# Lecture 4 -- Energy Band Theory II -- Diodes to FET I

```From Semiconductor Physics to
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R. W&ouml;rdenweber
SS 2015, Tuesday 12:00-13.30
Seminarraum des II. Physikalischen Instituts
Energy Bands II
&amp; from Diodes to CMOS
Si crystal
CNT
28.4.2015
organic molecule
e-mail: r.woerdenweber@fz-juelich.de
Bloch-Theorem -- Boundary conditions
• Periodic potential
• Periodic boundary conditions
Quasi free electrons
 ( x)   ( x  Na)
 eikNa ( x)  ei 2n ( x)
with n=1, 2, 3, ….
• Kronig-Penney model
1D
Strongly bounded
electrons
Kronig-Penney model
Kronig-Penney Model -&gt;
Band theory
Solution:
f1()
1  2
sin  o a  sinh  ob 1    cos  o a  cosh  ob 1    cos k (a  b)
2  (1   )
f2()
1  2
sin  o a  sin  ob   1  cos  o a  cos  ob   1  cos k (a  b)
2  (  1)
[-1,1]
1.005
1.000
2
f()
E&lt;U
f1()
0.995
0.990
2.10
2.15
2.20
= E/U
k=0
f()
1
1
3
0
2
4
5
k=/(a+b)
-1
E &gt; Uo
f2()
-2
0
1
2
3
4
= E/U
5
6
7
oa=2
ob=3.14
0  E  Uo
E  Uo
Band theory &amp; Brillouin-Zone
E&lt;U
2
4
f()
1
3
1
3
0
2
4
5
-1
2
1
Band theory: Energy band in crystal
E &gt; Uo
-2
0
1
2
3
4
5
6
7
= E/U
Reduced-zone
representation
1. Brillouin-Zone: primitive Wigner-Seitz-Cell
of the reciprocal lattice of the single crystal
Extended-zone representation
Band theory- Dynamic description:
Particle velocity, Phase velocity, effective Mass
Phase velocity: rate, at which the phase of the
wave propagates in space
(= velocity of the wave front)
vp 
Group velocity: the velocity with which the
overall shape of the waves' amplitudes — known as
the modulation or envelope of the wave —
propagates through space.
vg 

k
 f
d 1 dE

dk  dk
center
frequency:  = E/ħ
The effective Mass is defined as the analogy to
the 2. Newton’s Law (F=ma).
The quantummechanic description of electrons of
the crystal in external electric field E gives the
equation of motion:
F m
Compare to the classic
(Newton‘s) mechanics
*
dvg
dt
with
with
m* 
effective Mass
1
1 d 2E
 2 dk 2
Band theory- Dynamic description:
Particle velocity, Phase velocity, effective Mass
• For the free particle the Dispersion relation is quadratic
E k2 =&gt; M*=const. (=me)
m* &lt; 0
• In the crystal the situation is more complex:
The dispersion relation is in general not quadratic
=&gt; the effective Mass depends on the velocity
=&gt; ħk = &lt;p&gt; is the crystal momentum (Kristallimpuls)
m* &gt; 0
• Important for semiconductors:
- Minimum of the conductivity band (curvature is positive =&gt; m* &gt;0)
- Maximum of the valence band (curvature is negative =&gt; m* &lt; 0)
m* &gt; 0
Reflection of electrons at the ‚ion-frames‘:
Far away from the Band edge the effective mass can also become
negative or infinite, analog to the Bragg-Reflection (2dsinθ = nλ) in the 1D
lattice:
 for small values of k, the electrons move according to their free mass
me
 for greater values of k they will be reflected, until no effective
acceleration due to the electric filed is possible: m* -&gt; 
 for more greater k-values the acceleration due to the external electric
field leads to the acceleration in the opposite direction: m* &lt; 0
m* &lt; 0
Effective Mass:
m* 
1
2
d E
 2 
 dk 
2
1
Band theory - Remarks
Charge carriers and current:
- k-values:
E
N per Band (2 spin states -&gt; 2N states (conditions))
Distance: k  2 N (a  b)
- completely full or empty energy levels do not
contribute to the charge transfer
Conductance band
Valence band
- The contribution of the almost full band (valence
band) is equivalent to the transport of positively
charged charge carriers (holes with mass m*)
(Alternatively one could take the negative mass, but this will make
the calculation (e.g. band borders) more complex)
E-k Diagrams
GaAs
Si
Ge
a) direct / indirect transition
b) Valence band:
Conduction band
- maxima at k=0
- 3 subbands:
- 1st &amp; 2nd are degenerated
m* via energy-band curvature
 heavy-hole band
&amp; light-hole band
- 3rd subband slightly reduced
energy :
split-off band
- at k=0 shape and curvature of all
subbands are essentially
orientation independent
Valence bands
kz
x
1st Brillouin
zone
fcc structure
(diamond, zinc blende,
e.g.: Si, Ge, GaAs, …)
L
ky
x
L kx
L
E-k Diagrams
Ge
GaAs
Si
c) Conduction band:
- similar features for Ge, Si, GaAs
- subbands with local and absolute
minima:
Ge: - mimimum along &lt;111&gt;
- indirect
- 8 equivalent minima
Si: - at k=0.8(2/a) towards &lt;100&gt;
- indirect
- 6-fold symmetry
( 6 equivalent minima)
GaAs: - minimum at zone center
- direct
- local minimum at &lt;111&gt; only 0.29eV higher
can not be ignored for larger fields
(e.g. Gunn-effect diode)
Conduction band
Valence bands
GaAs
Lecture 4:
Energy Band Theory II
- Fermi-Dirac statistics
- band bending
- doping
Semiconductor Devices
- Diode
- different types and applications
- basic principle of pn-diodes
- fabrication
- optical applications
- FET to CMOS
Fermi-Dirac statistics (1926):
Average occupation of a system by Fermions
(Pauli Exclusion Principle) at given energy:
f E  
1
1  e  E
f E  
EF
kT
1
and  
kT
with   
1
Enrico Fermi
(1901-1954)
f(E)
1  e( E  EF ) / kT
E – EF (eV)
Paul Dirac
(1902–1984)
Fermi-Dirac Statistics and Fermi Level:
Equilibrium distribution of carriers:
g c E   f E 
Electrons:
g v E   1  f E 
Holes:
E
EC
E
E
1-f(E)
Electrons
EC
gc(E)
gc(E)f(E)
EF
EV
gv(E)
EV
f(E)
Energy-bands
conduction
band
gv(E)(1-f(E))
Holes
Density of
states
Occupation
factor
Carrier
distribution
(Zustandsdichten)
(Verteilungsfunktionen)
(Besetzungsverteilung)
valence
band
Equilibrium Distribution
of Carriers:
Energy band
diagram
Density of
states
Occupation
factor
Electrons: g c E   f E 
Holes: g v E   1  f E 
EF above midgap
Variation of Fermi level
 occupation ratio
holes/electrons
EF near midgap
EF below midgap
Carrier
distribution
 from intrinsic charge carrier concentration to
small for applications :
e.g. Ge at 300K:
ni  2.9 x 1013 cm-3
Si at 300K:
ni  1.5 x 1010 cm-3
GaAs at 300K: ni  2.3 x 106 cm-3
(necessary:
ni &gt; 1015 cm-3 )
Doping with electric active defects (atoms)
Donor
Acceptor
in conductance band
free holes
in valence band
e.g. for Si: atom with higher valence
e.g. P or As
atom with lower valence
e.g. B or Al
typical doping density:
ND &gt;
1015
cm-3
Intrinsic carrier concentration [cm-3]
Intrinsic charge carrier
concentration:
T [K]
Doping:
n-dopted Si
e.g. P or As
E
Doping with electric
active defects
atoms with higher/lower
valence
x
Donor
p-dopted Si
B or Al
E
x
Acceptor
Doping:
 Ionization energies of impurities in Ge, Si, GaAs:
Ge
Donor
0.66 eV
Acceptor
Si
1.12
GaAs
1.42
Doping elements
S.M. Sze, Physics of Semiconductor Devices, 1967
Doping:
n-doped Si
ND = 1015/cm3
 carrier concentration in
doped semiconductors:
Freeze
out
Extrinsic T-region
n / ND
ND :=density of
donors
Intrinsic
T-region
n / ND
Temperature [K]
Doping:
- Fermi level
doped Si
‚relative‘
Fermi level:
ED
EF – Ei
[eV]
Ec
Conduction
band
Ei
Intrinsic
Fermi level
P
n-type
p-type
B
ND
Ev
EA
Temperature [K]
experiment
theory
Valence
band
Fermi function and Fermi Level:
‚Band bending‘:
E(x)
:
 electrostatic potential
valence band
E x   q   (x)
Electric field:
(1D):

E  
d
E  
dx
Charge density  :
(Poisson equation)
conduction
band
dE

dx

E

Electrostatic
potential
Electric field
Charge density
Fermi function and Fermi Level:
‚Band bending‘:
Example:
pn-junction
p-type
n-type
Ei := intrinsic Fermi level
(without defects and doping)
EF := Fermi level
(with doping)
pn-junction
Remark:
Ei roughly in the middle between
Ev and Ec (in equilibrium same
number of electrons and holes)
Position shifted by doping!
E

Electric field
Charge density
Lecture 4:
Energy Band Theory II
- Fermi-Dirac statistics
- doping
- band bending
Semiconductor Devices
- Diode
- different types and applications
- basic principle of pn-diodes
- fabrication
- optical applications
- FET to CMOS
Diodes:
The word Diode comes from (gr.) δίοδος (d&iacute;odos) - „Pass“, „Way“
Mechanical analog:
back-pressure valve
(R&uuml;ckschlagventil)
closed
open
History:
1874 First semiconductor component:
rectifier (Gleichrichter) diode,
Metal needle
made of PbS by Ferdinand Braun
1906 G.W. Pickard’s Patent for crystal detector
(rectifier diode, also called the Cat’s-whisker)
Semiconductor
(PbS)
Diodes:
The IV-curve of an ideal diode:
Applications:
- rectifiers ( Forward current)
- stabilizators &amp; limiters
( reverse voltage , Zener Diode)
- High-frequency signal detection
- Optical applications
- …
Types of diodes
pn-diodes (e.g. rectifiers)
PIN-Diodes (positive-intrinsic-negative diode, highfrequency applications)
Varactors (p-i-n diodes, with controlled barriers)
p
Zener-Diodes (stabilization, limitation)
n
Schottky-Diodes (metal-semiconductor junction, fast
rectifier)
Gunn-diode (negative differential resistance NDR,
microwave generation)
p
i
n
Tunnel-diode (high-frequency applications, Phonondetektor, -emitter)
…
Light diodes (LED), Laser diodes
Photodiode
OLED
….
Solarcells
LED
Laser diode
Fermi-Energy in Metals
Separate metals
Metals in contact
Electron-transfer
Fermi-Energy is equalized
Example: Thermocouple
‚Band bending‘
pn – Diodes:
Fermi-Energy in Semiconductors
Fermi-Energy is equalized
Distance
EL-EF is
predefined
in volume
pn – Diodes:
Formation of the depletion zone
Electrons
Holes
(Akzeptor-R&uuml;mpfe)
(Verarmungszone)
(Donor-R&uuml;mpfe)
pn – junction:
Space-charge region
pn – junction:
Currents
Drift current jR(L)
=
Diffuse current jF(L)
Conductance band:
a) Diffuse- or Recombination current jF(L):
- forward current
- driven with the concentration gradient
- against electric field
- Recombination after time in p-region
b) Drift-, Field- or Generation current jR(L)
- reverse current
- Thermal generation
- Minor charge transfer
p
Diffuse current jF(V)
n
=
Drift current jR(V)
Valence band:
(analog, but: movement of holes to higher energies is energetically favorable!)
a) Diffuse current jF(V)
b) Drift current jR(V)
Drift- and Diffuse currents compensate each other!
pn – junction:
Field-, Potential and Band Structure
Precondition: - thermodynamic balance (no external fields or temperature gradients)
Schottky-approximation:
SCZ
- in the depletion zone (SCZ) all Donors and
Acceptors are ionized
- Transition of the charge density  from the SCZ
to the material’s volume is stepwise (stufenf&ouml;rmig)
- out of the SCZ the total charge is  = 0
0 for x &lt; -wp
 = -eNA for -wp &lt; x &lt; 0
eND for 0 &lt; x &lt; -wp
0 for x &gt; wn
- out of the SCZ the semiconductor has no field :
E(x) = 0 for x &lt; -wp
x &gt; wn
- out of the SCZ EL and EV are defined in the
volume through the doping
-wp
0
wn
pn – junction:
Poisson equation
Electric field:
Parallel plate capacitor
Q
E 
F
Q – charge
F – surface
Addition of the plate with the
thickness 𝛥x with the charge 𝛥Q:
Density of the charge carriers:
Transfer to the continuum:
E ( x) 
1
 o r
  ( x)dx
pn – junction:
Electric field
E ( x) 
1
 o r
  ( x)dx
Qualitative: (x) = 0
 E(x) = constant = 0
(x) = const.  E(x) linear in x
The solution of the Poisson equation for
Integration constant
E(x)
Boundary conditions:
-wp 0
E(-0)=E(0)
Analog for n:
wn
wp N A  wn N D
pn – junction:
Schematic representation
pn-diode
Holes or Electrons
concentration
Charge carriers density
Electric Field
Electric Potential
pn – junction:
Field or Drift-current
Dynamic equilibrium
(schematically)
Drift-current
Diffuse-current
Diffuse-current
Drift-current
pn – junction:
(dynamic equilibrium)
 Diffuse-current (through the concentration gradient)
dn
Particle current Concentration gradient: j ( x)   D
dx
j
diff
 j
diff
n
j
diff
p
dn p 
 dnn


 e Dn
 Dp
dx
dx 

 Drift-current (through the el. field in SCZ)
j drift  jndrift  j pdrift  enn n  n p  p E
Mobility
 Equillibrium:
n
j diff  j drift  0
 no external current
Charge carrier are thermally
injected in the SCZ
SCZ
pn – junction:
with bias voltage
 Current ballance of Electrons:
(analog for holes)
p
n: Drift-current:
Electrons in p-SC are generated and transferred
to the n-SC
Limitation: thermal generation
I Gen  I drift
n
p: Diffuse-current:
Does not depend
on the potential
Electrons from the n-SC diffuse over the
potential barrier into the p-SC
Limitation: Potential barrier VD-U
I diff  exp  eVD  U  / kT 
- only a part travels over the barrier
- Idiff depends on the applied voltage U!
pn – junction:
with bias voltage
 Diffuse current at the applied potential U:
I ndiff (U )  Ae eVD U / kT  Ae eVD / kT eeU / kT  I ndiff (0)eeU / kT
 Diffuse current at U=0:
I ndiff (0)  Ae eVD / kT  I ndrift (0)
 Drift-current (approx.) independent on U:
(possible because of the thermal generation)
I ndrift (U )  I ndrift (0)
I ndiff (U )  I ndrifteeU / kT
 Total current of electrons:


I n  I ndiff  I ndrift  I ndrift eeU / kT  1
 Total current of electrons and holes:
(Holes analog to electrons)





I ges  I ndrift  I pdrift eeU / kT  1  I gen eeU / kT  1
pn – junction:
with bias voltage
Diffuse current
Drift-current


I ges  I gen eeU / kT  1
Pass
direction
12
normalized current I/Igen
10
Block
direction
8
6
4
2
0
-2
-10
-8
-6
-4
-2
0
2
4
6
8
normalized voltage U/(kT/e)
10
pn – junction:
Production
Doping of the metal
Diffusing of the doped materials
and structuring
Structuring and then diffusing of the
doped materials
Structuring and Implantation
Doping atoms have to plug into the crystal structure
and be electrically active!
pn-diodes for optical applications:
- optical sensors (light detector/meter,
position sensor, …. CCD camera)
- solar cells
- light emitting diode (LED)
- standard LED
LED
- Laser diode (LLED)
similar to LED
electronically pumped laser
- organic light-emitting diode (OLED)
(Light Emitting Polymer (LEP) or Organic ElectroLuminescence (OEL)
- ‘organic LED’ (polymer + transparent cond. (ITO)
- flat, bendable, cheap
- to be used for:
monitors and screens (inst. of LCD or Plasma)
electronic paper
novel illumination
Laser diode
OLED
SCZ
pn-diodes for optical applications:
- recombination of electrons
and holes
- indirect transition:
 phonon assisted
e.g., Si-diode
- direct transition:
 emission of photon
e.g., GaAs, GaP (green)
 wavelength:
- direct transition
- photon
h  c 1240nm


EG
EG [eV ]
Semiconductor: 1eV &lt; EG &lt; 3 eV
- indirect transition
- phonon assisted
 = 400-1200nm
(visible light: 400-700nm)
pn-diodes for optical applications:
- recombination of electrons
and holes
Ge
Si
GaAs
- indirect transition:
 phonon assisted
e.g., Si-diode
- direct transition:
 emission of photon
e.g., GaAs, GaP (green)
 wavelength:
h  c 1240nm


EG
EG [eV ]
Semiconductor: 1eV &lt; EG &lt; 3 eV
 = 400-1200nm
(visible light: 400-700nm)
pn-diodes for optical applications:
h  c 1240nm


EG
EG [eV ]
Examples:
GaAlAs : red (650nm) to IR (up to 1000 nm)
GaAsP, AlInGaP: red, orange, yellow
GaP: green
SiC: 1st comm. blue LED (low efficiency)
ZnSe: blue (no commercial use)
InGaN/GaN: UV, blue and green
CuPb: near IR (NIR)
White LED:
- combination of different colors
- blue LED + luminescent converter (P layer)
Engineering of material properties
Spectra of red, green, blue, white LED
blue LED (InGaN)
pn-diodes for optical applications:
Examples of application