CS代考 PN junction

PN junction
(I) Fundamentals (this lecture) (II) Fabrication (after midterm
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Last lecture: Distribution of Electrons
⚫ Obtainn(E)bymultiplyinggc(E)andf(E)
Energy band diagram
Density of States
Probability
of occupancy
Carrier distribution
gc(E) gv(E)

Last lecture: Distribution of Holes
⚫ Obtain p(E) by multiplying gv(E) and 1-f(E)
Energy band diagram
Density of States
Probability
of occupancy
Carrier distribution
gc(E) gv(E)

N-type Material
Energy band diagram
Density of States
Probability of occupancy
Carrier distribution
gc(E) orgv(E)
gv(E)[1-f(E)] 4
Carrier distribution

P-type Material
Energy band diagram
Density of States
Probability of occupancy
Carrier distribution
gv(E)[1-f(E)]
Carrier distribution
gc(E) orgv(E)

Last lecture: total current
The total current flowing in a semiconductor is the sum of drift current and diffusion current:
Jtot =Jp,drift +Jn,drift +Jp,diff +Jn,diff
Jp,drift =qppE,
Jn,drift =qnnE
=−qD dp, p dx
J =qD n,diff n

The characteristic constants for drift and diffusion are
Note that kT  26mVat room temperature (300K) q
This is often referred to as the “thermal voltage”.

PN junction – (I)
⚫ Formation of depletion region (DR)
⚫ Built-in potential of DR
⚫ Distribution of electric field and electric potential in DR
⚫ Effect of applied voltage on DR
⚫ Depletion capacitance of DR*
Reference Reading
▪ Chapter 3.1 (page 92-116)

Junctions of n- and p-type Regions
p-n junctions form the essential basis of all semiconductor devices. A silicon chip may have 108 to 109 p-n junctions today.
What happens to the electrons and holes if
n and p regions are brought into contact ?
aluminum ? aluminum

PN Junction Diode
⚫ When a P-type semiconductor region and an N- type semiconductor region are in contact, a PN
junction diode is formed.

Carrier concentration distribution in thermal equilibrium
⚫ Because of the difference in hole and electron concentrations on each side of the junction, carriers diffuse across the junction:
majority carrier
minority carrier
majority carrier
minority carrier
nn  electron concentration on N-type side (cm-3) ≈ND pnholeconcentrationonN-typeside(cm-3) ≈ni2/ND
pp  hole concentration on P-type side (cm-3) ≈NA npelectronconcentrationonP-typeside(cm-3) ≈ni2/NA

n or p (cm-3)
x1x2 x Carrier Depletion Region

Carrier Diffusion across Junction

Carrier Drift across Junction
⚫ Because charge density ≠ 0 in the depletion region, an
electric field exists, hence there is drift current.
Thermal equilibrium: balance between drift and diffusion 14

Carrier Drift across the Junction Thermal equilibrium: balance between drift and diffusion
aluminum aluminum wire

PN junction – (I)
⚫ The formation of depletion region
⚫ Built-in potential (two methods for Vbi)
⚫ Distribution of electric field and electric potential
⚫ Effect of Applied Voltage
⚫ Depletion capacitance
Reference Reading
▪ Chapter 3.1 (Page 92-116)

PN Junction in Equilibrium
⚫ In equilibrium, the drift and diffusion components of current are balanced; therefore the net current flowing across the junction is zero.
Jp,drift +Jp,diff =0 Jn,drift + Jn,diff = 0
Jp,drift =qppE,
Jtot = J p,drift + Jn,drift + J p,diff + Jn,diff = 0
=−qD dp, p dx
Jn,drift =qnnE
J =qDdn n,diff n dx

Built-in Potential, Vbi
⚫ Because of the electric field in the depletion region, there exists a potential drop across the
qp E=qD dp  p −dV=D dp p pdx pdx pdx
 V(x)−V(x)=Dplnpp =kTln NA
 −pdV=Dp x1 pn
 p q (n2/N) pniD
V0=kTlnNAND =Vbi
E = − dV dx
(Unit: Volts)

V0 =kTlnNAND

Built-In Potential Example
⚫ Estimate the built-in potential for PN junction below.
➢ Note that
ND = 1018 cm-3
NA = 1015 cm-3
ln(10)  26mV  2.3  60mV, at RT
V0 =kTlnNAND

Energy bands of n- and p- type

If n-type and p-type are in the same thermal equilibrium system, they have the same Fermi level.

V =kTlnNAND

V =kTlnNAND

V =kTlnNAND

V =kTlnNAND
E−E n=ND=NCexp fn Cn
p=NA=NVexp Vp fp  kT 
 kT  E−E

E−E n=ND=NCexp fn C
 kT  E−E
p=NA=NVexp V fp  kT 
V =kTlnNAND

PN junction – (I)
⚫ The formation of depletion region
⚫ Built-in potential
⚫ Distribution of electric field and electric potential
⚫ Effect of Applied Voltage
⚫ Depletion capacitance

Depletion Approximation
Charge is stored in the depletion region.
acceptor ions
donor ions
– – – – – –
quasi-neutral p region depletion region quasi-neutral n region r0 chargedensity(C/cm3)
(x−x ) po
(−x  x  0) po
(0xx ) no
− qN r (x)=
(x x)  no
Nd ≡ ND Na ≡ NA

E=−dV or E=−d dx dx
Two Governing Laws
Gauss’s Law describes the relationship of charge (density)
and electric field.   1 Q EdA= rdV= encl
E(x)−E(x )=1x r(x)dx
dE = r dx 
Poisson’s Equation describes the relationship between electric field distribution and electric potential
(x)−(x0)=xx −E(x)dx 0
d (x) dE(x) r(x)
dx2 =−dx=−

Depletion Approximation 1 (Electric field)
Gauss’s Law 1 x E0(x)−E0(x0)=Si r0(x)dx
-qNa E0(x)
1x E0(x)−E0(xn0)=Si qNddx
E0(x)=qNd (x−xno) (0xxno) Si
E0(x)=−qNa (x+xpo) (−xpo x0) s
E0(0)=−qNaxpo =−qNdxno s s

Depletion Approximation 2 (Electrostatic potential)
E0(x) x n no
E (0) = − qNa xpo = − qNd xno 0 s s
 (x)− (x )=−E (x)dx
0 Pois0so0n’sEqu0ation x0
qN dx2+ ax2

Depletion Approximation 3
0(x)=x −E0(x)dx+0(−xpo)=x qNa (x+xpo)dx+0 −xpo −xpo s
(x)= (x+x ) (−x x0) −x −x 
xx a qNa xdx+2 xpo dx
0s2po popo po s
0(x)=x −E0(x)dx+0(0)=x −qNd (x−xno)dx+qNa (0+xpo)2 0 0s 2s
= d − xdx+ xdx+ ax
) qN s 0 0 2s
(x)=qNd x(2x −x)2+qNa x 2 (0xx ) 0 2no 2po no

Built-in Potential, B (x)=qNa(x+x )2 (−x x0)
02po po s
(x)=qNd x(2x −x)2+qNa x 2 0 2no 2po
At x = xno
 = =qNd x2 +qNa x2
0 B 2 no 2 po ss
(0xx ) no
 =V =kTlnNdNa B bi q n2

Still don’t know xno and xpo
1. Require overall charge neutrality: qNa xpo = qNd xno
2. Require (x) continuous at x = 0:  =qNd x2 +qNa x2
B 2 no 2 po ss
– – – – – –
Two equations with two unknowns. Solution:
2  N sBd
q(Na + Nd )Na
2  N sBa
q(Na + Nd )Nd

Depletion Region Width Wdep E
B =kTlnNDNA q n2
Si=r,Si0 depletion region width Wdep
charge density (C/cm3)
– – – – – –
Wdep = xpo + xno = 2 N +N 
qNNB AD
Si 10−12 F/cm
is the permittivity of silicon.
xpo xno

PN junction – (I)
⚫ The formation of depletion region
⚫ Built-in potential
⚫ Distribution of electric field and electric potential
⚫ Effect of Applied Voltage
⚫ Depletion capacitance

Effect of Applied Voltage
The quasi-neutral N-type and P-type regions have low
resistivity, whereas the depletion region has high resistivity. Thus, when an external voltage VD is applied across the
diode, almost all of this voltage is dropped across the depletion region. (Think of a voltage divider circuit.)
If VD < 0 (reverse bias, or VR), the potential barrier to carrier diffusion is increased by the applied voltage. If VD > 0 (forward bias, or VF), the potential barrier to carrier diffusion is reduced by the applied voltage.
V (V0) V D =  R D
V (V0) FD

+Bias effect on electrons in depletion zone
⚫ Forward bias
⚫ An applied voltage in the forward direction as indicated assists electrons in overcoming the coulomb barrier of the space charge in depletion region. Electrons will flow with very small resistance in the forward direction.
E (built-in)
E_(external) +
Force on electron from externally applied voltage 38

+Bias effect on electrons in depletion zone
To forward bias the p-n junction, the p side is made more positive, so that it is “downhill” for electron motion across the junction. An electron can move across the junction and fill a vacancy or “hole” near the junction. It can then move from vacancy to vacancy leftward toward the positive terminal, which could be described as the hole moving right. The conduction direction for electrons in the diagram is right to left, and the upward direction represents increasing electron energy.
q(Vbi – VF) qVF EFn

Forward Biased Conduction
When the p-n junction is forward biased, the electrons in the n- type material which have been elevated to the conduction band and which have diffused across the junction find themselves at a higher energy than the holes in the p-type material. They readily combine with those holes, making possible a continuous forward current through the junction.

Forward Biased Conduction
The forward current in a p-n junction when it is forward-biased (illustrated below) involves electrons from the n-type material moving leftward across the junction and combining with holes in the p-type material. Electrons can then proceed further leftward by jumping from hole to hole, so the holes can be said to be moving to the right in this process.

PN Junction under Forward Bias
A forward bias decreases the potential drop across the junction. As a result, the magnitude of the electric field decreases, and the width of the depletion region narrows.
E0 (built-in) – E (external)

Depletion Approx. – with VD>0 forward bias
E (0) = − qNa xpo = − qNd xno 0 s s
Lower barrier and large hole (electron) density at the right
places lead to large current! Built-in potential B
 -V B BD
n=1017 p=103

Depletion Region Width Wdep At VD=0
2 N +N  W=x+x=SiA D
dep po no qNNB AD
2 N +N 
Wdep =xp +xn = Si  A D (B −VD)
The width of the depletion region is a function of the bias voltage and is dependent on NA and ND.

-Bias effect on electrons in depletion zone
Reverse bias
An applied voltage with the indicated polarity further impedes the flow of electrons across the junction. For conduction in the device, electrons from the N region must move to the junction and combine with holes in the P region. A reverse voltage drives the electrons away from the junction, preventing conduction.
E0 (built-in)
E (external)

Bias effect on electrons in depletion zone
⚫ To reverse-bias the p-n junction, the p side is made more negative, making it “uphill” for electrons moving across the junction. The conduction direction for electrons in the diagram is right to left, and the upward direction represents increasing electron energy.
q(Vbi – VR) qVR

Reverse Biased P-N Junction
The application of a reverse voltage to the p-n junction will cause a transient current to flow as both electrons and holes are pulled away from the junction. When the potential formed by the widened depletion layer equals the applied voltage, the current will cease except for the small thermal current.

PN Junction under Reverse Bias
A revers bias increases the potential drop across the
junction. As a result, the magnitude of the electric field
increases, and the width of the depletion region widens.
E0 (built-in) + E (external)
-xpo V(x) xno V0

Depletion Approx. – with VD<0 reverse bias -xp -xpo xno xn x E (0) = − qNa xpo = − qNd xno 0 s s Higher barrier and few holes in n-type lead to little current! Built-in poXtential B B-VD n=1017 The width of the depletion region is a function of the bias voltage and is dependent on NA and ND. If one side is much more heavily doped than the other (which is commonly the case), this can be simplified then: Depletion Region Width Wdep AtV =0 2Si NA +ND  D W=x+x=   dep po no qNNB 2 N +N  Wdep =xp +xn = Si  A D (B −VD) Wdep 2Si(B−VD) qN where N is the doping concentration on the more lightly doped side. PN junction – (I) ⚫ The formation of depletion region ⚫ Build-in potential ⚫ Distribution of electric field and electric potential ⚫ Effect of Applied Voltage ⚫ Depletion capacitance parallel-plate capacitor: ⚫ Capacitance per unit area: r’S is the relative dielectric constant of insulators. 0 is the permittivity of free space. Depletion capacitance Change in ΔV across diode causes: change of ΔQj at −xp change of −ΔQj at xn W dep >> DWdep
change in r

Depletion capacitance per unit area (depletion approx.) ⚫ In analogy, in pn junction:
C j (V ) =  S = Wdep(V)
2(B −V)(Na +Nd)
Cj(V)= S Wdep(V)
Cjo 1−V /B
siq NaNd 1 2 Na +Nd B

Alternative view of capacitance: depletion charge
Within depletion approximation:
Cj is slope of Qj vs.V characteristics:
Cj(V)= qsNaNd =Cjo/ 1−V 2(Na +Nd)(B −V) B
Differential capacitance

A depletion region (in which n and p are each much smaller than the net dopant concentration) is formed at the junction between p- and n-type regions
kT N N  0= lnA D
A built-in potential barrier (voltage drop) exists across the depletion region, opposing carrier diffusion (due to a concentration gradient) across the junction:
At equilibrium (VD=0), no net current flows across the junction Width of depletion region
decreases with increasing forward bias (p-type region biased at higher potential than n-type region)
increases with increasing reverse bias (n-type region biased at higher potential than p-type region)
Wj 2Si(0−VD) qN
Charge stored in depletion region→capacitance
Cj = ADSi Wj

Current flowing in a semiconductor is comprised of drift anddiffusioncomponents:J =qp E+qn E+qD dn−qD dp
tot p n ndxpdx
A region depleted of mobile charge exists at the junction between P-type and N-type materials.
➢ A built-in potential drop (V0) across this region is established by the charge density profile; it opposes diffusion of carriers across the junction. A reverse bias voltage serves to enhance the potential drop across the depletion region, resulting in very little (drift) current flowing across the junction.
➢ The width of the depletion region (Wdep) is a function of the bias voltage (VD).
si  + (V0 −VD )
q NA ND 
V =kTlnNAND

Electron and hole concentrations
n=N exp−(EC −EF) C  kT 
p=N exp−(EF −EV) V  kT 
n p = n2 i
n=N exp−(EC−Ei) i C  kT 
n=N exp−(Ei−EV) i V  kT 
At RT n=ND
n=nexp(EF −Ei) i kT
p=nexp−(EF −Ei) i  kT 

Electron and hole concentrations
n=n exp(EF −Ei), i  k T 
p=n exp−(EF −Ei)
i  k T  p = N A
N =n=nexp(E −E ) D iFin
N =p=nexp−(EF−Eip)
 V =kTlnNDNA
qV0=Eip – Ein

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