# EE 30357 Semiconductors II Devices Lecture Note #4

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"EE 30357 Semiconductors II Devices Lecture Note #4"

```					                                EE 30357: Semiconductors II: Devices
Lecture Note #4 (01/21/09)

Review of semiconductors p-n junctions
Grace Xing

Outline:
Concept – Graph – Equation (CGE learning)
1. Current in homogeneous semiconductors (carrier concentration, mobility,
saturation velocity) (from last class)
2. Band diagram of pn junctions (built-in voltage, depletion width, junction voltage…)
3. Currents in pn junctions (next class)

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Grace Xing---EE30357 (Semiconductors II: Devices)
Current density J in homogeneous semiconductor:
J e (electron current density)  (charge per electron) (density of electrons) (average electron velocity)
J h (hole current density)  (charge per hole) (density of holes) (average hole velocity)
J total  J e  J h  qnvn  qpv p  qn(  n E)  qp(  p E)    (Equation)

 : mobility

(concept)
Both electrons and holes contribute to the total
current density, units: Amperes/area (typically A/cm2)

This linear increase and then saturation of                             I                  (Graph)
current is pretty unique to semiconductors!

Metals tend to render a linear I-V (small resistance)
before they burns out (fuses). Insulators tend to have
super linear I-Vs (very large resistance).
V

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Grace Xing---EE30357 (Semiconductors II: Devices)
Mobility,velocity and scattering
(Concept)
Mobility describes how easy electrons/holes can be accelerated by an electric field.
- The less scattering electrons/holes suffer, the larger their mobility.
- The smaller electron/hole effective mass, the larger their mobility.

vdn           Velocity
n       
E            Electric field

qE
with   v dn      *
tn
mce

qt n
(Graph)
 n  *           (Equation)
     m ce
13
In Si, “mean free time” t n  2 10 s  200 fs


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Grace Xing---EE30357 (Semiconductors II: Devices)
Mobility (Temperature & doping concentration dependence)

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Grace Xing---EE30357 (Semiconductors II: Devices)
This is a Log-log plot
High Field Effects: velocity saturation                                     Can you plot it in linear-linear plot?

V
Linear-linear V-E

E

Velocity Saturation
(Due to strong electron-lattice
or phonon collision)

Velocity overshoot in GaAs

                
1
qt n                                                                                            2
v dn  * E                                                                                m  d E
*
mce                              Recall Jn=-qnvdn (2.32)                              ce
dK 2
The smaller effective mass, the larger the mobility, generally speaking
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Grace Xing---EE30357 (Semiconductors II: Devices)
Two similar semiconductors, doped differently, before being joined in a junction.
Figure 5.2

1
Vbi  |  p  n |
q
Built-in voltage (contact potential) = difference in Fermi levels of the two materials

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Grace Xing---EE30357 (Semiconductors II: Devices)
A prototype homojunction: (a) The physical diagram; (b) the distribution of charge; (c) the electric field,
obtained by integrating the charge; (d) the voltage, obtained by integrating the field; (e) the energy band
diagram, with the same shape as the voltage but inverted.
Figure 5.11

Depletion                p-n junction
regions                                                             n-p junction (Anderson)

Concept – Graph
Common mistake 1: direction or sign of E field
Common mistake 2: difference between electrostatic potential V and
electron energy (E = -qV) on (electron) band diagram
Challenge: what is V and E for holes for the n-p junction on the right?
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Grace Xing---EE30357 (Semiconductors II: Devices)
P-i-n junction (recitation this week)
• Can you draw band diagram of p-i-n junction and its properties?

Neutral              Neutral
i-region
n-region              p-region

n-p junction
Q

e

E

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Grace Xing---EE30357 (Semiconductors II: Devices)
The pn homojunction under reverse bias. Solid line: equilibrium energy band diagram; dashed line: energy
band diagram under reverse bias. The field increases; this requires more ionized acceptors and donors, so the
depletion region gets wider under reverse bias.
Figure 5.6

Junction voltage

Junction voltage at
equilibrium: Vbi

Ef   At equilibrium, Ef is flat!

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Grace Xing---EE30357 (Semiconductors II: Devices)
The space charge region width under equilibrium and forward bias, and the corresponding energy band
diagrams. The depletion width decreases under forward bias.
Figure 5.7

Junction voltage

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Grace Xing---EE30357 (Semiconductors II: Devices)
E(electric energy : eV )

E  QV (qV for electrons)
Parabolic band diagram
V (electric potential :V )

e  V (i.e. delV)
integral
e (electric field :V / cm)
Linear electric field
e   /  (i.e. divergence of electric field vector)

integral              unit check :
(1 / cm)e (V / cm)  Q(C / cm3 )/ (F / cm)
Constant charge density           LHS :V / cm2
RHS : (C / F ) / cm2  (C / (C / V )) / cm2  V / cm2

(charge density : C / cm3 )

4 most important concepts in “EE”
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Grace Xing---EE30357 (Semiconductors II: Devices)
Knowns: ND, NA, ni, Vbi, Vj
Unknowns: wn, wp

Setup equations:
1. Charge neutrality
2. Gauss’s law (Poisson equation)
3. Electric field vs. electric potential
1. wn N D  wp N A

de n       qN D                       de p        qN A
2.                    ,                          
dx                                    dx          
Boundary conditions : e p (wp )  0, e n (wn )  0

notice that electric field points to -x direction

qN D wn            qN A wp
e max                                  , consistent with eqn.1
                
1 dEvac
3. e           
q dx
area :
1

wn  wpqN D
w  Vj            integration of e (x)  junction voltage
2             n                  i.e. the area enclosed by the triangle  V j
wn         wn           NA
and,                     
wn  wp        N        N A  ND
wn  D wn
NA                                                   2 V j (N A  N D )
 w  wn  wp 
qN A N D
Equations
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Grace Xing---EE30357 (Semiconductors II: Devices)
How carrier concentrations on both sides of the junction are related?

Equilibrium energy band
diagram for a step
EF           homojunction
(n+/p).

ND > N A

Therefore,

Depletion on the n side
<
that on the p side
(recall W nND = W pNA)

Concept – Graph
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Grace Xing---EE30357 (Semiconductors II: Devices)
How carrier concentrations on both sides of the junction are related?

Equilibrium energy band
Electric field                                 diagram for a step
junction.
Figure 5.14

( ECn  E f )/kT                      ( ECp  E f )/kT            ( ECp  E f )/kT
nn0  N C e                            n p0  N C e                        NC e

ECp  ECn  qV j ( qVbi at equalibrium)

nn0
             e
qVbi /kT

n p0

This junction function holds at non-equalibrium too

nn (depletion edge at n  side)
e
qV j /kT
i.e.
Equations                                   n p (depletion edge at p  side)
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Grace Xing---EE30357 (Semiconductors II: Devices)

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