ρc
W lc
(628)
For a long contact (lc >> (ρc /ρSD )1/2 ), instead:
(ρSD ρc )1/2
Rco ≈
,
WS
(629)
independent of lc since the current flows mainly at the edges of the contact.
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A similar analysis can be done for the drain resistance. The experimentally measured characteristics (ID − VD ,
ID − VG , gm , etc.) are called ’extrinsic characteristics’. When corrected for parasitic effects (mainly series
resistances for dc characteristics) they are called ‘intrinsic’. Typically, RS and RD are extracted by obtaining
the extrinsic channel resistance (proportional to L) as a function of channel length for a given technology.
Extrapolating to L = 0 one obtains the series resistance (assuming that nothing else changes as L is varied...)
• Various MOSFET structures and some advanced concepts.
GATE
GATE
OXIDE
OXIDE
low NA
high NA
BURIED OXIDE
RETROGRADE DOPING
SOI
GATE
GATE
OXIDE
OXIDE
BURIED OXIDE
OXIDE
BOTTOM GATE
FULLY–DEPLETED SOI
DOUBLE GATE
The figure above shows some MOSFET structures designed in order to minimize short-channel effects. All of
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these designs put a limit to the maximum depth of the depletion region, thus pushing the ‘threshold-voltage
roll-off’ to smaller channel lengths. Retrograde substrate doping achieves this goal with very heavy doping
of the substrate. Since this damages transport via Coulomb scattering, the doping profile is minimized near
the semiconductor/insulator interface, but is increased at an optimum depth. However, there are limits to the
maximum doping gradient which can be mantained during the thermal cycles. An insulating layer at the bottom
(silicon-on-insulator, SOI) also achieves the same effects, while also reducing the junction capacitances CJ .
Fully depleted SOIs (also called ‘thin body’ or ’thin Si’ SOIs) push this idea to the limit and have the advantage
of allowing an undoped substrate (provided some other way of adjusting VT is found), thus improving carrier
transport. The ‘ultimate’ device design is the ‘double gate’ FET (DG MOSFET): A bottom gate not only
minimizes short-channel effects, but it can provide higher current (at the expense of a larger gate capacintance)
and, if the two gates are controlled independently, dynamic VT control. DG FETs come in the ‘planar’ design
illustrated (hard to fabricate) or in the FINFET (Stanford U) or Trigate (Intel) versions.
In the search for even faster devices, as conventional scaling becomes harder, not only alternative device design
but also alternative materials are being considered. We have seen high-κ insulators as possible replacements for
SiO2 . Strained Si, SiGe alloys, or even III-V compund semiconductors are being considered. In class we will have
a brief discussion of these attempts.
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Other FET structures.
In addition to the MOSFET, it is worth considering – at least briefly – two other types of field-effect
transistors: The ’Junction FET’ or JFET, the ‘MEtal-Semiconductor FET’ or MESFET, and the ‘High ElectronMobility Transistor’, or HEMT (also called ‘Modulation-Doped FET’, or MODFET).
• JFETs.
The JFET (illustrated above in the original drawing by Shockley having Shottky barriers at the
gate/semiconductor interface; often, as we shall consider here, thin p+ regions are added under the gate
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contacts) is essentially a resistor (n-type in the figure) whose resistivity is controlled by ‘squeezing’ the cross
section of channel by modulating –via an external bias applied to top/bottom gate contact – the width of the
depletion regions of the p+ -n junctions.
When the gate is grounded and a small gate bias, VD , is applied, the drain current will be given by the expression
valid for a resistor of length L, cross section (2a − 2xdepl )W , where W is the device width, and conductivity
eND µn:
2(a − ydepl )W
ID = eND µn
(630)
VD .
L
The width of the depletion region at equilibrium is, as usual,
ydepl =
2s Vbi 1/2
,
eND
(631)
where Vbi = (kB T /e) ln(NA ND /n2i ) is the built-in potential of the p+ -n-regions under the gates (or of
the Shottky contacts, if metal gates are put in direct contact with the channel). Applying now a negative gate
bias, VG < 0, the depletion width grows to
ydepl =
2s (Vbi − VG ) 1/2
,
eND
(632)
so that the cross section of the channel shrinks and the drain current drops according to Eq. (630). The channel
will by completely ‘squeezed’ when ydepl = a, or for a gate bias (which we’ll call the ‘thrshold voltage’):
eND a2
VT = Vbi −
.
2s
(633)
If we now apply a larger drain bias, we cannot ignore anymore the variation of the depletion-width along the
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channel (along, say, the x axis). Then:
ydepl (x) =
2s [Vbi − VG + V (x)] 1/2
.
eND
(634)
We can now proceed more or less following the path we have followed dealing with the drain current of a
MOSFET: In a small element of the channel of length dx we have a resistance
dR(x) =
dx
.
2eµn ND [a − ydepl (x)]W
(635)
Then by Ohm’s law the voltage drop dV (x) across this channel-element will be dV (x) = ID dR(x).
Integrating along the entire length of the channel and recalling that ID is independent of x we have:
1/2 V
D
2s
,
ID L = 2eµnND W
dV
a−
(Vbi − VG + V )
(636)
eND
VS
so that, for VS = 0:
ID = 2eµn ND
W
a
L
VD −
2
2s
3 eND a2
1/2 (Vbi − VG + VD )3/2 − (Vbi − VG )3/2
.
(637)
Note that when the drain voltage increase beyond the value such that ydepl (L) = a, the channel becomes
‘pinched-off’. This occurs for:
eND a2
VD,sat =
− (Vbi − VG ) = VG − VT .
2s
(638)
For VD > VD,sat the current, similarly to what happens in a MOSFET, saturates as well. Inserting the value
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for VD,sat of Eq. (638) into Eq. (637) we have:
ID,sat = 2eµn ND
W
a
L
2
2
eND a
− (Vbi − VG ) +
6s
3
2s
eND a2
1/2
(Vbi − VG )3/2
.
(639)
As we saw for the MOSFET, the output conductance in saturation is zero, according to this model. In practice,
this is never the case, mostly because the presence of series resistances. Also in saturation the transconductance
is
1/2
W
2s
1/2
..
gm,sat = 2eµn ND
(V
−
V
)
(640)
a 1+
G
bi
L
eND a2
The performance of JFETs (as well as MESFETs below) is limited by the fact that the gate has less control
over the channel-conductance than in a MOSFET. In the latter, the inversion charge is very close to the gate
contact, resulting in a large gate capacitance ≈ Cox = ox /tox . On the contrary, in JFETs the conductance
is modulated via the depletion width, which means that the gate must control charges farther away. This results
often in a lower dc or large-signal performance.
• MESFETs.
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The MESFET can be viewed as a JFET longitudinally split in half. As in a JFET, the gate controls the
conductance of the channel by modulating the width of the depletion region, and so of the channel.
Usually MESFETs are fabricated on compound III-V semiconductors (GaAs, AlInAs, etc.), since it has been
historically very difficult to grow or deposit a good insulator on these materials and Shottky contacts are the
preferred way to go. Because of the high low-field mobility of these semiconductors, MESFETs have been
traditionally preferred microwave devices, thanks to their low-signal speed. For large signal, the poor control of
the channel-charge by the gate (as we saw above for JFETs) and the need to move a large amount of charge
in-and-out of the depletion region hampers their performance.
The operation of a MESFET is very similar to the operation of a JFET: If Vbi is now the Shottky potential, in
analogy with Eq. (635) we can write:
dV = ID dR =
so that:
ID L = eµn ND W
V
D
VS
ID dx
.
eµn ND [a − ydepl (x)]W
dV
(641)
1/2 2s
,
a−
(Vbi − VG + V )
eND
(642)
or
W
ID = eµn ND
a
L
2
VD −
3
1/2
2s
3/2
3/2
.
(Vbi − VG + VD )
− (Vbi − VG )
2
eND a
(643)
Pinch-off occurs for VD > VD,sat where, exactly as in Eq. (638):
eND a2
VD,sat =
− (Vbi − VG ) = VG − VT .
2s
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(644)
217
In saturation we have:
ID,sat = eµn ND
W
a
L
2
2
eND a
− (Vbi − VG ) +
6s
3
2s
eND a2
1/2
(Vbi − VG )3/2
..
(645)
The transconductance is given by an expression identical to Eq. (640).
BAND EDGE ALIGNMENT TO Au ( eV )
2.0
1.5
GaP
AlAs
AlSb
1.0 AlP
Si
GaAs
Ge
0.5
Ev
Ec
Ec
Ec
(Γ)
(Γ)
(X)
(L)
GaSb
InP
InSb
GaSb
0.0
Ge
Si
–0.5
InSb
InAs
GaAs
GaP
InAs
–1.0
AlSb
InP
AlAs
–1.5
5.4
AlP
5.6
5.8
6.0
6.2
LATTICE CONSTANT ( )
6.4
6.6
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• HEMTs (or MODFETs).
Before introducing HEMTs, we must recall what hetero-structures are. We have seen previously – paged 128-135 of Lecture Notes,
Part2 – what heterojunctions are. Typically, a III-V semicondcutor can be alloyed to form a ‘ternary compound’ – such as Alx Ga1−x As.
This can be viewed a GaAs lattice in which a fraction x (the ‘mole fraction’) of Ga ions is replaced substitutionally by Al ions. The
structure of the lattice remains qualitatively unchanged. Its properties – lattice constant, band-gap, deformation potential, etc. – can
be linearly interpolated between AlAs and GaAs, at least in the simplest approximation called the ’virtual crystal approximation’, in
which the distribution of Al and Ga ions is seen as a regular sub-lattice, without fluctuations.
The attractive features of heterojunctions – and structures based on them, called ‘heterostructures’ – is the ability to control them
during growth at the level of single atomic layers via molecular-beam epitaxy – and the possibility of designing almost arbitray
configurations of band discontinuities – the so-called ‘band-gap engineering’. The figure above shows how the conduction and valence
bands of many cubic semiconductors line up as a function of their lattice constant.
HEMTs or MODFETs are basically MOSFETs based on III-V semiconductor devices. Two are the major
differences between HEMTs and Si MOSFETs:
1. Because of the lack of a good insulator for III-V materials (as we have mentioned above), the insulator
(typically, SiO2 ) is replaced by a larger band-gap III-V material. For example, Alx Ga1−x As on GaAs. This
material acts as a sort of insulator – albeit with a reduced barrier.
2. A high carrier mobility is obtained by leaving the channel undoped. Electrons are induced in the channel by a
method called ‘modulation doping’: During the growth of the AlxGa1−x As layer (usually carefully controlled
at the mono-layer level using molecular-beam epitaxy, MBE), the growth is interrupted, a monolayer of dopant
is deposited, and the growth of Alx Ga1−x As is resumed. This thin sheet of dopants (called ’delta-doping’,
since its profile approximates a Dirac delta-function) causes an excess of electrons in the otherwise undoped
layer. Because of the potential barrier at the AlxGa1−x As -GaAs interface, these electrons spill over into
the GaAs channel (as we have seen when considering hetero-junctions at equilibrium) and constitute the
conductive charge in the channel. The fact that the channel remains undoped enhances the electron transport
properties, since Coulomb scattering is highly reduced. Only scattering with the ‘remote’ ionized impurities
remains.
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These devices operate very much like a MOSFET, but have extremely large mobilities (up to 1.5 × 106 cm2 /Vs
at a temperature of 4.2 K, 8,000 at room temperature for GaAs channels) and are well suited to high-frequency
applications. (Here at UMass Sigfrid Yngvesson has used the properties of GaAs 2DEG to design ballistic
hot-electron bolometers). The high quality of MBE-grown interfaces also minimizes surface-roughness and the
related mobility reduction. Their major drawback is due to the low values of the semiconductor-‘insulator’
interfacial barrier, typically only a few tenths of an eV. The large gate current prevents large signal applications.
Also, an additional scattering process is present in alloyed channels (as in Inx Ga1−x As channels): The virtual
crystal approximation ignores fluctuations of the distributions of In and Ga ions. But these fluctuations, always
present, cause the electrons to feel a random potential due to the substitution of Ga ions by In ions at random
locations. This causes an additional scattering process whose rate is proportional to ∆V x(1 − x)ρ(E), where
ρ(E) is the DOS at energy E and ∆V is an average difference between the ionic potentials of In and Ga (or,
in other words, the conduction-band discontinuity between InAs and GaAs). For alloys with high mole fractions
(x ≈ 0.5), this process may be significant in depressing the mobility.
The figure on the next page shows schematically the HEMT structure and the band-edge configuration.
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• Frequency response of FETs
Let’s now briefly mention two figure-of-merits usually employed to gauge the frequency response of MESFETs.
A similar characterization may also be made regarding MOSFETs.
The fundamental characteristic time scale of intrinsic devices (that is, ignoring parasitic effects) is the ‘transit
time’ of carriers across the channel. Depending on whether we are in the linear region (velocity controlled by
the mobility µ times the drain-to-source field F ) or in velocity saturation, this is given by:
L
L2
τ =
,
≈
µF
µVD
(646)
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and
τ =
L
,
(647)
vsat
respectively. In saturation most semiconductors have vsat ∼ 107 cm/s, so that τ ∼ 10−11 s for L ≈ 1µm.
Therefore, the maximum frequency at which the device is able to repond to the ac signal by filling and emptying
the channel is:
1
vave
gm
fT =
,
(648)
=
=
2πτ
2πL
2πCGS
recalling that gm /CGS ≈ gm /Cox is an average carrier velocity in the channel. Clearly, in saturation
vave ≈ vsat . The frequency fT is called the ‘threshold or cut-off frequency’. Also commonly used is the
maximum frequency of the oscillation:
fmax =
2
fT
r1 + fT τ3
,
(649)
where r1 = (RG + Ri + RS )/Rch is the ratio between the input resistance (Ri = ∂IG /∂VG is the
gate input resistance, usually extremely large; RG is the resistance of the gate contact; RS is the S/D series
resistance; Rch , usually low when the device is on, is the channel resistance) and τ3 = 2πRG CGD is yet
another time constant. (The definitions of fT and fmax are given in terms of unit maximum gain and unilateral
gain, respectively, for forward power amplifiers.) Note that a reduction of gate-length L results in both a
reduction of the transit time as well as an increase of transconductance.
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Bipolar Devices
So far the transistors we have considered have been field-effect devices in which the currents of interest were
carrier by only one type of carriers: Electrons in n-channel devices, holes in p-channels. They were ‘unipolar’
devices. Now we shall consider transistors in which the relevant currents are carried by both electrons and holes,
‘bipolar’ transistors. We shall consider ‘bipolar junction transistors’ (BJTs) and ‘heterojunction bipolar transistors’
(HBTs).
• Bipolar Junction Transistors.
BJTs are the first type of transistors actually made, in 1947, by Bardeen, Shockley, and Brattain, at Bell
Laboratories. (For this they received the Nobel Prize in 1956.) The first transistor, a rather gigantic discrete
device by today’s standard, was essentially a pair of p-n junctions butted together. The essence of BJTs has
not changed since. They have been – and still are – the preferred devices for high-power applications, thanks to
their ability to carry large currents at high voltage. However, up untill the late 1980’s they were also used in the
logic circuits of large mainframe computers. It was only with the advent of CMOS technology and because of
the incredibly fast performance-growth of MOSFETs that it became economically and financially preferable to
use CMOS technology also in logic applications (read: computers). Also, BJTs are never fully ‘off’ devices, as
MOSFETs. They switch from ‘strongly on’ to ’less strongly on’. The high power dissipated by BJTs, compared
to MOSFETs, made them unstuitable for very large scale integration at the quarter-micron scale.
The principles of operation of BJTs are a little less intuitive than that of MOSFETs. Indeed MOSFETs were
conceived already in the early 1930’s. The first MOSFET was fabricated only some 30 years later because of the
difficulty in producing reliably a good gate insulator. BJTs, not having to depend on high-quality SiO2 , came
first, as we said above.
The first BJT of Bardeen, Shockley, and Brattain was a discrete device. In its LSI version, a ‘planar’ n-p-n BJT
is illustrated in the figure below (at left). The n-p-n doping (at right) of the two back-to-back junctions is also
shown. These figures show n-p-n BJTs. BJTs can also fabricated in a complementary arrangement, p-n-p.
We shall now consider in detail an n-p-n devices. Obviously, p-n-p will behave in a fully symmetric way.
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– Current gain.
The figure above shows the basic band-diagram of an n-p-n BJT. We now deal with the basic current-voltage
relationship of the BJT following the discussion of an ideal p-n junction (see Lecture Notes, Part 2, pages
105 and ff.). Looking at this figure, the n region at left is called the ‘emitter’, the central, narrower p
region is called ‘base’, the n region at right is the ‘collector’. Often, a heavily doped n+ region at the far
right is called ‘subcollector’. Basically, applying a bias VBE between the emitter and the collector, if the
emitter-base junction is unbiased we prevent the electrons from flowing from left to right. The device is ‘off’,
as for a MOSFET below threshold. If we now apply a positive bias, VBE , at the base, we forward-bias
the emitter-base junction and lower the barrier in the central base region and allow flow of electrons into
it. Simultaneously, holes flow from the base to the emitter. If both emitter and base are long – that
is, Lbase > Ln = (DBn τBn )1/2 and Lemitter > Lp = (DEp τEp )1/2 (where DBn , τBn , DEp ,
and τEp are the electron and hole diffusion constants and recombination lifetimes in the base and emitter,
respectively), then electrons will recombine with holes in the base, holes with electrons in the emitter and we
will have just a base and emitter current. If, however, the base is short enough, not all electrons will recombine
in the base and some may make it across it. Most modern BJTs are such that a vast majority of electrons
actually cross the base and enter the collector. In this case we see that the higher VBE will be, the higher
the flux of electrons from emitter to collector will be (and so the current ICE from collector to emitter will
increase). A negative (reverse) bias across the base-collector junction will make sure that electrons are swept
into the collector by a large field. The bias situation VBE > 0 and VBC < 0 is called ‘forward operation’
and it the commonly used mode of operation of a BJT. In this mode of operation the base bias, in a way,
behaves like the gate bias of a MOSFET. The major qualitative difference is that, unlike a ‘well-tempered
MOSFET’ which has negligible or no gate current, in the BJT there will be a significant hole current from
the base to the emitter. Since this current will be generally much smaller then the electron current ICE , we
can have a large ‘current gain’ IC /IB (where IB and IC are the total base and collector current. Also, IC
can be very large indeed. Hence, both the speed and power-handling capabilities of a BJT. From the usual
convention about the sign of the current (along the flow of positive charge, opposite to the electron flow, and
ECE609 Spring 2010
225
current being defined as positive when entering the contact), we have
IC + IB + IE = 0 .
(650)
Since BJTs are designed so that IB << IC , the so-called ‘common-base gain’ αF (the current gain when
the base is grounded) is defined via:
IC IC = −αF IE , that is αF = −
IE V
,
(651)
BC =0
or
IC =
αF
I B = βF I B ,
1 − αF
(652)
where βF – also denoted by hF E , where F stands for ‘forward’ and E for common-emitter – is called the
(forward) ‘common-emitter gain’ (the current gain with emitter grounded). The common-base current-gain
αF – often denoted as hF B , where F refers to ‘forward’ and B to common-base – can also be expressed in
differential form as:
∂IC αF = hF B =
∂IE V
∂IEn ∂ICn ∂IC =
= γF αT M ,
∂I
∂I
∂I
E
En
Cn
VBC =0
BC =0
(653)
where γF is called the ‘emitter efficiency’, αT the ‘base transport factor’, and M the ‘collector multiplication
factor’. We shall discuss below (page 237 ff.) in turn each of these factors.
ECE609 Spring 2010
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– The BJT as an amplifier.
To see how an n-p-n bipolar transistor may be used as an amplifier, let’s look at the diagram above: Consider
a voltage source applying an input Vin between base and emitter. The output voltage Vout is measured
between collector and emitter. Looking at the figure above we obviously have:
Vin = RS IB + VBE ,
(654)
VCC = RLIC + VCE .
(655)
and
Since IC = βF IB , we have:
Vout = VCE = VCC − βF
RL
(Vin − VBE ) .
RS
(656)
For a small variation of the input voltage, VBE will not change significantly, since the voltage across a forwardbiased p-n junction (the base-emitter junction) is almost constant (the curent grows exponentially, so a large
ECE609 Spring 2010
227
change in the current causes a negligibe change of voltage). For Si diodes, VBE ≈ 0.7 V. Therefore, we see
that a small variation δVin of the input voltage is converted into a variation δVout = −βF (RL/RS )δVin ,
so it is amplified by a factor βF (RL/RS ) with a phase-shift of π.
– Basic current-voltage relationship.
Let’s get into some detail following a model proposed by Ebers and Moll. This model is based on the same
set of assumptions we have employed to calculate the current-voltage characteristics of an ideal p-n junction,
mainly: No generation/recombination in the depletion regions, low-level injection, and Boltzmann statistics.
As we can see in the figure above showing the band diagram of a BJT, most of the voltage-drop occurs in
the depletion regions. The neutral base region is almost field-free and the equations controlling the transport
of electrons (the minority carriers in the base) will be dominated by diffusion. If x is the coordinate along the
axis normal to the junctions, the electron continuity equation in the base at steady state will be:
0 =
∂n
1 ∂Jn
n(x) − nB
,
=
−
∂t
e ∂x
τBn
(657)
where nB is the equilibrium concentration of minority electrons in the p-type base,
Jn = eDBn
∂n
,
∂x
Jp = Jtot − eDBn
(658)
∂n
.
∂x
(659)
From Eqns. (657) and (658) we get
∂ 2n
n(x) − nB
=
.
∂x2
DBn τBn
Recalling that L2Bn = DBn τBn , the solution can be written as:
x
x
n(x) = nB + A exp
+ B exp −
.
LBn
LBn
ECE609 Spring 2010
(660)
(661)
228
The integration constants A and B are determined by the fact that at the edges of the base, x = 0 and
x = wB , the electron concentration is fixed by the Boltzmann factors:
eVBE
eVBC
n(0) = nB exp
(662)
n(wB ) = nB exp
kB T
kB T
(note that in the ‘normal’ forward operation n(wB ) ≈ 0, since VBC << 0) so that, after some cumbersome
algebra:
n(x) = nB +
[n(wB ) − nB ] sinh[x/LBn ] + [n(0) − nB ] sinh[(wB − x)/LBn ]
.
sinh(wB /LBn )
(663)
From Eqns. (658) and (659) we can calculate the total currents (shown in the figure above): The electron
current to the collector will be the electron diffusion current evaluated at x = wB , since we are assuming no
GR processes in the depletion regions:
dn enB DnB
ICn = eDBn
=
−A
{exp[eVBE /(kB T )] − 1} +
dx x=w
LBn sinh(wB /LBn )
B
ECE609 Spring 2010
229
+A
enB DnB
{exp[eVBC /(kB T )] − 1} ,
LBn tanh(wB /LBn )
(664)
where A is the cross-sectional area of the device. Similarly, the electron current to the emitter will be the
electron diffusion current evaluated at x = 0:
dn enB DnB
IEn = eDBn
=
−A
{exp[eVBE /(kB T )] − 1} +
dx x=0
LBn tanh(wB /LBn )
+A
enB DnB
{exp[eVBC /(kB T )] − 1} ,
LBn sinh(wB /LBn )
(665)
The hole concentration in the emitter and collector can be found easily from the theory of p-n junctions:
x + lE
eVBE
p(x) = pE + pE exp
for x < 0 ,
(666)
− 1 exp
kB T
LEp
and
p(x) = pC + pC exp
eVBC
kB T
−1
exp
lC − x
LCp
for x > wB ,
(667)
where pE and pC are the equilibrium minority-carrier (hole) concentrations in the emitter and collector
respectively and −lE and lC are the edges of the emitter-base and base-collector depletion regions,
respectively. Therefore the emitter and collector hole currents can be calculated by taking the derivatives with
respect to x of these expressions at −lE and lC , respectively:
eDEp pE
dp eVBE
IEp = − AeDEp
= −A
(668)
−1 ,
exp
dx x=−l
LEp
kB T
E
dp ICp = − AeDCp
dx x=l
C
= A
eDCp pC
LCp
exp
eVBC
kB T
−1
.
ECE609 Spring 2010
(669)
230
(Note that in the ‘normal’ forward operation the base-collector junction is strongly reverse-biased (VBC <<
0), so the ICp will be usually negligible. Yet, we give its expression (and we have sketched it as an arrow in
the figure above) for completeness.) Thus the total emitter current will be, from Eqns. (665) and (668):
IE = IEp + IEn = −
AeDEp pE
LEp
AeDBn nB
+
LBn tanh(wB /LBn )
AeDBn nB
+
LBn sinh(wB /LBn )
exp
eVBC
kB T
exp
eVBE
kB T
−1
+
−1
.
(670)
Similarly, for the collector current we get from Eqns. (664) and (669):
AeDBn nB
IC = ICn + ICp =
LBn sinh(wB /LBn )
AeDCp pC
AeDBn nB
+
−
LBn tanh(wB /LBn )
LCp
exp
exp
eVBE
kB T
eVBC
kB T
−1
−
−1
.
(671)
The difference IE − IC is the (smaller) base current IB .
ECE609 Spring 2010
231
We have so far assumed implicitly constant doping in the base. Let’s now consider a graded doping profile in the base, NB dropping
from x = 0 to x = wB . Since the acceptor concentration varies in the base, the hole density will vary as
p(x) ≈ NB (x) = ni exp [Ei − EF (x)]/(kB T ) ,
(672)
where Ei is the intrinsic Fermi potential and EF (x) the Fermi energy. Therefore, we have a built-in field
Fbi (x) = −
dEF
k T 1 dNB
= B
.
dx
e NB dx
(673)
Equation (658) expressing the electron current across the base will now become:
∂n
,
JBn = eµBn n Fbi (x) + eDBn
∂x
(674)
or, using Eq. (673):
JBn = eDBn
∂n
n dNB
+
∂x
NB dx
=
eDBn
p
∂n
∂p
p
+ n
∂x
∂x
=
eDBn ∂(np)
,
p
∂x
ECE609 Spring 2010
(675)
232
having assumed p ≈ NB . Ignoring recombination in base (OK if the base is thin enough), we can integrate this equation as follows:
x
JBn
x
pdx
=
eDBn
Now,
p(0) n(0) = n2
i exp
eVBE
x
d(pn)
= p(x )n(x ) − p(x)n(x) .
dx
x
kB T
2
and p(wB ) n(wB ) = ni exp
eVBC
kB T
(676)
.
(677)
Therefore, from Eqns. (676) and (677) we get:
en2
DBn
JBn = wi
B pdx
0
exp
eVBC
kB T
− exp
eVBE
kB T
.
(678)
Let’s now assume that the electron mobility, and so the electron diffusion constant DBn , do not depend on x. This ignores
the effect of scattering with ionized impurities which, obviously, is position-dependent if the doping is non-uniform. Defining
w
QB = e 0 B NB (x) dx (the total charge per unit area due to dopants in the base, called the base ‘Gummel number’) and
JS = e2 n2
i DBn /QB , we have finally:
eVBE
eVBC
− exp
.
JCn ∼ JEn = JBn = JS exp
(679)
kB T
kB T
Comparing this expression with the constant-doping expression, Eq. (671) when VBC → −∞ and VBE >> kB T ,
AeDBn nB
eVBE
,
IC ≈
exp
LBn sinh(wB /LBn )
kB T
(680)
we see that the current is modified by a factor en2
i LBn sinh(wB /LBn )/(nB QB ) ≈ ewB NB /QB for a thin base. A
steep doping profile will reduce QB , so it will enhance the current adding to the diffusion of electrons across the base also a drift
term in Eq. (675). Such a base makes the device a ‘drift transistor’.
– The Ebers-Moll model.
Equations (670) and (671) are rather cumbersome expressions. In order to simplify them and shed some
ECE609 Spring 2010
233
light on their physical meaning, let’s define the saturation current for the emitter/base junction, IES , as the
magnitude of the saturation current we get when we reverse-bias in saturation the emitter-base junction while
short-circuiting the collector to the base. From Eq. (670), setting VBC = 0 and letting VBE → ∞, IES
will be just the term inside the first square bracket at the right-hand side:
IES =
AeDEp pE
LEp
+
AqDBn nB
.
LBn tanh(wB /LBn )
(681)
Similarly let’s consider the magnitude ICS of the saturation current for the reverse-biased base-collector
junction when the base is short-circuited to the emitter. From Eq. (671), setting VBE = 0 and letting
VBC → ∞, ICS will be just the term inside the second square bracket at the right-hand side:
ICS =
AqDCp pC
AeDBn nB
.
+
LBn tanh(wB /LBn )
LCp
(682)
From the definition of αF we have:
IC αF =
IE V
BC =0
=
AeDBn nB /[LBn sinh(wB /LBn )]
=
AeDEp pE /LEp + AeDBn nB /[LBn tanh(wB /LBn )]
=
cosh
wB
LBn
1+
pE DEp LBn
nB DBn LEp
tanh
wB
LBn
−1
.
(683)
Similarly, defining a ‘reverse common-base gain’ αR :
IE αR =
I C VBE =0
=
cosh
wB
LBn
1+
pC DCp LBn
nB DBn LCp
tanh
wB
LBn
−1
,
ECE609 Spring 2010
(684)
234
so that the Ebers-Moll equations (670) and (671) can be written in the more compact form:
IE = −IES
exp
eVBE
kB T
−1
+ αR ICS
exp
eVBC
kB T
−1
,
(685)
eVBE
eVBC
IC = αF IES exp
− 1 − ICS exp
−1 .
kB T
kB T
In matrix form, Eqns. (685) and (686) can be recast as:
(686)
and
IE
IC
=
−IES
αF IES
αR ICS
−ICS
exp[eVBE /(kB T )] − 1
exp[eVBC /(kB T )] − 1
.
(687)
These equations constitute the Ebers-Moll model. Supplemented by Kirchoff law, Eq. (650), they can be used
to calculate the current-voltage characteristics of a bipolar transistor for any arbitrary bias configuration. The
model depends on four parameters, IES , ICS , αF , and αR , of which only three are independent thanks to
the ‘reciprocity relationship’:
αF IES = αR ICS =
AeDBn nB
,
LBn sinh(wB /LBn )
(688)
which can be immediately derived from the pair of Eqns. (681) and (683) and from the pair of Eqns. (682)
and (684).
We can interpret the Ebers-Moll equations as describing the current-voltage characteristics of a pair of diodes.
In the common ‘forward’ operation, the first diode is the forward-biased emitter-base n-p junction with forward
current:
eVBE
IF = IES exp
(689)
−1 .
kB T
ECE609 Spring 2010
235
The second diode is the reverse-biased base-collector p-n junction with reverse current:
eVBC
IR = ICS exp
−1 .
kB T
(690)
Inserting Eqns. (689) and (690) into Eqns. (685) and (686) we get:
IE = −IF
+ αR IR
IC = −IR
+ αF IF
and
(691)
This form of the Ebers-Moll model is more transparent than the expressions Eqns. (670) and (671), but
it is seldom used, since the ‘parameters’ IF
and IR
cannot be measured. In order to reformulate the
model in terms of quantities which are directly measurable, let’s consider the collector current IC0 when the
base-collector junction is strongly reversed-biased (VBC → −∞, as usual), but the emitter is left floating
(so that IE = 0). In this case the Ebers-Moll equations, (687), read:
0
=
IC0
−IES
αF IES
αR ICS
−ICS
exp[eVBE /(kB T )] − 1
−1
,
(692)

 IES {exp[eVBE /(kB T )] − 1} = − αR ICS
or
.

(693)
IC0 = αF IES {exp[eVBE /(kB T )] − 1} + ICS
Substituting the expression for {exp[eVBE /(kB T )] − 1} from the first equation into the second, we get
IC0 = −αF αR ICS + ICS , or IC0 = (1 − αF αR )ICS . Similarly, let’s consider the emitter current
IE0 flowing through the emitter when the emitter-base junction is strongly reversed-biased (VBE → −∞)
and the collector is left floating (IC = 0):
IE0
0
=
−IES
αF IES
αR ICS
−ICS
−1
exp[eVBC /(kB T )] − 1
,
ECE609 Spring 2010
(694)
236
or:

 IE0 = IES + αR ICS [exp[eVBE /(kB T )] − 1]
,

(695)
αF IES = −ICS [exp[eVBE /(kB T )] − 1]
so that IE0 = IES − αR αF IES or IE0 = (1 − αR αF )IES . Note that the following reciprocity
relationship holds:
IE0
IES
=
= αF IE0 = αR IC0 .
(696)
IC0
ICS
With these definitions the Ebers-Moll equations can be rewritten in terms of easily measurable currents in the
following form:
1
IE
−IE0
exp[eVBE /(kB T )] − 1
αR IC0
(697)
,
=
exp[eVBC /(kB T )] − 1
IC
αF IE0
−IC0
1 − αF αR
or:

−IE0{exp[eVBE /(kB T )]−1}+αR IC0 {exp[eVBC /(kB T )]−1}


 IE =
1−αF αR


 I = αF IE0 {exp[eVBE /(kB T )]−1}−IC0 {exp[eVBC /(kB T )]−1}
C
1−αF αR
.
(698)
Multiplying the first of these equations by αF and adding the result to the second equations we get:
IC = −αF IE − IC0 {exp[eVBC /(kB T )] − 1} .
(699)
Multiplying the second of the equations (698) by αR and adding it to the first we get:
IE = −αR IC − IE0 {exp[eVBE /(kB T )] − 1} .
(700)
These expressions show that both the collector and the emitter curents are the sum of two contributions: a
diode-like component (reverse for IC , forward for IE ), and the current due to a ‘current source’.
ECE609 Spring 2010
237
– Components of the current gain: Emitter efficiency and transport base factor.
We have already defined the common-base current gain αF = hF B via Eq. (653) which we reproduce for
convenience:
∂IC ∂IEn ∂ICn ∂IC αF = hF B =
=
= γF αT M .
(701)
∂IE V
∂I
∂I
∂I
E
En
Cn V
=0
=0
BC
BC
Similarly, we have defined the ‘common-emitter current gain’ βF = hF E indirectly in Eq. (652):
∂IC βF =
.
∂IB V
=0
(702)
BC
In Eq. (701) the ‘emitter efficiency’ γ is essentially the ratio of the electron current from the emitter into the
base to the total emitter current. Clearly, in good devices, since the base current is small and the diffusion
of holes from the base into the emitter is small, the emitter efficiency is very close to unity. We have from
Eqns. (665) and (668):
∂IEn
γ =
∂(IEn + IEp ) VBC =0
IEn
≈
IEn + IEp =
VBC =0
1+
pE DEp LBn
nB DBn LEp
tanh
wB
LBn
−1
.
(703)
Since in modern well-designed devices wB << LBn , we can approximate the tanh-term as wB /LBn , so
that
−1
−1
pE DEp wB
µEp pE wB
γ ≈ 1+
= 1+
.
(704)
nB DBn LEp
nB µBnLEp
Note that the emitter efficiency increases at high emitter doping.
In case of non-uniform doping in the emitter, in order to evaluate the emitter efficiency we may follow closey the approach followed
ECE609 Spring 2010
238
above (see Eqns. (672) and ff.): The nonuniform doping will cause the presence of a built-in field
Fbi =
kB T 1 dNE
.
e NE dx
(705)
As we found before for the electron current in the non-uniformly-doped base, we now have:
or
k T 1 dNE
∂p
JEp = −eµEp B
− eDEp
,
e NE dx
∂x
(706)
eDEp d(pN )
∂p
1 dNE
E .
− eDEp
= −
JEp = −eDEp p
NE dx
∂x
NE
dx
(707)
Let’s assume a linear dependence of p on x (as it will be the case when the emitter is short), let’s assume a perfect Ohmic contact
ensuring p = n2
i /NE at the emitter contact, and let’s ignore recombination (OK for short emitters). Thus, the hole current is
constant in the emitter. We can then integrate Eq. (707) obtaining:
IEp = −
Ae2 DEpn2
i
QE
exp
eVBE
−1
kB T
,
(708)
where QE = em NE (x)dx is the emitter ‘Gummel number’.
From the definiton Eq. (703) of the emitter efficiency, and noticing that when VBC = 0 from Eq. (679):
IEn = −
we have:
γ
=
IEn + IEp Ae2 DBn n2
i
QB
exp
=
1 + IEp /IEn eVBE
kB T
VBC =0
− 1
,
1
IEn
≈
VBC =0
1+
(709)
DEp QB −1
DBn QE
ECE609 Spring 2010
(710)
239
The ‘base transport factor’ αT in Eq. (701) is a measure of the probability that an electron injected into the
base from the emitter will make it to the collector. From Eqns. (664) and (665):
ICn αT =
IEn V
2
wB
1
=
,
≈ 1 −
cosh(wB /LBn )
2L2Bn
=0
BC
(711)
having assumed, in the last step, wB << LBn , as done before.
The calculation of αT in the case of non-uniform doping in the base requires additional work. From the continuity equation for the
electron current in the base,
n(x) − nB
∂n
1 ∂Jn
=
−
(712)
,
∂t
e ∂x
τBn
at steady-state (∂n/∂t = 0) and recalling that nB = n2
i /NB , we have
n − n2
n − nB
∂Jn
i /NB .
= e
= e
∂x
τBn
τBn
(713)
Both n and NB are now functions of x. We can then calculate the current due to recombination in the base, IBr , by integrating
the change of the electron current density over the width of the base:
IBr = A(JEn − JCn ) = − eA
w
B
0
dx
n(x) − n2
i /NB (x)
τBn
.
The concentration of the electrons injected from the emitter is much larger than nB , so, negleting the nB -term:
w
eA
B
IBr ≈ −
dx n(x) .
τBn 0
Linearizing the electron density in the base – which is an exact assumption in the limit of very thin bases – we can write:
eVBE
x
−1 ,
n(x) ≈ nB 1 −
exp
wB
kB T
ECE609 Spring 2010
(714)
(715)
(716)
240
so that Eq. (715) becomes:
Finally,
n2
w
eV
eA
i
B exp
BE − 1 .
IBr ≈ −
τBn NB (0) 2
kB T
IEn − IBr IBr αT =
= 1 −
.
IEn
IEn VBC =0
VBC =0
(717)
(718)
Using now Eq. (679), we have:
JEn = JBn =
so that
e2 n2
D
i Bn
QB
IEn V
= −
BC =0
exp
eVBC
exp
− exp
kB T
Ae2 n2
i DBn
QB
eVBE
kB T
eVBE
.
kB T
(719)
− 1
.
(720)
Using finally this equation with Eqns. (717) and (718) we get:
αT = 1 −
wB QB
2eDBn τBn NB (0)
.
(721)
Finally, the ‘collector multiplication factor’ M in Eq. (701) is unity for VBC smaller than the threshold for
avalanche multiplication.
For large VBC breakdown can occur in the depleted collector region, much as we saw in the case of strongly reverse-biased p-n
junctions. Let’s recall that IC0 is the collector current when the emitter contact is left floating (open). When VBC is sufficiently
large, the collector current will be given by IC = M IC0 , where the multiplication factor can be written using the empirical
relationship:
1
,
M ≈
(722)
1 − (VBC /BV )n
where BV is the brekdown voltage and n is an exponent usually taking values between 4 and 6. In commmon-base configuration
ECE609 Spring 2010
241
we have:
M ≈
1
,
1 − (VBC /BVBC0 )n
(723)
where BVBC0 is the common-base collector-base breakdown voltage when the emitter contact is left open.
When we are below the breakdown regime, the total commomn-base current gain can be written as:
αF = hF B ≈ γF αT ≈
1+
pE DEp wB
nB DBn LEp
−1 2
wB
1 −
2L2Bn
,
in the case of uniform doping, or, more generally for non-uniform doping:
DEp QB −1
wB QB
,
1 −
αF ≈ 1 +
DBn QE
2eL2Bn NB (0)
(724)
(725)
Note that both γ and αT are smaller than unity. The deviations from unity actually represent the amount of
holes which must be supplied by the base contact.Typically in modern transistors with bases shorter than one
tenth of the diffusion length αT > 0.995 and αF is almost totally controlled by the emitter efficiency.
Considering now the current gain βF , assuming αT ≈ 1, we have from Eq. (703):
nB DBn LEp
nB DBn LEp
γ
wB
nB −1
NE
βF = hF E =
coth
∼
wB ∼
,
≈
≈
1−γ
pE DEp LBn
LBn
pE DEp wB
pE
QB
(726)
where QB is the Gummel number and NE is the donor concentration in the emitter.
– Factors affecting the current-gain βF .
The current-gain βF is arguably the most important parameter determing the performance of a BJT. Several
factors contribute to its value:
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1. Gummel number. Reducing the Gummel number QB (and so the doping concentration in the base)
we improve the gain, as seen from Eq. (726) above. From that equation we can also see that in the
common-emitter configuration an increase of VCE causes a shrinkage of the base width wB , a reduction of
the Gummel number and so an increase of the gain.
2. Emitter doping. Increasing NE also boosts the gain, as also seen from Eq. (726). Note the a low base
doping and a high emitter doping are to be expected when we want to maximize the emitter efficiency: In
a p-n junction in forward bias, the electron current is proportional to np0 and the hole current to pn0 (see
Eq. (26), Lecture Notes, Part 2). In our case for the emitter-base junction these quantities are nB and pE ,
respectively. A high emitter efficiency is achieved when IEp << IE ≈ IEn . With a large emitter doping
we lower pE , with a small base doping we increase nB , which is what we want.
3. Sah-Noyce-Shockley recombination. At very low collector current, the recombination-generation current
in the emitter is large compared to the diffusion current of minority carriers across the base, so the emitter
efficiency is low. In particular, at low current βF varies with collector current. If we assume that the
base-emitter junction is characterized by a non-ideality factor m, we have
βF =
∂IC
exp[eVBE /(kB T )]
1−1/m
∼
.
∼ IC
∂IB
exp[eVBE /(mkB T )]
(727)
Minimizing the density of traps in the emitter – and so rendering m closer to unity – we improve the gain at
low current levels.
4. Webster effect. At high collector current (high injection levels) numerical simulations show that the gain
drops with IC :
∂IC
exp[eVBE /(2kB T )]
−1
βF =
∼
.
(728)
∼ IC
∂IB
exp[eVBE /(kB T )]
This drop of efficiency at large collector currents is called ‘Webster effect’. It is due to the fact that at high
injection the electron density in the base approaches the hole density, effectively increasing the base doping
(Gummel number), thus reducing the emitter efficiency.
5. Band-gap narrowing. This is an effect due to the electron-hole, electron-electron, and electron-dopant
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Coulomb interactions in heavily-doped semiconductors. As the density of free carriers increases, the electrical
and optical gap of the semiconductor decreases. This is a very complicated many-body problem which has
been studied at length. Many texts make use of an empirical relation to shows that the gap shrinks according
to
1/2
e2 NE
3e2
∆EG ≈ −
,
(729)
16πs
s kB T
or, at room temperature:
NE 1/2
∆EG ≈ − 22.5
meV ,
(730)
1018
where NE is measured in cm−3 . More recent studies show that we should expect a relation of the form
e2
1/3
∆EG ≈ − σ
NE ,
4πs
(731)
where σ is approximately 3, including also quantum corrections: When evaluating the strength of the
Coulomb interactions one should account for the finite size of the electron wave-packet.
The magnitude of band-gap narrowing is also dependent on whether we consider the ‘electrical’ or the
‘optical’ gap. In any event, since a change of the gap causes a change of the intrinsic carrier concentration,
2
2
niE = NC NV exp[−(EG + ∆EG )/(kB T )] = ni exp[∆EG /(kB T )] ,
(732)
since nB = n2i /NB and pE = n2iE /NE , we have
βF ∼
nB
∝ exp[∆EG /(kB T )] ,
pE
(733)
so that the gain decreases as the gap shrinks with increasing emitter doping. We should also note that at
very high doping concentrations the energy levels of the dopants ‘merge’ to form an ‘impurity band’. It acts
as a modification of the band structure near the band-edges and causes an additional narrowing of the gap.
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6. Auger recombination. As we saw in Lecture Notes, Part 1, page 92 and ff., Auger recombination is a
process in which a hole can recombine with an electron while transfering the released energy to another
electron. The lifetime for this process – which involves two electrons and a hole and occurs when a hole is
injected from the base into the heavily-dope n-type emitter – τA , is given by 1/Gn n2E , where Gn is of
the order of 10−31 cm−6 /s for Si at room temperature. Therefore, the total hole lifetime in the emitter
will be 1/τ = 1/τp + 1/τA , where τp is the lifetime for the Sah-Noyce-Shockley recombination. As NE
increases, the the minority carriers recombine faster in the emitter and the emitter efficiency drops.
7. Kirk effect (or base push-out). At very high levels of injection, minority electrons in the base may flood
the base and a high electron concentration may extend beyond the base-collector junction. Indeed the charge
in the collector-side of the base-collector depletion region will be:
ρ(x) ≈ e[ND (x) − NA (x)] −
ICn
IC
,
≈ e[ND (x) − NA (x)] −
Av(x)
Avsat
(734)
having assumed the the high field pushes the electrons at the saturated velocity and that the collector current
is mostly due to electrons (which usually is the case). At low levels of injection (small IC ) this charge is
mostly due to the ionized collector donor dopants, so it is positive. But at sufficiently large IC the sign of
the charge may actualy flip. Poisson equation actually implies that the collector-side of the base depletion
region moves towards the sub-collector. We can see this by noticing that the high conductivity of the region
– now flooded by electrons – will flatten the potential beyond the collector junction and push the collector
high-field region deeper towards the subcollector. This effectively increases the width of the base, increases
the Gummel number, and reduces both the emitter efficiency and the base transport factor. It is called ‘Kirk
effect’ or ‘base push-out effect’.
– dc output characteristics.
From the discussions in the previous sections it it clear that the terminal currents in a BJT are determined
by the minority carrier (electron) distribution in the base region. For devices with a high emitter efficiency,
we see from Eqns. (670) and (671) that the dc expressions for the emitter and collector currents depend on
∂n/∂x at x = 0 and x = wB , respectively, or, equivalently, on nB /[LBn sinh(WB /LBn )] ≈ nB /wB
for short bases. Thus, we see that the BJT is controlled:
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1. By the applied bias, which determines the electron densities at the base-edges via the Boltzmann factors
exp[eV /(kB T )].
2. By the minority-carrier density-gradients at the base-edges x = 0 and x = wB .
3. By the emitter efficiency and base-transport factors which determine how large will be the difference between
emitter and collector currents. The emitter efficiency reflects the size of the undesired hole current from
the base to the emitter. This current can be minimized via the proper emitter and base doping (large and
small, respectively) and via a reduced recombination (Sah-Noyce-Shockley and Auger) in the emitter. The
base-transport factor, instead, reflects the loss of electrons via recombination in the base. This can be
minimized by reducing the base thickness and minimizing the density of traps in the base itself.
We can now analize qualitatively the current-voltage characteristics of a device operating in forward active
mode. As we mentioned above, we can operate the device either in common-base mode (base grounded), or
in common-emitter mode (emitter grounded).
The figure below shows (qualitatively) the IC -VBC characteristics of an n-p-n BJT operated in the
common-base configuration. The emitter current parametrizes the curves.
Note that collector and emitter currents are practically identical. The base-collector bias has no effect on the
collector current. Indeed IC is fixed almost completely by the barrier at the emitter-base junction, and so
by the emitter-base bias (or, equivalently, by the emitter current). To reduce the collector current to zero a
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small forward bias must be applied to the base-collector region: This increases the electron concentration at
x = wB untill the gradient ∂n/∂x vanishes and the electron diffusion across the base stops. At large enough
VBC impact ionization will set-in, resulting in a fast growth of IC (not shown in the figure). Breakdown may
also be caused by punch-trhough: At large enough VBC the base may deplete, causing an unimpeded flow of
electrons directly from the emitter to the collector.
The figures above (left frame) show the IC -VCE characteristics in the common-emitter configuration. The
base current parametrizes the curves (indeed the base-emitter bias is just about ‘pinned’ at 0.7 V (in Si BJTs)
and it is not a useful parameter, given its limited range of variability: IB is a better parameter). The base
current (or base bias) fixes the electron-flow over the emitter-base barrier. For a fixed IB , as VBC increases
the base-width shrinks electrostatically because of the reduction of the base-emitter depletion width with
increasing forward bias. This amounts to a reduction of the Gummel number, so to an increasing gain, and
the collector current grows accordingly. This is called ‘Early effect’ (name we have already encountered before
dealing with MOSFETs with high output conductance). For a BJT with wB >> LBn the Early voltage VA
2
(shown in the figure above, right frame) is given by ≈ eNB WB
/s .
A more accurate expression for the Early voltage can be obtained from Eq. (679). In the forward active mode,
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that expression reads
Ae2 DBn n2i
eVBE
IC ≈
exp
(735)
.
QB
kB T
Thus, the variation of IC caused by a change of wB (the upper limit of integration in QB ) – in turn induced
by a change of VBC – will be:
Ae3 DBn n2i
dIC
≈ −
exp
dVBC
Q2B
eVBE
kB T
p(wB )
dwB
.
dVBC
(736)
Defining the Early voltage VA as
VA =
QB
,
ep(wB )(dwB /dVBC )
(737)
Eq. (736) becomes the following simple expression for the output conductance:
dIC
IC
= −
.
dVBC
VA
(738)
Note that ep(wB )(dwB /dVBC ) = dQB /dVBC . In practice the Gummel number does not change much
with VBC , so the Early voltage can be considered a constant.
For very small VCE , IC falls to zero quite rapidly. This is because the emitter-collector voltage drops over
two junctions. But in order to maintain a constant IB , the drop across the emitter-base junction remains
essentially constant. Thus, it is the base-collector bias which must drop. At VCE ≈ 1V in Si the base-collector
junction is actually at zero bias and as VCE is reduced further it actually becomes forward-biased. As soon
the gradient of the electron density in the base vanishes, the collector current vanishes as well.
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– Modes of operation.
A BJT can work in two main configurations: 1. As an amplifier, as we saw above (page 226), under a
large dc bias VBC < 0 driving the bias-collector junction in reverse, and a constant VBE > 0, driving the
base-emitter junction in forward bias. A small ac signal Vin added to the dc bias VBE is amplified by the
large VBC . 2. As a switch, in which the transistor is brought from an ”on” state of large IC to an ”off”
state of small IC . This is accomplished in various modes. To classify them, it is useful to divide the output
characteristics of a BJT into three regions:
∗ Region I: Cut-off region of zero or small collector current, both emitter-base and base-collector junction
reverse biased.
∗ Region II: (Forward) active region, emitter-base junction forward-biased, base-collector junction reversebiased. This is the mode of operation we have considered so far.
∗ Region III: Saturated region, both junctions forward-biased.
∗ Region IV: Inverse active region, emitter-base junction in reverse, base-collector junction under forward bias.
For all switching mode, the BJT follow a switching trajectory on the I-V plane along a load line. The most
common mode of swithing is a transition from the saturated to the cut-off region. Switching from cut-off to
active (the so-called current-mode) is suited to high-speed swicthing, since the smaller swing of VBC induces
a smaller delay due to the charging/discharging of the base-collector depletion region.
The figure below (left) illustrates these two swicthing modes on a resistive load (a straight-line in the
IC − VCE plane) as well as the distribution of minority carriers in the base (left). We shall discuss below
how small-signal and large-signal model deal with the amplifying and switching modes, respectively. Before
doing that, however, we must discuss the ac behavior of the BJT.
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– ac behavior.
So far we have considered only the dc (static) behavior of the device. We now consider its ac behavior. Before
discussing a model commonly used to describe the time-transient behavior of BJTs, it is useful to consider its
expected frequency response from the point of view of simple transport properties.
∗ Frequency response.
We saw above (pages 220 and 221) that the frequency cut-off a MOSFET is given by fT ≈ 1/(2πτ ),
where τ is the transit time across the channel. The situation for a BJT is quite similar: The cut-off frequency
is given by:
1
fT =
,
(739)
2πτec
where τec is the total delay-time suffered by electrons flowing from emitter to collector. There are 4 main
contributions to this delay:
τec = τE + τB + τC + τC
.
(740)
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1. The emitter depletion-layer charging time τE is given by
τE = RE (CE + CC + CP ) ≈
kB T
(CE + CC + CP ) ,
eIE
(741)
where RE is the emitter resistance, CE the emitter capacitance, CC the collector capacitance, and CP
any other parasitic capacitance connected to the base lead, IE ≈ IC . The fact that RE ≈ kB T /(eIE )
comes from differentiating the expression for IE given by the Eber-Moll expression, Eq. (670), with respect
to the emitter voltage.
2. The base-layer transit time τB is the electron transit time across the base. We can estimate it by using
the linear approximation Eq. (716): The electron current across the base in the uniform-doping case is:
∂n nB
eVBE
JBn ≈ eDBn
≈
eD
.
(742)
−
1
exp
Bn
∂x w
kB T
wB
B
Therefore the electron drift velocity in the base will be:
JBn
DBn
vD (x) =
≈
en(x)
wB
x
1 −
wB
−1
.
(743)
Then, the electron transit time across the base will be:
τB =
wB
0
dx
wB
≈
vD (x)
eDBn
wB 0
x
1−
wB
dx =
2
wB
2DBn
.
(744)
2
/(ηDBn ) where the
For a inhomogeneously-doped base one gets the more general expression τB = wB
parameter η depends on the doping profile. Note that when the inhomogeneous doping introduces a large
built-in field, one can show (in a way similar to what we have done above) that η ≈ 2[1 + (Fbi /F0 )3/2 ],
where F0 = 2DBn wB /µBn . Thus, the built-in field helps in reducing the transit time.
Finally, it is interesting to note that as the base gets thinner, ballistic transport – rather than diffusive – will
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result in a linear relationship τB ∼ wB /vinj where vinj is the velocity at which electrons are injected
from the emitter into the base. This transition from diffusive to ballistic transport occurs for wB ∼ 10 nm.
3. The collector depletion-layer transit time τC will be
τC ≈
xC − wB
,
2vsat
(745)
where vsat is the electron saturation velocity in the collector. This is approximately valid for long collectors
and large fields in the collector (that is, in the forward active mode.
4. Finally, the collector charging time τC
will be
τC
= RC CC ,
(746)
where RC is the collector series resistance and CC the collector capacitance.
Putting all together, the cut-off frequency will be given by:
−1
2
wB
1
kB T (CE + CC + CP )
xC − wB
fT =
=
2π
+
+
.
2πτec
eIC
ηDBn
2vsat
(747)
We see from the expression that in order to enhance the cut-off frequency we must use thin bases, thin
collectors, and we should operate the transistor at a high collector current. However, for thin collectors and
high currents we heve a reduced breakdown voltage, so, as usual, compromises are necessary.
∗ Charge Control Model.
Let’s now discuss the time-transient behavior of a BJT using a model in which charge, rather than currents
or voltages, are the independent variables. The basic idea consists in reformulating all components of the
base, collector, and emitter currents in terms of a charge divided by a characteristic transit or charging time.
Analyzing the change of these charges as the applied biases charge with time provides an expression for the
time-dependence of the currents.
BJTs are modulated by the base-emitter bias VBE . The basic charges controlled by this bias are:
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1. The charge due the excess minority carriers (electrons in n-p-n BJTs) in the base:
QBn = −eA
w
B
[n(x) − nB ] dx .
(748)
2. The charge due to excess minority carriers (holes) in the emitter:
QEp = eA
[p(x) − pE ] dx .
(749)
0
emitter
3. The depletion charge QV E due to a variation of the emitter depletion region due to VBE .
4. The depletion charge QV C due to a variation of the collector depletion region due to VBC .
Using these charges, let’s re-express the collector and base current following the spirit of the ‘charge control’
model.
Collector current. For thin base, using the usual linearization of the electron carge in the base,
eAwB n2i
qF = QBn =
2NB
exp
eVBE
kB T
−1
.
(750)
Defining:
eAwB n2i
qF 0 =
,
2NB
we can write
eVBE
qF = qF 0 exp
−1 ,
kB T
so that, ignoring recombination in the base:
IC =
(751)
(752)
qF
,
τF
(753)
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where τF is the transit time in the base τB previously defined (the reason for adding the subscript F will
be apparent below).
Base current. The dc base current consists of a hole current injected into the emitter, IEp , and a
recombination current, IBr . From Eq. (668), making use of the definition of qF above, Eq. (750), we have:
IEp = qF
2DEp pE
LEp wB nB
.
(754)
The second component can be reformulated as follows (see Eq. (717)):
IBr = −
qF
.
τBn
(755)
Adding these two equations,
IB = IEp − IBr = qF
2DEp pE
LEp wB nB
+
1
τBn
=
qF
,
τBF
(756)
where the charging time τBF is implicitly defined by this equation. Note that the common-emitter forward
gain can be expressed as ratio of the times we have just defined:
βF =
IC
τBF
=
.
IB
τF
(757)
Since IB = −IC − IE , Eqns. (753) and (756) allow us to express all currents as {charge}/{time}, as
desired.
In order to account for the time dependence, we must include the displacement currents of all charges listed
above. Thus:
qF
qF
dQV E
dQV C
IB =
+
(758)
+
+
,
τBF
dt
dt
dt
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and
IC =
and, since IE = −IC − IB ,
IE = qF
From Eq. (681) we see that
qF
dQV C
−
,
τF
dt
1
+
τBF
τF
1
−
(759)
dQV E
dqF
−
.
dt
dt
(760)
1
IES = qF 0
+
(761)
.
τBF
τF
Note that the term dqF /dt is a diffusion capacitance due to the variation of the charge of minority carriers
stored in the base, while dQV E /dt and dQV C /dt are the depletion capacitances due to the charge stored
in the emitter and collector depletion regions, respectively.
The charge-control model is useful in establishing the basis on which we can analyze the amplifying (small
signal) and switching (large signal) behavior of the BJT, as we can analyze how the BJT makes transitions
from one region of operation (I, II, or III) to another.
The figure below illustrates the equivalent circuit we can associate with the model developed so far.
1
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Large Signal Model.
In the switching mode, we can view the characteristics of the BJT as a superposition of forward-active
and inverse-active mode. We can define all charges in inverse-active mode in a symmetric fashion and
superimpose linearly the time-variation of all the charges, thus extending the validity of Eqns. (758)-(761),
and obtaining a model which is valid in all regions (I-IV):
IB =
qF qF
qR qR
dQV E
dQV C
+
+
+
,
τBF dt
τBR dt
dt
dt
qF
dQV C
IC =
−
− qR
τF
dt
and
IE = qF
1
+
τBF
τF
1
−
1
1
+
τR
τ BR
(762)
dqR
,
dt
(763)
dQV E
qR
dqF
.
−
+
dt
dt
τR
(764)
−
The analogous of Eq. (761) is:
ICS = qR0
1
+
τBR
τR
1
,
(765)
where qR = qR0 {exp[eVBC /(kB T )] − 1}.
The figure below (taken from Fig. 8.41 of the Colinge-Colinge text) illustrates the equivalent circuit
corresponding to Eqns. (762)-(765).
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Small Signal Model.
Let’s now consider the amplifying mode of a BJT. From Eq. (719) we have
JEn = JS
where
exp
eVBC
kB T
− exp
eVBE
kB T
,
e2 n2i DBn
JS =
.
QB
In forward active mode
(767)
IC = −AeJEn = IS exp
(766)
eVBE
kB T
(768)
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If we bias a BJT in dc, but superimpose to a small ac bias to VBE , we have
dIC
eIS
=
exp
dVBE
kB T
eVBE
kB T
=
eIC
= gm ,
kB T
(769)
which is the transconductance of the transistor. Since IC = βF IB ,
dIB
gm
τF
=
=
gm .
dVBE
βF
τBF
(770)
Note that the change of the base-charge qF is given by:
dqF
dIC τF
=
= gm τF = CD ,
dVBE
dVBE
(771)
where CD is the capacitance associated with the change of the charge of minority carriers injected by the
emitter into the base and by the base into the emitter.
Finally, regarding the effect of an ac small signal on the output conductance, note that
dIC
IC
gm kB T
= −
= −
dVCB
VA
eVA
(772)
The figure below (taken from Fig. 8.41 of the Colinge-Colinge text) illustrates the equivalent circuit (the
so-called hybrid π-model) corresponding to Eqns. (768)-(772).
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– Results from Monte Carlo simulations.
The four figures below show results from 2D self-consistent Poisson/Monte Carlo simulations of an n-p-n
BJT. The first pair of figures show the electron potential energy in the device (left) and the Monte Carlo
particles employed in the simulation (right). The next two figures show the electron (left) and hole (right)
density in the BJT. The next figure shows the average electron kinetic energy in the device. The final pair of
figures show the variation of the average electron kinetic energy (left) and electron drift velocity (right) from
emitter to collector.
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• Heterojunction Bipolar Transisitors.
Heterojunction bipolar transistors make use of ‘band-gap engineering’ to improve two basic properties of BJT:
Gain and speed.
We have seen that in order to achieve a high gain it is necessary to optimize the emitter efficiency
γF ≈
1+
DEp QB −1
DBn QE
,
(773)
by using a high emitter doping (high emitter Gummel number QE ) and a low base doping (low base Gummel
number QB ). This ensures that the current flowing through the emitter-base forward-biased junction is mainly
due to electrons.
In Heterojunction Bipolar Transistors (HBTs), a material of a smaller band-gap is used in the base, so that
holes injected from the base are prevented from entering the emitter by a potential barrier ∆EV . Materials
typically used include group IV or III-V compound semiconductors: A group-IV HBT may consist of a Si emitter
and a Si1−x Gex base (alloy whose gap is about 0.87 eV for x = 20%, most of the gap discontinuity falling
actually in the valence band) A III-V HBT may make use of an AlxGa1−x As emitter with a GaAs base or of an
Inx Ga1−x As base with a GaAs emitter. The use of strained-base layers will also result in the desired band-gap
discontinuity. A rough estimate of the improvement achievable can be obtained by noting that, from Eqns. (73)
and (74) (Lecture Notes, Part 2):
IC
IEn
∆EG
(0)
βF =
≈
≈ βF exp −
(774)
,
IB
IEp
kB T
(0)
where βF is the common-emitter current gain in the absence of a band-gap discontinuity and ∆EG =
∆EC + ∆EV is the band-gap discontinuity between base and emitter.
A second advantage of HBT stems from the possibility of setting a built-in field in the base without using strong
doping-gradients but taking advantage of band-gap engineering. For example, by grading the mole-fraction of
Ge in the Si1−x Gex base (such that we progress from a low mole-fraction near the emitter-base junction towards
a higher x as we approach the base-collector junction), we create a built-in field Fbi which improves the base
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2
transit time τB = wB
/(ηDBn ) with η ≈ 2[1 + (Fbi /F0 )3/2 ] (see the discussion after Eq. (744).
A final advantage of HBT lies in the high mobilities afforded by some base materials (such as an Inx Ga1−x As
base with a GaAs emitter).
The excellent transport properties of the base of HBTs render them suitable for high-frequency applications. It
should be noted, however, that as wB is scaled, the importance of τB in determining the overall frequency
response of the device is reduced. For very thin bases, the collector transit time begins to dominate.
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Optical and Quantum devices
In this section we discuss optical and quantum devices.
The former are devices which employ
electroluminescence to convert electrical signals to light (such as ligh-emitting diodes – LEDs –, diode or
‘injection’ lasers), light to electric signals either for detection (photodetectors such as avalanche photodiodes) or
for power-production, such as solar cells. We shall restrict our discussion mainly to LEDs and lasers. Quantum
devices fall into the broad categories of tunnel diodes (Esaki diodes, resonant-tunneling diodes – RTDs), mainly
employed as high-frequency oscillators, and devices based on quantum effects (such as quantum interference) to
perform logic operations. These are still ‘experimental’ devices whose application to the ‘real world’ is still being
investigated – not to say controversial. Therefore, we shall deal mainly with tunnel diodes. Devices now being
studied for their potential application to quantum computing (interference devices, spintronic devices, etc.) are
dealt with in a course by prof. Anderson.
• Optical (photonic) devices.
– Radiative transitions.
Before discussing LEDs and injection lasers we must review some properties of radiative processes in
semiconductors. (It will be helful to review the discussion in the Lecture Notes, Part 1, pages 89-92). The
figure below illustrates schematically basic recombination processes across the gap of a semiconductor.
These processes are divided into 3 groups: The red processes are: (a) intrinsic inter-band processes usually
involving the emission (or, in the reverse processes, the absorption) of light, often mediated by phonons
and excitons (electron-hole pairs weakly bound together as in a hydrogen atom); (b) higher-energy processes
involving hot carriers, as in avalanche processes. The green processes involve defect-assisted transitions: (a)
band-to-defect/impurity; (b) defect/impurity-to-band; (c) defect/impurity to defect/impurity. Finally, the
black processes are intra-band processes involving excitation or relaxation of hot carries
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a
EC
ED
a
b
a
b
d
c
EA
a
EV
– Emission spectra.
Let’s recall that the rate at which photons (which are bosons) are emitted and absorbed is, similarly to phonons
(which are also bosons) proportional to 1 + nb and nb, respectively, where nb is the Bose-Einstein thermal
occupation of bosons of energy h̄ωb:
nb =
exp
h̄ωb
kB T
− 1
−1
.
(775)
The term ”1” in the factor 1 + nb appearing in the emission rate is called ‘spontaneous emission’. The term
nb is called ‘stimulated emission’ and has no semiclassical counterpart.
Spontaneous emission depends only on the density of filled states in the conduction band at energy E , fc (E)
(electrons of kinetic energy E ) and of empty states, 1 − fv (E ) in the valence band (holes of kinetic energy
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E ). The emission rate for photons of frequency ν can be roughly expressed as:
2
|Hphot| fC (E) [1 − fV (hν − E − EG )] ∼
I(hν) ∼ ν NC NV
hν − EG
exp
(776)
∼ ν (hν − EG )
,
kB T
the last expression being valid in the nondegenerate limit and at thermal equilibrum. The term Hphot
(strictly speaking a function of the electron and hole wavevectors) is the optical matrix element which we shall
review below. Clearly, the larger the gap, the shorter the wavelength of the emitted light.
Regarding defect/impurity-assisted transitions, LEDs and lasers have rather large impurity densities. The
simple picture of discrete impurities with energy levels in the gap does not hold. Rather, a picture similar to
the impurity-band responsible (in part) for bandgap-narrowing is more appropriate. Thus, the effect of the
impurities is that of causing the presence of a ‘Gaussian tail’ of electronic states in the gap.
It is interesting to see how this happens. Suppose that we have a random distribution of traps. These are
Coulomb centers (attractive or repulsive) which locally shift the energy of the bottom of the conduction band,
EC . For a completely random distribution of traps, we can assume that locally EC has a probability
2
1
EC − E0
dEC ,
P (EC )dEC =
(777)
exp −
1/2
1/2
(2π) ∆
2 ∆
2
1/2
of taking a value between EC and EC + dEC . In this expression E0 is the average energy of the CB-bottom
and ∆ the variance of the distribution. The choice of a Gaussial distribution obviously stems from the fact
that in nature random processes are described by Poisson distributions and these tend to be Gaussians in the
limit of infinitely large ensembles.
Let’s now consider the density-of-states (DOS) in the conduction band in Si in the absence of traps:
3/2
ρ(E − EC ) = 6
2md
(E − EC )
2π2h̄3
1/2
.
(778)
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Now, Eq. (777) describes the fluctuations of EC , so that the DOS in the presence of the random traps will
be:
3/2
E
6md
∆1/2
3
ρtraps (E) =
ρ(E − EC ) P (EC ) dEC =
Γ
2
π2 h̄3 (2π)1/2
−∞
(E − E0 )2
E − E0
× exp −
D−3/2 −
(779)
,
4∆2
∆
where the ‘parabolic cylinder function’ Dν (z) is defined as:
exp(−z 2 /4)
Dν (z) =
Γ(−ν)
∞
0
2
dx e−zx−x /2 x−ν−1 ,
(780)
for ν < 0. At large positive energies (|E − E0 | >> ∆; E > E0 ),
ρtraps (E) → ρ(E)
∆2
1 +
16(E − E0 )2
,
(781)
that is, the DOS approaches the unperturbed DOS. At large negative energies (|E − E0 )| >> ∆; E < E0 ),
3/2 ρtraps (E) →
6md
2π2h̄3
3/2
∆ 1/2
∆
−(E−E0 )2 /(2∆2 )
e
,
2π
E0 − E
(782)
which exhibits a Gaussian tail in the gap of the semiconductor.
– Dipole approximation.
We reproduce here the discussion (Lecture Notes, Part 1, pages 91 and 92) about the basic properties of
radiative processes in the ‘dipole approximation’.
To first order in the electromagnetic field, the Hamiltonian describing the interaction of an electron with the
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electromagnetic field is, written in terms of the vector potential, A(r, t) = A0 ei(q·r−ωt) :
Hphot =
ieh̄
A·∇,
m
(783)
or (in Gaussian units) Hphot = ie/(mc)A · ∇. The matrix element between two states in the valence and
conduction bands thus has the form:
ieh̄
−iq·r
ik ·r v
< k, c|Hphot |k, v > ≈
dr e−ikc ·r uc∗
A0 · ∇ e v ukv (r) .
(784)
kc (r) e
m
V
If the wavelength of the photon is much longer than the electron wavelength, we can set e−iq·r ≈ 1 and
ieh̄
< k, c|Hphot |k, v > ≈
A0 ·
m
V
ikv ·r v
dr e−ikc ·r uc∗
ukv (r) .
kc (r) ∇ e
(785)
This is called the ‘dipole approximation’. Now:
ikv ·r v∗
dr e−ikc ·r uc∗
ukv (r) =
kc (r) ∇ e
V
i(kv −kc )·r
v
v
ikv ·r
=
dr e
uc∗
dr e−ikc ·r uc∗
.
(786)
kc (r) ∇ ukv (r) +
kc (r) ukv (r)∇e
V
V
The Bloch functions uckc (r) and uvkv (r) are orthogonal and the exponential is slowly varying. Therefore the
v
second integral is usually neglected. Now notice that the product uc∗
kc (r)∇ ukv (r) is the same in each unit
cell and it is multiplied by a slowly varying function which we can take outside the integral. Thus:
ieh̄
< k, c|Hphot|k, v > ≈
A0 ·
m
all cells
i(kv −kc )·r
e
c∗
Ω
v
dr ukc (r)∇ ukv (r) =
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ieh̄
1
i(k −k )·r
c∗
v
A0 ·
dr e v c
dr ukc (r)∇ ukv (r) .
m
Ω Ω
V
Now let’s define the ‘momentum matrix element’ between two Bloch functions:
ih̄
c∗
v
< c|p|v > = −
dr ukc (r)∇ ukv (r) .
Ω Ω
(787)
(788)
e
< k, c|Hphot |k, v > ≈ −
A0 · < c|p|v >
dr ei(kv −kc )·r .
(789)
m
V
From this we see that the optical matrix element vanishes unless kc = kv . Only ‘vertical (or direct) transitions’
can happen.
Transitions which involve a change in (crystal) momentum can only happen at higher orders in perturbation
theory, requiring extra momentum which must be provided by phonons, traps, etc.
Then:
– Quantum efficiency.
Radiative recombination processes compete with non-radiative recombinations, especially in indirect-gap
semiconductors (such as Si, Ge, GaP, etc.). The ‘quantum efficiency’ ηQ is the defined as the raction
of excited carriers which recombine radiatively to the total recombination. In terms of radiative (τr ) and
non-radiative (τnr ) lifetimes:
Rrad
1/τnr
ηQ =
=
.
(790)
Rtot
1/τr + 1/τnr
Thus, we need a very short radiative lifetime τr to achieve a high quantum efficiency. This is quite obvious:
If the radiative lifietime is long, carriers will recombine non-radiatively before having the chance of emitting light.
– Methods of excitation.
Electroluminescence – that is, optical processes triggered by electric signals, as we are discussing now – can
be stimulated in various ways:
1. Intrinsic: Exciting electron-hole pairs across the gap: A powder of a semiconductor (such as ZnS) is
embedded in a dielectric (glass or plastic) and placed between two semi-transparent conductive electrodes.
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An ac electric field applied to the electrodes accelerate carriers which can impact-ionize. The generated pairs
recombine radiatively. This process is now becoming efficient enough to be used in several types of displays
(automotive, in consumer electronics, etc.)
2. Avalanche: p-n junctions or metal-semiconductor junctions are reverse-biased into avalanche breakdown.
Strong intraband and/or interband radiative processes can take place as hot carriers are generated in
sufficiently high densities. The reverse of this process is used in avalanche photodiodes, or photomultiplyers:
A single photon absorbed in a reverse-bised junction on the verge of breakdown can trigger an avalanche of
electrons. They recombine, thus multiplying the number of photons.
3. Injection: In a forward-biased p-n junction large concentrations of minority carriers are injected across the
junction. Electrons in the p-type region and holes in the n-type region can recombine radiatively with the
majority carriers. These are the processes used in LEDs.
– LEDs.
In light-emitting diodes (LEDs), electrons and holes ‘flood’ the junction region of a forward-biased p-n
junction. Typical materials employed for emission in the visible region of the spectrum – for LEDs as well as
lasers – are many phosphides (AlP, GaP) and ternary alloys (GaAsx P1−x ) in the green, SiC or GaN – a material
of great current interest – in the blue, GaAs or CdSe in the red/infrared. For communication applications
where low absorption in optical fibers is required, Inx Ga1−x As (lattice-matched to the InP substrates for
x=0.53) is commonly employed. Typically the junctions are heavily n-doped, so that the electron current is
much larger than the hole current. The efficiency of the device is called the ‘injection efficiency’ and it is the
ratio of the electron current (which determines the emission intensity) over the total current:
ηi =
In
.
In + Ip + Irec
(791)
Values of 30% to 60% are common for the injection efficiency.
Materials with indirect gap (such as GaP or SiC) have been mentioned above. A high temperature the high
phonon population can provide the missing momentum to assist radiative recombination. Since this is a
second-order process (the product of two matrix elements relative to two perturbation Hamiltonian terms must
be considered – electron-phonon together with electron-phonon interactions), the emission is weak. However,
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other properties of the materials (a large bandgap, so emission at short wavelength; radiation hardness;
operation at high temperature, etc.) make them preferable in particular applications. Radiative recombination
through particular impurities (such as Er in Si) has also been investigated to enhance the emission efficiency in
indirect-gap semiconductors. Finally, hetero-junctions and band-gap engineering can be employed to increases
the quantum efficiency. But these efforts have been most intensive in the context of injection lasers and will
be discussed below.
– Laser diodes.
The operation of a laser is illustrated schematically in the figure in the next page. Consider a simple
2-level quantum system, with energies E0 and E1 , assuming E1 > E0 . Denote by n(0) and n(1) the
occupation of the levels. At thermal equilibrium at a sufficiently low temperature (kB T << E1 − E0 )
electrons occupy prevalently the lowest-energy state, since the occupation of the high-enery level will be
n(1) ∼ n(0) exp[−(E1 − E0 )/(kB T )] << n(0). If we are now able to ‘invert’ the population of
the levels so that n(1) > n(0) by ‘exciting’ somehow the system (for example, electrical ‘pumping’ gases,
such as Ne, so that collisions among ions result in electrons jumping into the excited level, as it happens
in fluorescent ligh), electrons will decay into the ground-state level. If the system is such that this decay
is prevalently radiative, photons of energy hν = E1 − E0 will be emitted by spontaneous emission. If,
however, in addition to ‘pumping’, we manage to confine the emitted photons inside the volume containing
the population-inverted medium (usually done by placing the gas/solid inside a cavity with reflecting walls, or
‘mirrors’), then the presence of these many emitted photons will trigger ‘stimulated emission’: The photons
emitted in spontaneous decay will have the correct energy to stimulate emission whenever they approach an
excited electron. The photons emitted by stimulated emission will be ‘coherent’ with the stimulating photons;
that is, they will be in phase.
Such a lasing action can be achieved in a diode: The population inversion is obtained by injecting carriers, as
in an LED. Electrons recombining radiatively with holes will emit the stimulating photon. Placing the junction
inside a cavity will increase the density of photons, and so the probability of triggering stimulated emissions.
Above a critical injected current (called the ‘threshold current’), lasing will occur.
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hν
hν
Thermal
equilibrium
Population Spontaneous Stimulated
inversion
emission
emission
Unfortunately, unless proper actions are taken, the threhold current would be too large (of the order of 104
A/cm2 or more) to sustain lasing without heating the material. This would result in increased non-radiative
recombination, and so in an even higher threshold current for lasing, thus in additional heating, and so on in
a divergent unstable feedback. This problem is commonly bypassed in the following way (schematically shown
in the next figure): A layer of a material of high refractive index n2 , such as GaAs, is grown on top of a layer
of lower refractive index n1 and larger bandgap, such as n-type AlxGa1−x As. On top of the GaAs another
similar AlxGa1−x As layer – but p-type doped – is also grown. Forward biasing the resulting p-n junction
causes two major effects: 1. The electron-hole pairs injected in the junction are confined in the GaAs quantum
well, and so will have a higher probability of recombining. 2. Light emitted in the GaAs layer finds itself in a
waveguide: By Snell’s law, photons generated in the GaAs layer incident on the GaAs/AlxGa1−x As interfaces
will be reflected back into the GaAs layer for angles of incidence larger than the critical angle θc given by:
sin θc =
n1
.
n2
(792)
For the GaAs/AlxGa1−x As system, at 300 K the dependence of the energy gap EG and refractive index n
on the mole fraction x are approximately given by:
EG ≈ 1.42 + 1.247x eV ,
(793)
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and
n ≈ 3.59 − 0.71x + 0.091x2 .
(794)
For example, for x=0.3 the refractive index of Alx Ga1−x As is about 6% smaller than that of GaAs,
n1 /n2 ≈ 0.94, so that θc ≈ 70 degrees from the normal. The GaAs layer surrounded by the AlxGa1−x As
layers acts as an efficient waveguide.
electron injection
n–type AlxGa1–xAs
hν
GaAs Quantum Well
hole injection
p–type AlxGa1–xAs
Let’s discuss these effects in more detail.
∗ Confinement factor.
The figure below shows the coordinate system we use. Let’s consider a symmetric 3-layer system and
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assume that the waveguide extends infinitely along the y -direction, so that ∂/∂y = 0. Let’s also consider
transverse-electric (TE) polarized waves, so that the z -component of the electric field, Fz , vanishes. Then,
in the case of even TE waves the wave equation simplifies to:
∂ 2 Fy
∂ 2 Fy
∂ 2 Fy
+
= µ0 s
.
∂x2
∂z 2
∂t2
(795)
where µ0 is the vacuum permeability and s the permittivity of the layer.
x
Normal plane
z
p
Plane of the
junction
n
y
Cleaved mirror
Separating of variables we can obtain the solutions of this equation in the active GaAs layer −d/2 < x <
d/2:
Fy (x, z, t) = Ae cos(κx) exp[i(ωt − βz)] ,
(796)
where
2
2 2
2
κ = n2 k0 − β ,
(797)
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k0 = (ω/n2 )(µ0 s )1/2 and β is a separation constant. The only nonzero component of the magnetic
field can be obtained from the Maxwell equation ∇ × F = −µ0 (∂ H/∂t):
Hz (x, z, t) =
i ∂Fy
iκ
Ae sin(κx) exp[i(ωt − βz)] .
= −
ωµ0 ∂x
ωµ0
(798)
Outside the active layer the field must decay if we are going to have a waveguide. Thus,
Fy (x, z, t) = Ae cos(κd/2) exp[i(ωt − βz)] exp[−γ(|x| − d/2)] ,
(799)
and
Hz (x, z, t) = −
x
|x|
iγ
Ae cos(κd/2) exp[i(ωt − βz)] exp[−γ(|x| − d/2)] ,
ωµ0
where
γ 2 = β 2 − n21 k02 .
(800)
(801)
Since both γ and κ are positive numbers, Eqns. (797) and (801) imply that we must have n2 > n1 in order
to achieve confinement. Quite obviously, this is equivalent to the result we obtained above using Snell’s law.
In order to determine the separation constant β , let’s consider the condition imposing the continuity of Hz
across the dieletric interfaces: Requiring that Eq. (798) and (800) be equal at |x| = d/2 we have:
tan
κd
2
=
γ
=
κ
2 2 1/2
β − n1 k0
.
n22 k02 − β 2
2
(802)
There are multiple solutions to this equation, since the tangent function acquires the same value when
the argument changes by 2πm, for integers m which indentifies the fundamental TE-mode (m=0), the
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first-order mode (m=1), etc. Solving this equation numerically or graphically and inserting the results into
Eqns. (796)-(800), we finally obtain the values of the electric and magnetic fields.
The important quantity we must define is the ‘confinement ratio’ Γ, defined as the ratio between the light
intensity within the active layer to the total intensity, both within and outside the active layer. Since the
light intensity is proportional to the Poynting vector F × H which, in turn, is proportional to |Fy |2 , we have
in the case of even TE waves:
d/2
Γ = d/2
0
0
cos2 (κx) dx +
=
∞
cos2 (κx) dx
2
d/2 cos (κx) exp[−γ(|x| − d/2)]dx
cos2 (κd/2)
1 +
γ[(d/2) + (1/κ) sin(κd/2) cos(κd/2)]
=
−1
.
(803)
The figure on the next page shows the dependence of Γ on the thickness d of the active layer in the case of
a GaAs/AlxGa1−x As laser.
Similar expressions can be obtained for odd-TE and also TM waves.
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CONFINEMENT FACTOR Γ
1.0
x=0.6
0.8
x=0.1
0.6
GaAs/AlxGa1–xAs
0.4
0.2
0.0
0.0
0.5
1.0
1.5
ACTIVE LAYER THICKNESS d (µm)
2.0
∗ Far-Field pattern.
Usually gas lasers (such as the HeNe laser) are characterized by highly collimated beams, thanks to the large
aspect-ratio of the cavity. Not so for laser diodes. It is therefore quite important to analyze the spatial
distribution of the field associated with the emitted radiation. This is called the ’Far-Field Pattern’ of the
diode. We can obtain it by considering TE waves in free space (that is, outside the diode for z > 0). The
equation describing the field is similar to Eq. (795) with the vacuum permittivity 0 replacing s . Separating
variables and using the continuity of Fy (x, z) at z = 0 (that is, at the vacuum/GaAs interface), we find for
the light intensity I(θ) (where θ is the angle between the chosen direction and the z -axis on the x, z -plane)
with respect to the intensity at θ = 0:
2
∞
θ
F
(x,
z)
exp(i
sin
θk
x)
dx
cos
y
0
I(θ)
−∞
.
=
2
I(0)
∞
−∞ Fy (x, z = 0) dx
2
(804)
For a symmetrical 3-layer diode and even TE waves, one can simply substitute the expressions for the electric
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FULL–ANGLE AT HALF–POWER θ
field given by Eqns. (796)-(800) above and compute the Far-Field Pattern. The beam divergence is quite
significant, since the entire area on the plane of the junction emits. For example, for the GaAs/AlxGa1−x As
system, the full-angle at half-power ranges between 35 and 60 degrees for x in the range 10-to-60% (see the
sketch in the figure below). For this reason lasers are often built in ‘stripe geometries’, in which only a small
portion of the area emits.
Accounting also for the spread along the angle φ (the angle between the chosen direction and the z -axis on
the y, z -plane), it can be shown that the beam shape is roughly ellipsoidal, with the ‘longitudinal axis’ along
the x axis.
80
x=0.60
GaAs/AlxGa1–xAs
60
40
20
0
0.0
x=0.10
0.5
1.0
1.5
ACTIVE LAYER THICKNESS d (µm)
2.0
∗ Threshold current.
As we have briefly mentioned above, lasing starts occurring at a sufficiently large injection current. We shall
not calculate this current, since the calculation is quite involved, but we shall outline the basic ingredients of
such a calculation.
First, note that when photons are emitted in the cavity, they can stimulate emission only if they are not
absorbed first. Thus, we must estimate the emission and absorption rates and see under which conditions
absoprtion is smaller than emission.
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Under strong enough injection, the density of free carriers will be sufficiently large to cause strong interparticle Coulomb scattering. This results in a sort of ‘equlibrium’ among the carriers themselves (not to be
confused with thermal equilibrium with the lattice and the environment). Therefore, we can describe the
‘inverted’ populations of electrons and holes as ‘equilibrium’ Fermi distributions, albeit at some larger carrier
temperature determined by the injection process. Thus, using Eqns. (776) and (789) above, we can express
the rates for spontaneous emission and absorption as:
2
Wsp−em (hν) = B
|Hphot| ρC (E)fC (E) ρV (hν − E − EG )[1 − fV (hν − E − EG )] dE ,
(805)
and
Wabs (hν) = B
2
|Hphot| ρC (E)[1−fC (E)] ρV (hν−E−EG )fV (hν−E−EG ) dE , (806)
where ρC and ρV are the density of states in the conduction and valence bands, respectively, and B is a
coefficient involving the photon density of states. For net amplification we must have Wsp−em > Wabs .
In terms of electron and hole quasi Fermi levels, it can be shown that this is equivalent to requiring
EF C − EF V > hν .
The gain g is the incremental increase of the energy flux per unit length in the active layer. Its calculation is
complex, since it depends on the device geometry, doping, semiconductor band structure, etc. For samples
with the band-tail described by Eq. (782) above, the gain can be calculated and one has:
g =
g0
(Jnom − J0 ) .
J0
(807)
where (looking at the figure below) g0 /J0 = 5 × 10−2 cm-µm-A and J0 = 4.5 × 103 A/cm2 − µm. In
Eq. (807) Jnom is the ‘nominal’ current density required to excite at unit quantum efficiency a 1 µm-thick
active layer. Therefore the actual current will be J = Jnom d/ηQ , where d is the layer thickness and
ηQ the quantum efficiency. The gain is superlinear at small currents (feature not captured by the equation
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above), but it grown linearly with J . For GaAs the gain ranges between 50 and 400 1/cm for Jnom in the
range 6-13 ×103 A/cm2 − µm, as sketched qualitatively in the figure below. Threshold occurs when the
gain satisfies the condition that a photon crosses the entire length of the cavity without attenuations, that is:
R exp[(Γg − α)L] = 1 ,
(808)
where R is the reflectance of the mirrors at the end of the cavity, Γ is the confinement factor, α the
absorption length, and L the length of the cavity. This equation may be recast as:
1
Γgth = α +
ln
L
1
R
,
(809)
in terms of the gain g at threshold, gth . Using the fact that J = Jnom d/ηQ , Eq. (807) and (809) we
have, finally:
J0 d
J0 d
1
1
Jth =
+
(810)
α +
ln
.
ηQ
g0 ηQ Γ
L
R
In order to reduce the threshold current one can increase ηQ (by minimizing the role of non-radiative modes
of decay, such as traps, for instance), Γ (by optimizing the discontinuities of refractive indices, for example),
L (via a different design of the diode), or R (improving the cleaving process or coating the free surfaces).
Also, one can reduce α (a material property, possibly also affected by impurities) and d (which requires the
excitation of a smaller volume of material). Progress is still being made in reducing the value of Jth .
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500
GaAs 300 K
ND=NA=4x1017 cm–3
g (cm–1)
400
300
200
100
0
0
2
4
6
8
10
3
2
Jnom (10 A/cm –µm)
12
14
• Quantum-effect Devices
– Esaki diode.
In 1957 Leo Esaki, working with heavily doped Ge p-n diodes, observed the unexpected current-voltage
characteristics shown in the figure below: Instead of the usual I -V characteristics of a diode (shown as a
dashed line), the current in forward bias increased quickly with voltage, reached a maximum (‘peak current’
density JP ) at a ‘peak voltage’ VP , went through a minimum (called ‘valley’) before finally merging with the
expected behavior. Biased in reverse, the diode also exhibited a very large current.
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J
JP
JV
VP
VV
V
The explanation – which in 1974 brought the Nobel Prize to Easki – is based on the diagram shown in the next
figure. Let’s follow the sequences shown in that figure. In reverse bias a large tunneling current flows from
valence to conduction band, thanks to the small bandgap of Ge (leftmost two frames, the top frame showing
schematically the band diagram, the bottom frame the corresponding position in the J -V plot. As the reverse
bias is reduced, the field across the junction drops and so does the tunneling current. At zero bias, obviously,
no current flows (second pair of frames from the left). But as a forward bias is applied, electrons in the
degenerate n-side of the junction can tunnel into the available empty states in the degenerately-doped p-side
of the junction. The current reaches a maximum when the ‘band’ of occupied conduction states lines-up with
the ‘band’ of empty states in the valence band (third pair of frames from the left). As the bias is increased even
more, an increasing fraction of electrons are prevented from tunneling since the face the band-gap (rightmost
frames). The tunneling current then begins to decrease, until, eventually, the ‘usual’ forward current of a p-n
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junction takes over.
Jn
–Jn
J
J
J
V
V
J
V
V
The ‘Esaki diode’ is just one example of tunneling devices exhibiting ‘negative differential resistance’ (ndr), the
resonant tunneling diode (RDT) discussed below being another notable example. This region is intrinsically
unstable: If we bias the diode in forward in the region of ndr, a random fluctuation causing a temporary
increase of the current will reduce the voltage drop across the device. In turn, this yields a further reduction
of the voltage resulting in a further boost to the current and so on. This unstable situation will end up with
the device having climbed ‘backwards’ to the peak current, at which point the situation will reverse itself. An
oscillation will result. Thus, devices exhibiting ndr are oscillators (usually in the microwave range), although
with a low power output.
∗ Tunneling probability.
In order to analyze a little more quantitatively the current-voltage characteristics of the Esaki diode we should
go back to our discussion of the Zener tunneling probability (Lecture Notes, Part 2, pages 126 and 127) and
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of the tunneling currents (pages 144-146).
Approximating the barrier for tunneling across the semiconductor bandgap as a triangular barrier (see figure
at page 126, right), we saw that the tunneling amplitude in the WKB approximation was (see Eq. (424)):


∗ 1/2 3/2
4(2m ) EG
 = e−βt /F .
Pt(E) ≈ exp 
(811)
3eh̄F
Accounting for momentum conservation, we should amend this expression. In this equation E is the total
electron energy in a one-dimensional model. In three dimensions, assuming an isotropic and parabolic
dispersion, we can write
h̄2
2
2
E =
(k
+
k
) = E + Ez ,
(812)
z
2m∗
where the z -axis has been assumed to be along the direction normal to the plane of the interface. Then, the
magnitude of the imaginary wavevector κ(z) in the gap will be κ(z) = [2m∗(EG − eF z + E )]1/2h̄,
which amounts to replacing E with Ez in Eq. (811) and adding E to the barrier height EG . Thus

Pt(E) ≈ exp 
∗ 1/2
4(2m )
(EG + E )
3eh̄F
where
E0 =
3/2

 ≈ e−βt /F exp
−2
E
E0
,
eF
h̄
.
4 (2m∗EG )1/2
(813)
(814)
∗ Tunneling current.
The tunneling current can be obtained as at pages 144-146 of the Lecture Notes, Part 2.
The current flowing from right to left (conduction band to valence band) when the occupied CB states are
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lined-up with the empty VB states will be:
2e
JCV =
(2π)3
h̄kz
dkz dk
fC (k)[1 − fV (k)] Pt ,
m∗
(815)
where fC and fV are, respectively, the Fermi functions for electrons in the n-side of the junction and for
holes on the p side of the junction. Similarly, for the current flowing from VB to CB:
h̄kz
2e
dk
JV C =
d
k
fV (k)[1 − fC (k)] Pt ,
(816)
z
(2π)3
m∗
so that the total current will be:
2e
Jtun = JCV − JV C =
(2π)3
h̄kz
dkz dk
[fC (k) − fV (k)] Pt .
m∗
(817)
Let’s now use the variables Ez and E defined above and use Eq. (813). We get:
Jtun =
em∗
2π2 h̄3
β /F
e t
dE
−2E /E0
e
E
0
dEz [fC (E + Ez ) − fV (E + Ez )] . (818)
Introducing as new integration variable the total energy E = Ez + E in place of Ez , we have:
Jtun =
em∗
2π2h̄3
eβt /F
E
V
EC
dE [fC (E) − fV (E)]
E
0
−2E /E0
.
dE e
(819)
The integration over E is trivial and we get, finally:
Jtun =
em∗
2π2h̄3
E0
eβt /F
2
E
V
EC
dE [fC (E) − fV (E)] e−2E/E0 .
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285
As a rough approximation we can replace the field F as the average field across the junction,
V −V
F ≈ bi
≈ (Vbi − V )
wd
−1/2
2s (Vbi − V ) −1/2
eN ∗ (Vbi − V )
=
,
eN ∗
2s
(821)
where N ∗ = NA ND /(NA + ND ). The integration in Eq. (820) extends from the region where emptyVB-states and occupied-CB-states overlap and it has been indicated as extending from EC to EV .
A compact expression for this current is:
V
V
Jtun = JP
exp 1 −
(822)
,
VP
VP
where JP is the peak current and VP the peak voltage.
∗ Currrent-Voltage characteristics.
In addition to the ‘direct’ tunneling current, there will be also the ‘usual’ diode current
eV
Jpn = JS exp
−1 ,
kB T
(823)
and the ‘excess current’, Jx , due to electrons tunneling via defects/traps in the gap. This current, responsible
for the large values of the valley current usually observed, can be calculated only when the density and
distribution (spatial and energetic) of the traps is known. Empirically, it can be written as:
∗ 1/2
4
m s
Jx = JV exp
(V − VV ) ,
(824)
3
N∗
where JV and VV are the valley current and voltage, respectively. Therefore, total current flowing across
the Esaki diode will be:
Jtot = Jtun + Jnp + Jx .
(825)
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– Resonant Tunneling Diode (RTD).
Another notable example of a diodes with negative differential resistance are the Resonant Tunneling Diodes
(RTDs). These are based on a remarkable property of quantum tunneling.
E
1
a
c
f
r
b
d
g
k1
κ2
k3
κ4
t
k5
E
V
∗ Double-barrier structure.
Let’s consider the double-barrier structure shown above and consider an electron of energy E smaller than
the height of either barriers, ∆E1 and ∆E2 (let’s stick to the simpler one-dimensional case for simplicity)
incident from the left. We wish to calculate the transmission probability T across the double-barrier. Let t2
be the thickness of the first barrier (region 2 in the potential-vs-position plot), t3 the thickness of the well,
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t4 the thickness of the second barrier. Let’s introduce the wavevectors:
k1
κ2
k3
κ4
k5
=
=
=
=
=
(2m∗E)1/2 /h̄
[2m∗ (∆E1 − E]1/2 /h̄
(2m∗E)1/2 /h̄
[2m∗ (∆E2 − E]1/2 /h̄
(2m∗E)1/2 /h̄
.
(826)
In this example k1 = k3 = k5 , but we can easily generalize to a situation in which the potentials at the
left, right, and in the well are different. Note that the ‘wavevectors’ κ2 and κ4 describe the decay-rate of
the wavefunction inside the barriers. Note, finally, that we have assumed that the effective mass is the same
in all regions. Therefore, the electron wavefunction will have the form
 ik x

e 1 + r e−ik1 x



κ x
−κ x

 ae 2 + be 2
ψ(x) =
c eik3 x + d e−ik3 x



f eκ4 x + g e−κ4 x



t eik5 x
(x ≤ 0)
(0 < x ≤ t2 )
(t1 < x ≤ t2 + t3 )
(t2 + t3 < x ≤ t2 + t3 + t4 )
(x > t2 + t3 + t4 )
,
(827)
so that the incident flux will be jin = h̄k1 /m∗ and the transmitted flux will be jout = |t|2h̄k5 /m∗ . The
transmission probability will be:
jout
k5
T =
=
|t|2 .
(828)
jin
k1
In principle, we could solve directly Scrödinger equation in each region using Eq. (827) and obtaining the 8
coefficients r , a, b, c, d, f , g , and t from the 8 equations implying the continuity of ψ(x) and dψ(x)/dx
at x = 0, x = t2 , x = t2 + t3 , and x = t2 + t3 + t4 . The algebra will be quite complex. In practice
another technique (based on transmission-reflection matrices in each region of constant potential) can be
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used to obtain the desired result:
T =
28k1 κ22 k32 κ24 k5
|A|2 (k12 + κ22 )(κ22 + k32 )(k32 + κ24 )(κ24 + k52 )
.
(829)
The factor A is given by the rather messy expression:
A=
where:
exp(κ2 t2 + κ4 t4 ){exp[i(−φ1 + φ2 + φ3 + φ4 + φ5 ] − exp[i(φ1 + φ2 − φ3 − φ4 + φ5 ]} +
exp(κ2 t2 − κ4 t4 ){− exp[i(−φ1 + φ2 + φ3 − φ4 − φ5 ] + exp[i(φ1 + φ2 − φ3 + φ4 − φ5 ]}
exp(−κ2 t2 + κ4t4 ){− exp[i(−φ1 − φ2 − φ3 + φ4 + φ5 ] + exp[i(φ1 − φ2 + φ3 + φ4 − φ5 ]
exp(−κ2 t2 − κ4 t4 ){exp[i(φ1 − φ2 − φ3 − φ4 − φ5 ] + exp[i(−φ1 − φ2 + φ3 + φ4 − φ5 ]}
(830)
φ1 = k3 t3 κ
φ2 = atan k2
1
κ
φ3 = atan k2
3
κ
φ4 = atan k4
3
κ
φ5 = atan k4
.
(831)
5
The remarkable property emerging from this equation is the following: Consider the first term of the factor
A. When
−φ1 + φ2 + φ3 + φ4 + φ5 = φ1 + φ2 − φ3 − φ4 + φ5 + 2πn ,
(832)
that is, when
κ2
κ4
−φ1 + φ3 + φ4 = φ1 − φ3 − φ4 + 2πn → k3 t3 = atan
+ atan
+ πn , (833)
k3
k3
where n is an integer, this term vanishes. But this is the only term containing in the exponential positive
‘damping’ terms k2 t2 and κ4 t4 . Thus the transmission probability reaches a maximum. If the double-barrier
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is symmetric (t2 = t4 and ∆E1 = ∆E2 ), one can see that T reaches unity (see the figure below): At
this ‘resonant energy’ the structure becomes completely transparent, as if the barriers were not there! This
happens when the electron energy matches the energy of the resonance (or ‘quasi-bound’ level) in the well.
TRANSMISSION PROBABILITY T
100
10–1
Symmetric double–barrier
10–2
m*=0.063m0,
∆E=0.3eV
t2=t4=2.5nm, t3=5.0nm
10–3
0.00
0.05
0.10
0.15
ENERGY E (eV)
0.20
0.25
∗ Resonant Tunneling Diode.
This condition is called ’resonant tunneling’. One possible way to interpret what’s happening and see how
this property can give raise to current-voltage characteristics with an ndr region is to consider the well-region
as a quantum well with ‘quasi-bound’ levels. The lowest-energy level is sketched with a red segment in
the figure below. (The bottom frames illustrate the corresponding point along the I -V curve with the
ndr region.) These are the same levels we encounter in quantum wells. However, their finite lifetime
(due to tunneling-induced leakage outside the well because of the finite barrier height) ‘broadens’ these
levels. Thus, the DOS in the well will be described not by delta-functions at the energy E0 of the level,
ρ(E) ∼ δ(E − E0 ), but by a Lorentzian, ρ(E) ∼ Γ/[(E − E0 )2 + Γ2 ], where Γ is the level-width
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given by Heisenberg principle: If τres is the lifetime of the state, then Γ ∼ h̄/τres . The lifetime of the
‘resonant’ state, τres can be estimated heuristically: In order to build a resonance, the electron wave has to
travel back-and-forth in the well of thickness w at least once. So, if vg is the group velocity of the electron
at the resonant energy E0 , the electron will hitthe barriers every 2w/vg (E)0 ) seconds. At each ‘hit’ it will
have a probability (Tl + Tr )/2 of escaping, so that
τres ∼
2w
1
.
vg (E0 ) Tl + Tr
(834)
As bias is applied to the diode, a tunneling current begins to flow. This can be viewed as ‘sequential
tunneling’ in which electrons tunnel first through the left barrier, with transmission probability Tl , propagate
in the well region and subsequently tunnel through the right barrier with transmission probability Tr , so that
the total tunneling probability will be approximately:
Ttot ≈ Tl Tr .
(835)
[In ‘sequential’ tunneling the two tunneling processes are considered uncorrelated. It is assumed that phase-breaking scattering
processes destroy the interference in the well region, so that the total tunneling probability is the product of the tunneling
probabilities for the two barriers. The two tunneling processes, in other words, are treated as independent and uncorrelated. In
‘coherent’ tunneling, instead, it is assumed that no phase-breaking scattering occurs in the well. Then the two tunneling processes
are correlated. In this case, keeping track of the phases in the well, one gets from Eq. (829):
Ttot =
α 0 Tl Tr
,
α1 T 2 Tt2 + α2 T 2 + α3 Tr2 + α4
l
l
(836)
where the coefficients αi reflect the effect of the phase-induced interference between the two barriers as functions of bias.
‘Coherent tunneling’ is the correct description in the case of low-temperature, short-well devices, since the weak scattering makes
it unlikely that electrons will lose memory of their phases when transiting the well region.]
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E
E
E
x
J
x
J
V
x
J
V
V
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As the bias increases, electrons in the Fermi see of the the left contact will line-up energetically with the
quasi-bound level in the well. In such a case, the electron will tunnel through the large local DOS of the
quasi-bound state, ‘resonating’ with this state. As we saw above, in the ideal limit of symmetric structures
at zero bias, at the resonant energy the double-barrier structure becomes completely transparent.
At resonance, α4 → 0 in Eq. (836), and the denominator is largely controlled by either the α2 or α3 term,
so that
α0 Tr
α0 Tl
Ttot,res ≈
or
Ttot,res ≈
,
(837)
α2 Tl
α3 Tr
depending on the properties (thickness, effective mass, heights, bias) of the barriers. For barriers sufficiently
‘transmissive’ one has approximately
Tmin
Ttot,res ≈
,
(838)
Tmax
where Tmin (Tmax ) is the smaller (larger) of the transmission probabilities Tl and Tr . Off resonance,
instead, the transmission probabilities are small (that is, Tl Tr << 1), so that
Ttot,of f −res ≈
α0
Tl Tr ≈ Tl Tr ,
α4
(839)
as we saw in Eq. (835).
∗ Tunneling current.
The calculation of the tunneling current flowing through the diode proceeds as in Eqns. (815) and following.
The current flowing from left-to-right will be
2e
JLR =
(2π)3
h̄kz
dkz dk
fL (E)[1 − fR (E + eV )] Ttot(E) ,
m∗
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293
where V is the applied bias. Similarly for the right-to-left current:
h̄kz
2e
dk
JRL =
d
k
fR (E)[1 − fL (E + eV )] Ttot(E) ,
z
(2π)3
m∗
(841)
so that, switching as usual the integration variables to energy:
Jtot =
em∗
2π2 h̄3
dEz dE [fL (E) − fL (E + eV )] Ttot (Ez , E ) .
(842)
At high temperature the witdh Γ of the resonance can be ignored compared to kB T , so that
Ttot (Ez , E ) ≈ T (E0 ) Γ δ(Ez − E0 ) ,
(843)
where E0 is the energy of the quasi-bound state (resonance) and T (E0 ) is a parameter fixed by the
properties of the barriers. Then, we can integrate Eq. (842):
Jtot ≈
em∗
2π2 h̄3
T (E0 ) Γ kB T ln 1 + exp
EF − E 0
kB T
.
(844)
At low-T , instead, we can approximate Ttot (Ez ) around the peak of the transmission as a Lorentzian,
Ttot (Ez ) ≈ Tmax Λ(Ez − E0 ), where
Λ(Ez − E0 ) =
1
.
1 + (Ez − E0 )2 /Γ2
(845)
We are left with a simple numerical integration of Eq. (842).
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