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From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 1From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 1
JF, SCO 1617
Semiconductor Science
S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education© 2013 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is prote cted by Copyright and written permission should be obtained from the
publisher prior to any prohibited reproduction, sto rage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photo copying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearso n Education, Inc., Upper Saddle River, NJ 07458.
16-Mar-2017
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 2From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 2
JF, SCO 1617
Intrinsic Semiconductors
S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education© 2013 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is prote cted by Copyright and written permission should be obtained from the
publisher prior to any prohibited reproduction, sto rage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photo copying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearso n Education, Inc., Upper Saddle River, NJ 07458.
16-Mar-2017
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 3
(a) Energy levels in a Li atom are discrete. (b) The energy levels corresponding to outer shells of isolated Li atoms form an energy band inside the crystal, for example the 2s level forms a 2s band.
Energy levels form a quasi continuum of energy within the energy band. Various energy bands overlap to give a single band of energies that is only partially full of electrons. There are states with
energies up to the vacuum level where the electron is free. (c) A simplified energy band diagram and the photoelectric effect.
Energy Bands in Metals
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 4
(a) Above 0 K, due to thermal excitation, some of the electrons are at energies above EF. (b) The density of states,g(E) vs. E in the band. (c) The probability of occupancy of a state at an energy E is f(E). (d) The product g(E)f(E) is the number of electrons per unit
energy per unit volume or electron concentration per unit energy. The area under the curve with the energy axis is the concentration of electrons in the band, n.
Energy Bands in Metals
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 5
Energy Bands in Metals
−+=
Tk
EEEf
B
Fexp1
1)(
2/12/132/3)2(4)( AEEhmE e == −πg
Density of states Fermi-Dirac function
dEEfEnFE
∫Φ+
=0
)()(g
3/22 3
8
=
πn
m
hE
eFO
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 6From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 6
JF, SCO 1617
(a) A simplified two dimensional view of a region of the Si crystal showing covalent bonds. (b) The energy band diagram of electrons in the Si crystal at absolute zero of
temperature. The bottom of the VB has been assigned a zero of energy.
Energy Bands in Semiconductors
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 7From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 7
JF, SCO 1617
(a) A photon with an energy hυ greater than Eg can excite an electron from the VB to the CB. (b) Each line between Si-Si atoms is a valence electron in a bond. When a photon breaks a Si-Si bond, a free electron and a hole in the Si-Si bond is created. The result is
the photogeneration of an electron and a hole pair (EHP)
Energy Bands in Semiconductors
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 8From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 8
JF, SCO 1617
A pictorial illustration of a hole in the valence band (VB) wandering around the crystal due to the tunneling of electrons from neighboring bonds; and its eventual recombination with a wandering electron in the conduction band. A missing electron in a bond represents a hole as in (a). An electron in a neighboring bond can tunnel into this empty state and thereby cause the hole to be displaced as in (a) to (d). The hole is able to wander around in the crystal as if it were free but with a different effective mass than the electron. A wandering electron in the CB meets a hole in the VB in (e), which results in the recombination and the filling of the empty VB state as in (f)
Hole Motion in a Semiconductor
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 9From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 9
JF, SCO 1617
(a) Energy band diagram.
(b) Density of states (number of states per unit energy per unit volume).
(c) Fermi-Dirac probability function (probability of occupancy of a state).
(d) The product of g(E) and f (E) is the energy density of electrons in the CB (number of electrons per unit energy
per unit volume). The area under nE(E) versus E is the electron concentration.
Semiconductor Statistics
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 10
Electron and Hole Drift Velocities
vde = Drift velocity of the electrons, µe = Electron drift mobility, Ex = Applied electric field, vdh = Drift velocity of the holes, µh = Hole drift mobility
vdh = µhEx
Conductivity of a Semiconductor
σ = Conductivity, e= Electronic charge, n = Electron concentration in the CB, µe = Electron drift mobility, p = Hole concentration in the VB, µh = Hole drift mobility
σ = enµe + epµh
vde = µeEx
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 11
Electron Concentration in CB
dEEfEnc
c
E
E∫+
=χ
)()(CBg
−−≈Tk
EEEf
B
Fexp)(2/1
2/132/3CB
)(
)()2(4)(
c
ce
EEA
EEhmE
−=
−= −πg
Density of states in the CBFermi-Dirac function ≈Boltzmann function
−−=Tk
EENn
B
Fcc
)(exp
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 12
Electron Concentration in CB
n = Electron concentration in the CB Nc = Effective density of states at the CB edge Ec = Conduction band edge, EF = Fermi energy kB = Boltzmann constant, T = Temperature (K)
−−=Tk
EENn
B
Fcc
)(exp
Effective Density of States at CB Edge
Nc = Effective density of states at the CB edge, me* = Effective mass of the electron
in the CB, k = Boltzmann constant, T = Temperature, h = Planck’s constant
2/3
2
*22
=
h
TkmN Be
c
π
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 13
Hole Concentration in VB
p = Hole concentration in the VBNv = Effective density of states at the VB edge
Ev = Valence band edge, EF = Fermi energykB = Boltzmann constant, , T = Temperature (K)
−−=Tk
EENp
B
vFv
)(exp
Effective Density of States at VB Edge
Nv = Effective density of states at the VB edge, mh* = Effective mass of a hole in
the VB, k = Boltzmann constant, T = Temperature, h = Planck’s constant
2/3
2
*22
=
h
TkmN Bh
v
π
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 14
−==
Tk
ENNnnp
B
gvci exp2
Mass Action Law
Thenp product is a “constant”, ni2, that depends on the material properties Nc, Nv,
Eg, and the temperature. If somehow n is increased (e.g. by doping), p must decrease to keep npconstant.
Mass action law applies
in thermal equilibrium
and
in the dark (no illumination)
ni = intrinsic concentration
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 15
Fermi Energy in Intrinsic Semiconductors
EFi = Fermi energy in the intrinsic semiconductorEv = Valence band edge, Eg = Ec − Ev = Bandgap energy
kB = Boltzmann constant, T = temperatureNc = Effective density of states at the CB edgeNv = Effective density of states at the VB edge
−+=
v
cBgvFi N
NTkEEE ln
2
1
2
1
me* = Electron effective mass (CB), mh
* = Hole effective mass (VB)
−+=
*
*
ln4
3
2
1
h
eBgvFi m
mTkEEE
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 16
Average Electron Energy in CB
= Average energy of electrons in the CB, Ec = Conduction band edge, kB = Boltzmann constant, T = Temperature
TkEE Bc 2
3CB +=
CBE
(3/2)kBT is also the average kinetic energy per atom in a monatomic gas (kinetic molecular theory) in which the gas atoms move around freely and randomly inside a container.
The electron in the CB behaves as if it were “free” with a mean kinetic energy that is (3/2)kBT and an effective mass me*.
dEEfE
dEEfEEE
c
c
E
E
∫
∫∞
∞
=)()(
)()(
CB
CB
CBg
g
Average is found from
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 17
Fermi EnergyFermi energy is a convenient way to represent free carrier concentrations (n in
the CB and p in the VB) on the energy band diagram.
However, the most useful property of EF is in terms of a change in EF.
Any change ∆EF across a material system represents electrical work input or output per electron.
For a semiconductor system in equilibrium, in the dark, and with no applied voltage or no emf generated, ∆EF = 0 and EF must be uniform across the system.
For readers familiar with thermodynamics, its rigorous definition is that EF is the
chemical potential of the electron, that is Gibbs free energy per electron. The
definition of EF anove is in terms of a change in EF.
∆EF = eV
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 18From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 18
JF, SCO 1617
Extrinsic Semiconductors
S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education© 2013 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is prote cted by Copyright and written permission should be obtained from the
publisher prior to any prohibited reproduction, sto rage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photo copying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearso n Education, Inc., Upper Saddle River, NJ 07458.
16-Mar-2017
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 19
Extrinsic Semiconductors: n-Type
(a) The four valence electrons of As allow it to bond just like Si but the fifth electron is left orbiting the As site. The energy required to release to free fifth-electron into the CB is very small. (b) Energy band diagram for an n-type Si doped with 1 ppm As. There are
donor energy levels just below Ec around As+ sites.
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 20
Extrinsic Semiconductors: n-Type
edhd
ied eN
N
neeN µµµσ ≈
+=
2
Nd >> ni, then at room temperature, the electron concentration in the CB will nearly be equal to Nd, i.e. n≈ Nd
A small fraction of the large number of electrons in the CB recombine with holes in the VB so as to maintain np= ni
2
n = Nd and p = ni2/Nd
np= ni2
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 21
Extrinsic Semiconductors: p-Type
(a) Boron doped Si crystal. B has only three valence electrons. When it substitutes for a Si atom one of its bonds has an electron missing and therefore a hole. (b) Energy band
diagram for a p-type Si doped with 1 ppm B. There are acceptor energy levels just above Ev around B− sites. These acceptor levels accept electrons from the VB and
therefore create holes in the VB.
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 22
Extrinsic Semiconductors: n-Type
haea
iha eN
N
neeN µµµσ ≈
+=
2
Na >> ni, then at room temperature, the hole concentration in the VB will nearly be equal to Na, i.e. p≈ Nd
A small fraction of the large number of holes in the VB recombine with electrons in the CB so as to maintain np= ni
2
p = Na and n = ni2/Na
np= ni2
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 23From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 23
JF, SCO 1617
Intrinsic, i-Sin = p = ni
Semiconductor energy band diagrams
n-typen = Nd
p = ni2/Nd
np = ni2
p-typep = Na
n = ni2/Na
np = ni2
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 24From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 24
JF, SCO 1617
Semiconductor energy band diagrams
Energy band diagrams for (a) intrinsic (b) n-type and (c) p-type semiconductors. In all cases, np= ni
2. Note that donor and acceptor energy levels are not shown. CB = Conduction band, VB = Valence band, Ec = CB
edge, Ev = VB edge, EFi = Fermi level in intrinsic semiconductor, EFn = Fermi level in n-type semiconductor, EFp = Fermi level in p-type semiconductor. χ is the electron affinity. Φ, Φn and Φp are the work functions for the intrinsic, n-
type and p-type semiconductors
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 25From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 25
JF, SCO 1617
Semiconductor Properties
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 26From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 26
JF, SCO 1617 Compensation DopingCompensation doping describes the doping of a semiconductor with both donors and acceptors to control the properties.
Example: A p-type semiconductor doped with Na acceptors can be converted to an n-type semiconductor by simply adding donors until the concentration Nd exceeds Na.
The effect of donors compensates for the effect of acceptors.
The electron concentration n = Nd − Na > ni
When both acceptors and donors are present, electrons from donors recombine with the holes from the acceptors so that the mass action law np = ni
2 is obeyed.
We cannot simultaneously increase the electron and hole concentrations because that leads to an increase in the recombination rate which returns the electron and hole concentrations to values that satisfy np= ni
2.
n = Nd − Na
p = ni2/(Nd − Na)
Nd > Na
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 27
Summary of Compensation Doping
ad NNn −=ad
ii
NN
n
n
np
−==
22
iad nNN >>−More donors than acceptors
More acceptors than donorsida nNN >>−
da NNp −=da
ii
NN
n
p
nn
−==
22
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 28From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 28
JF, SCO 1617
Degenerate Semiconductors
S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education© 2013 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is prote cted by Copyright and written permission should be obtained from the
publisher prior to any prohibited reproduction, sto rage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photo copying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearso n Education, Inc., Upper Saddle River, NJ 07458.
16-Mar-2017
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 29From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 29
JF, SCO 1617 Degenerate Semiconductors
(a) Degenerate n-type semiconductor. Large number of donors form a band that overlaps the CB. Ec is pushed down and EFn is within the CB. (b) Degenerate p-type semiconductor.
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 30From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 30
JF, SCO 1617
Energy bands under an applied electric field
S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education© 2013 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is prote cted by Copyright and written permission should be obtained from the
publisher prior to any prohibited reproduction, sto rage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photo copying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearso n Education, Inc., Upper Saddle River, NJ 07458.
16-Mar-2017
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 31From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 31
JF, SCO 1617
Energy bands bend in an applied fieldEnergy band diagram of an n-typesemiconductor connected to a voltagesupply of V volts.
The whole energy diagram tilts becausethe electron now also has an electrostaticpotential energy.
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 32From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 32
JF, SCO 1617Example: Fermi levels in semiconductors
An n-type Si wafer has been doped uniformly with 1016 phosphorus (P) atoms cm–3. Calculate the position of the Fermi energy with respect to the Fermi energy EFi in intrinsic Si. The above n-type Si sample is further doped with 2×1017 boron atoms cm–3. Calculate position of the Fermi energy with respect to the Fermi energy EFi in intrinsic Si at room temperature (300 K), and hence with respect to the Fermi energy in the n-type case above.
Solution
P (Group V) gives n-type doping with Nd = 1016 cm–3, and since Nd >> ni ( = 1010
cm–3 from Table 3.1), we haven = Nd = 1016 cm–3. For intrinsic Si,
ni = Ncexp[−(Ec − EFi)/kBT]
whereas for doped Si,
n = Ncexp[−(Ec − EFn)/kBT] = Nd
where EFi and EFn are the Fermi energies in the intrinsic andn-type Si. Dividing the two expressions
Nd /ni = exp[(EFn − EFi)/kBT]
so that EFn − EFi = kBTln(Nd/ni) = (0.0259 eV) ln(1016/1010) = 0.358 eV
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 33From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 33
JF, SCO 1617Example: Fermi levels in semiconductors
Solution (Continued)
When the wafer is further doped with boron, the acceptor concentration, Na = 2×1017 cm–3 > Nd = 1016 cm–3. The semiconductor is compensation doped and compensation converts the semiconductor to a p-type Si. Thus,
p = Na − Nd = 2×1017−1016 = 1.9×1017 cm–3.
For intrinsic Si,
p = ni = Nvexp[−(EFi − Ev)/kBT],
whereas for doped Si,
p = Nvexp[−(EFp − Ev)/kBT] = Na − Nd
where EFi and EFp are the Fermi energies in the intrinsic and p–type Si respectively Dividing the two expressions,
p/ni = exp[−(EFp − EFi)/kBT]
so that
EFp − EFi = −kBT ln(p/ni)
= −(0.0259 eV)ln(1.9×1017/1.0×1010) = −0.434 eV
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 34From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 34
JF, SCO 1617Example: Conductivity of n-Si
Consider a pure intrinsic Si crystal. What would be its intrinsic conductivity at 300K? What is the electron and hole concentrations in an n-type Si crystal that has been doped with 1016 cm–3
phosphorus (P) donors. What is the conductivity if the drift mobility of electrons is about 1200 cm2
V-1 s-1 at this concentration of dopants.
Solution
The intrinsic concentration ni = 1×1010 cm-3, so that the intrinsic conductivity is
σ = eni(µe + µh) = (1.6×10-19 C)( 1×1010 cm-3)(1450 + 490 cm2 V-1 s-1)
= 3.1×10-6 Ω-1 cm-1 or 3.1×10-4 Ω-1 m-1
Consider n-type Si. Nd = 1016 cm-3 > ni (= 1010 cm-3), the electron concentration n = Nd = 1016 cm-3 and
p = ni2/Nd = (1010 cm-3)2/(1016 cm-3) = 104 cm-3
and negligible compared to n.
The conductivity is 1111231619 cmΩ 92.1)sVcm 1200)(cm 101)(C 106.1( −−−−−− =××== edeN µσ
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 35From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 35
JF, SCO 1617
Electrons in Semiconductor Crystals
Direct and Indirect Bandgap Semiconductors
S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education© 2013 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is prote cted by Copyright and written permission should be obtained from the
publisher prior to any prohibited reproduction, sto rage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photo copying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearso n Education, Inc., Upper Saddle River, NJ 07458.
16-Mar-2017
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 36From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 36
JF, SCO 1617
The electron PE, V(x), inside the crystal is periodic with the same periodicity as that of theCrystal, a. Far away outside the crystal, by choice, V = 0 (the electron is free and PE= 0).
Electrons in Semiconductor Crystals
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 37From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 37
JF, SCO 1617
Electrons in Semiconductor Crystals
0)]([2
22
2
=−+ ψψxVE
m
dx
d e
h
V(x) = V(x + ma) ; m = 1, 2, 3…
Periodic Potential Energy
ψk(x) = Uk(x)exp(jkx)Bloch electron wavefunction
Electron wave vectorQuantized
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 38From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 38
JF, SCO 1617
Electrons in Semiconductor Crystals
Periodic Potential Energy
ψk(x) = Uk(x)exp(jkx)
k is quantized. Each k represents a particular ψk
Electron’s crystal momentum = hk
Each k and ψk have an energy Ek
×exp(−jωt)
Ψk(x,t)
External force = d(hk) / dt
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 39From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 39
JF, SCO 1617
Electrons in Semiconductor Crystals
Periodic Potential Energy in 3 D
k is quantized. Each k represents a particular ψk
V(r) = V(r + mT) ; m = 1, 2, 3…
Ek vshk
Crystal structure is very important
ψk(x) = Uk(r)exp(jk⋅r)
Crystal momentum
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 40From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 40
JF, SCO 1617
The E-k diagram of a direct bandgap semiconductor such as GaAs. The E-k curve consists of many discrete points with each point corresponding to a possible state, wavefunction ψk(x), that is
allowed to exist in the crystal. The points are so close that we normally draw the E-k relationship as a continuous curve. In the energy range Ev to Ec there are no points, i.e. no ψk(x) solutions
E vs. k Diagrams
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 41From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 41
JF, SCO 1617
Direct and Indirect Bandgap Semiconductors
S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education© 2013 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is prote cted by Copyright and written permission should be obtained from the
publisher prior to any prohibited reproduction, sto rage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photo copying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearso n Education, Inc., Upper Saddle River, NJ 07458.
16-Mar-2017
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 42From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 42
JF, SCO 1617
(a) In GaAs the minimum of the CB is directly above the maximum of the VB. GaAs is therefore a direct bandgap semiconductor. (b) In Si, the minimum of the CB is displaced
from the maximum of the VB and Si is an indirect bandgap semiconductor. (c) Recombination of an electron and a hole in Si involves a recombination center .
E vs. k Diagrams and Direct and Indirect
Bandgap Semiconductors
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 43From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 43
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pn Junctions
S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education© 2013 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is prote cted by Copyright and written permission should be obtained from the
publisher prior to any prohibited reproduction, sto rage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photo copying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearso n Education, Inc., Upper Saddle River, NJ 07458.
16-Mar-2017
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 44From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 44
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Properties of the pn junction
(a) The p- and n- sides of the pn junction before the contact.
(b) The pn junction after contact, in equilibrium and in open circuit.
(c) Carrier concentrations along the whole device, through the pn junction. At all points, npoppo = nnopno = ni
2.
(d) Net space charge density ρnet across the pn junction.
pn Junction
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 45From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 45
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(e) The electric field across the pn junction is found by integrating ρnet in (d).
(f) The potential V(x) across the device. Contact potentials are not shown at the semiconductor-metal contacts now shown.
(g) Hole and electron potential energy (PE) across the pn junction. Potential energy is charge potential = q V
pn Junction
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 46
Ideal pn Junction
Depletion Widths
ndpa WNWN =
ερ )(net x
dx
d =E
Field (E) and net space charge density
Field in depletion region
∫−=x
Wp
dxxx )(1
)( netρε
E
Acceptor concentration Donor concentration
Net space charge density
Permittivity of the medium
Electric Field
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 47
Ideal pn JunctionBuilt-in field
εnd
o
WeN−=E
=
2ln
i
dao n
NN
e
kTV
Built-in voltage
Depletion region width
( ) 2/12
+=da
odao NeN
VNNW
ε
ε = εo εr
where Wo = Wn+ Wp is the total width of the depletion region under a zero applied voltage
ni is the intrinsic concentration
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 48
Law of the JunctionApply Boltzmann Statistics (can only be used with nondegenerate semiconductors)
N1 = ppo N2 = pno
E1 = PE2 = 0 E2 = PE1 = eVo
−−=Tk
EE
N
N
B
)(exp 12
1
2
−−=Tk
eV
p
p
B
o
po
no )0(exp
−=
Tk
eV
p
p
B
o
po
no exp
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 49
Law of the Junction
Apply Boltzmann Statistics
−=
Tk
eV
p
p
B
o
po
no exp
−=
Tk
eV
n
n
B
o
no
po exp
=
=
2lnln
i
daB
no
poBo n
NN
e
Tk
p
p
e
TkV
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 50From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 50
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Forward biased pn junction and the injection of minority carriers (a) Carrier concentration profiles across the device under forward bias. (b). The hole potential energy with and
without an applied bias. W is the width of the SCL with forward bias
Forward Biased pn Junction
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 51
Forward Bias: Apply Boltzmann Statistics
Note: pn(0) is the hole concentration just outside the depletion region on the n-side
N1 = ppo
N2 = pn(0)
−−=Tk
EE
N
N
B
)(exp 12
1
2
−=
−−=Tk
eV
Tk
eV
Tk
VVe
p
p
BB
o
B
o
po
n expexp)(
exp)0(
pno/ppo
=
Tk
eVpp
Bnon exp)0(
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 52
Forward BiasLaw of the Junction: Minority Carrier Concentrations and Voltage
( )
( )
=
=
Tk
eVnn
Tk
eVpp
Bpop
Bnon
exp0
exp0
np(0) is the electron concentration just outside the depletion region on the p-side
pn(0) is the hole concentration just outside the depletion region on the n-side
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 53
Forward Bias: Minority Carrier Profile
hnn L
xx'px'p
′∆−=∆ δδ )()(
AssumptionsNo field in the neutral region (pure diffusion)Neutral region is very long
1/Lh = Mean probability of recombination per unit distance; or mean distance diffused
Change in excess hole concentration due to recombination over a small distance δx′would be
−∆=∆
hnn L
x'px'p exp)0()(
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 54
Forward Bias: Diffusion Current
'
)(hole, dx
x'pdeD
dx'
(x')dpeDJ n
hn
hD
∆−=−=
−∆
=
hn
h
hD L
x'p
L
eDJ exp)0(hole,
−
= 1exp
2
hole, Tk
eV
NL
neDJ
Bdh
ihD
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 55
Forward Bias: Diffusion Current
'
)(hole, dx
x'pdeD
dx'
(x')dpeDJ n
hn
hD
∆−=−=
−∆
=
hn
h
hD L
x'p
L
eDJ exp)0(hole,
−
= 1exp
2
hole, Tk
eV
NL
neDJ
Bdh
ihD
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 56From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 56
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The total current anywhere in the device is constant. Just outside the depletion region it is due to the diffusion of minority carriers.
Forward Bias: Total Current
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 57
Forward Bias: Diffusion Current
Hole diffusion current in n-side in the neutral region
−
= 1exp
2
hole, Tk
eV
NL
neDJ
Bdh
ihD
There is a similar expression for the electron diffusion current density JD,elec in the p-region.
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 58
Forward Bias: Diffusion Current
Ideal diode (Shockley) equation
−
= 1exp
Tk
eVJJ
Bso
2i
ae
e
dh
hso n
NL
eD
NL
eDJ
+
=
Reverse saturation current
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 59
Forward Bias: Diffusion Current
−
+= 1exp2
Tk
eVn
NL
eD
NL
eDJ
Bi
ae
e
dh
h
Intrinsic concentration
−=
Tk
ENNn
B
gvci exp2
Jso depends strongly on the material (e.g.bandgap) and temperature
ni depends strongly on the material (e.g. bandgap) and temperature
Ideal diode (Shockley) equation
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 60From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 60
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Schematic sketch of the I−V characteristics of Ge, Si, and GaAs pn junctions.
V
I
0.2 0.6 1.0
Ge Si GaAs
Ge, Si and GaAs Diodes: Typical Characteristics
1 mA
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 61
Forward Bias: Diffusion CurrentShort Diode
ln and lp represent the lengths of the neutral n- and p-regions outside the depletion region.
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 62
Forward Bias: Diffusion Current
Short Diode
−
+= 1exp2
Tk
eVn
Nl
eD
Nl
eDJ
Bi
ap
e
dn
h
ln and lp represent the lengths of the neutral n- and p-regions outside the depletion region.
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 63From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 63
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Forward Bias: Recombination Current
Forward biased pn junction, the injection of carriers and their recombination in the SCL. Recombination of electrons and holes in the depletion region involves the current
supplying the carriers.
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 64From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 64
JF, SCO 1617
Forward Bias: Recombination Current
A symmetricalpn junction for calculating the recombination current.
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 65From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 65
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Forward Bias: Recombination Current
he
eBCDeABCJ
ττ+≈recom
−−=Tk
VVe
p
p
B
o
po
M
2
)(exp
=
Tk
eVnp
BiM 2exp
+≈
Tk
eVWWenJ
Bh
n
e
pi
2exp
2recom ττ
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 66
Forward Bias: Recombination and Total Current
Recombination Current
( ) ]12/[exprecom −= TkeVJJ Bro
+
=
Tk
eVJ
Tk
eVJJ
Bro
Bso 2
expexp
Total diode current = diffusion + recombination
e
TkV B>
The diode equation
+=
h
n
e
piro
WWenJ
ττ2 where
−
= 1exp
Tk
eVJJ
Bo η
where Jo is a new constant and is an ideality factor
General diode equation
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 67From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 67
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Typical I-V characteristics of Ge, Si and GaAs pn junctions as log(I) vs. V. The slope indicates e/(kBT)
Typical I-V characteristics of Ge, Si and GaAs pn junctions
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 68From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 68
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Reverse biased pn junction. (a) Minority carrier profiles and the origin of the reverse
current. (b) Hole PE across the junction under reverse bias
Reverse Bias
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 69
Reverse Bias
Total Reverse Current
g
ii
ae
e
dh
h eWnn
NL
eD
NL
eDJ
τ+
+= 2
rev
Mean thermal generation time
Diffusion current in neutral regionsthe Shockley reverse current
Thermal generation in depletion region
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 70From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 70
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Left: Reverse I-V characteristics of a pn junction (the positive and negative current axes have different scales). Right: Reverse diode current in a Ge pn junction as a function of temperature in a ln(Irev) vs
1/T plot. Above 238 K, Irev is controlled by ni2 and below 238 K it is controlled by ni. The vertical axis
is a logarithmic scale with actual current values. (From D. Scansen and S.O. Kasap, Cnd. J. Physics. 70, 1070-1075, 1992.)
Reverse Current
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 71From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 71
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(a) Depletion region has negative (−Q) charges in Wp and positive (+Q) charges in Wp, which are separated as in a capacitor. Under a reverse bias Vr, the charge on the n-side is +Q. When the reverse bias is increased by δVr, this charge increases by δQ. (b) Cdepvs voltage across an abrupt pn junction
for 3 different set of dopant concentrations. (Note that the vertical scale is logarithmic.)
Depletion Capacitance: Cdep
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 72From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 72
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(a) The forward voltage across a pn junction increases by δV, which leads to further minority carrier injection and a larger forward current, which is increases by δI. Additional minority carrier charge δQis injected into the n-side. The increase δQ in charge stored in the n-side with δV appears as a though
there is a capacitance across the diode. (b) The increase δV results in an increase δI in the diode current. The dynamic or incremental resistance rd = δV/δI. (c) A simplified equivalent circuit for a
forward biased pn junction for small signals.
Dynamic Resistance and Capacitance
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 73From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 73
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Forward Bias Dynamic Resistance and Capacitance
I = Ioexp(eV/kBT)
I
V
dI
dVrd
th== Vth = kBT/e Dynamic resistance
dV
dQC =diff
Diffusion capacitance
I = Q/τh
d
hh
rV
IC
ττ ==th
diff
pn junction current
Diffusion capacitance
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 74From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 74
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Forward Bias Diffusion Capacitance
dV
dQC =diff
Diffusion capacitance
d
hh
rV
IC
22 thdiff
ττ ==
d
tt
rV
IC
ττ ==th
diff
Diffusion capacitance, long
diode (ac)
Diffusion capacitance, short diode (ac)
Minority carrier
transit time
τt = ln2 / 2Dh
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 75From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 75
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Depletion Capacitance: Cdep
dV
dQ
V
QC ==
δδ
dep
2/1
2/1dep )(2
)(
)(
+−==
da
da
o NN
NNe
VV
A
W
AC
εε
Definition
2/1
dep )(2
−=
VV
NeAC
o
dε
Depletion layer capacitance for p+n
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 76
EXAMPLE : A direct bandgap pn junction
Consider a symmetrical GaAs pn junction which has the following properties. Na (p-side doping) = Nd (n-side doping) = 1017 cm- 3(or 1023 m- 3), direct recombination coefficientB ≈2×10-16 m3 s-1, cross sectional area A = 1 mm2. Suppose that the forward voltage across the diode V = 0.80 V. What is the diode current due to minority carrier diffusion at 27 °C (300 K)assuming direct recombination. If the mean minority carrier lifetime in the depletion region were to be the same as this lifetime, what would be the recombination component of the diode current? What are the two contributions at V = 1.05 V ? What is your conclusion?Note that at 300 K, GaAs has an intrinsic concentration (ni) of 2.1× 106 cm-3 and a relative permittivity (εr) of 13.0 The hole drift mobility (µh) in the n-side is 250 cm2 V-1 s-1 and electron drift mobility (µe) in the p-side is 5000 cm2 V-1 s-1 (these are at the doping levels given).
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 77
EXAMPLE : A direct bandgap pn junction
Solution
Assuming weak injection, we can calculate the recombination times τe and τh for electrons and holes recombining in the neutral p and n-regions respectively. Using S.I. units throughout, and taking kBT/e= 0.02585 V, and since this is a symmetric device,
To find the Shockley current we need the diffusion coefficients and lengths. The Einstein relation gives the diffusion coefficients as
Dh = µhkBT/e= (250×10-4) (0.02585)= 6.46×10-4 m2 s-1,
De = µekBT/e= (5000×10-4) (0.02585) = 1.29×10-2 m2 s-1.
s1000.5)m 10)(1sm 100.2(
11 83231316
−−−− ×=
××=≈=
aeh BN
ττ
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 78
Solution (continued)
where kBT /ewas taken as 0.02585 V. The diffusion lengths are easily calculated as Lh = (Dhτh)1/2 = [(6.46×10-4 m2 s-1)(50.0×10-9 s)]1/2 = 5.69×10-6 m,Le = (Deτe) 1/2 = [(1.29×10-2 m2 s-1)(50.0×10-9s)]1/2 = 2.54×10-5 m.Notice that the electrons diffuse much further in the p-side. The reverse
saturation current due to diffusion in the neutral regions is
A 104.4
)101.2)(106.1()10)(1054.2(
1029.1
)10)(1069.5(
1046.6)10(
21
21219235
2
236
46
2
−
−−
−
−
−−
×≈
××
××+
××=
+= i
ae
e
dh
hso en
NL
D
NL
DAI
µA 0.12orA 102.1V 02585.0
V 80.0exp)A 104.4(
exp
721
diff
×=
×=
=
−−
Tk
eVII
Bso
Thus, the forward diffusion current at V = 0.80 V is
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 79
Solution (continued)
Recombination component of the current is quite difficult to calculate because we need to know the mean electron and hole recombination times in the SCL. Suppose that, as a first order, we assume that these recombination times are as above.
The built-in voltage Vo is
Depletion layer width W is
As this is a symmetric diode, Wp = Wn = W /2. The pre-exponential Iro is
23 23
2 12 2
10 10ln (0.02585) ln 1.27 V
(2.1 10 )a dB
oi
N Nk TV
e n
= = = ×
µm. 0.116or m, 101.16
)m )(10m C)(10 10(1.6
V) 80.0)(1.27m 10)(10m F 102(13)(8.85
))((2
7
2/1
32332319
32323112
2/1
−
−−−
−−−
×=
×−+×=
−+=da
oda
NeN
VVNNW
ε
2 2pi n i
roe h e
WAen W Aen WI
τ τ τ
= + =
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 80
Solution (continued)
6 19 12 713
8
(10 )(1.6 10 )(2.1 10 ) 1.16 103.9 10 A
2 5.00 10
− − −−
−
× × ×= ≈ × ×
so that at V = 0.8 V,
The recombination current is more than an order of magnitude greater than the diffusion current. If we repeat the calculation for a voltage of 1.05 V across the device, then we would find Idiff = 1.9 mA and Irecom= 0.18 mA, whereIdiff
dominates the current. Thus, as the voltage increases across a GaAs pn junction, the ideality factor η is initially 2 but then becomes 1 as showm in Figure 3.20.
The EHP recombination that occurs in the SCL and the neutral regions in this GaAs pn junction case would result in photon emission, with a photon energy that is approximately Eg. This direct recombination of injected minority carriers and the resulting emission of photons represent the principle of operation of the Light Emitting Diode (LED).
µA 2.1orA 100.2)V 02585.0(2
V .80exp)A 109.3(
2exp
613
recom
−− ×=
×≈
≈
Tk
eVII
Bro
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 81From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 81
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LEFT: Considerp- andn-type semiconductor (same material) before the formation of thepn junction,, separated from each and not interacting. RIGHT: After the formation of thepn junction, there is a built-in voltage across the junction.
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 82From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 82
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Energy band diagrams for a pn junction under (a) open circuit and (b) forward bias
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 83From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 83
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Energy band diagrams for a pn junction under reverse biasLEFT: Shockley model
RIGHT: Thermal generation in the depletion region
From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 84From: S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education, USA 84
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Light-emitting diodes I
S.O. Kasap, Optoelectronics and Photonics: Principles and Practices, Second Edition, © 2013 Pearson Education© 2013 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This publication is prote cted by Copyright and written permission should be obtained from the
publisher prior to any prohibited reproduction, sto rage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photo copying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearso n Education, Inc., Upper Saddle River, NJ 07458.
21-Mar-2017