Pierret, Chapter 2 “Carrier Modeling” Jeff Davis FALL 2007 ECE3040
References • Prof. Alan Doolittle’s Notes – users.ece.gatech.edu/~alan/index_files/ECE3040index.htm
• Prof. Farrokh Ayazi’s Notes – users.ece.gatech.edu/~ayazi/ece3040/
• Figures for Require Textbooks – (Pierret and Jaeger)
Important Early Dates Leading to Quantum Mechanics • • • • • • • •
1901 - Planck models blackbody radiation assume that energy release from atoms is quantized 1905 - Einstein defines photons to describe photoelectric effect 1910 - Rutherford experiment to determine atom core 1913 - The Bohr atom is proposed (semi-classical) 1920s - Wave particle duality of EM radiation is broadly accepted 1925 - DeBroglie postulates that wave-particle duality applies to matter 1926 - Schrodinger proposes wave mechanics with his famous equation … Wave functions are born! 1927 - Davisson and Germer experiment with diffracting e-beam from nickel crystal to support DeBroglie’s hypothesis
Energy Quantization - READING APPENDIX A “The Bohr Atom (1913)”
Energy Hydrogen electron
mo q 4 13.6 eV =! = ! 2( 4" # 0 ! n) 2 n2
where mo = electron mass, ! = planks cons tan t / 2" = h / 2" q = electron ch arg e, and n = 1,2,3... n is the principle quantum number
Quantum Numbers
Designates shell
N = principle quantum number (N = 1,2,3,..) Size of orbital N>=(L+1) L = angular quantum number (L = 0,1,2,3,4,5,6,7 OR s,p,d,f,g,h, i, k) Designates subshell
M = magnetic quantum number ( orientation of the orbitals) M can be negative! |M|0 Kelvin) For (Ethermal=kT)>0 Electron free to move in conduction band Ec + Ev “Hole” free to move in valence band
Carrier Movement Under Bias For (Ethermal=kT)>0 Electron free to move in conduction band
Ec + Ev “Hole” movement in valence band
Direction of Current Flow
Direction of Current Flow
Carrier Movement Under Bias For (Ethermal=kT)>0 Electron free to move in conduction band
Ec + Ev “Hole” movement in valence band
Direction of Current Flow
Direction of Current Flow
Carrier Movement Under Bias For (Ethermal=kT)>0 Electron free to move in conduction band
Ec + Ev “Hole” movement in valence band
Direction of Current Flow
Direction of Current Flow
Material Classification based on Size of Bandgap: Ease of achieving thermal population of conduction band determines whether a material is an insulator, semiconductor, or metal
Energy Dependence of Bandgap Energy GaAs Eg(T=0) =1.519 [eV] α= 5.41x10-4 (eV/K) β= 204 [K] Silicon Eg(T=0) =1.166 [eV] α = 4.73x10-4 (eV/K) β = 636 [K]
"T 2 E g (T ) = E g (T = 0) ! T +#
Germanium Eg(T=0) =0.7437 [eV] α = 4.77x10-4 (eV/K) β = 235 [K]
Physically why does this occur?
Energy Dependence of Bandgap Energy - Example GaN
Ramirez-Flores, G., H. Navarro-Contreras, A. Lastrae-Martinez, R.C. Powell, J.E. Greene, Temperature-dependent optical band gap of the metastable zinc-blende structure beta -GaN, Phys. Rev. B 50(12) (1994), 8433-8438.
Definition of Intrinsic Carrier Concentration, ni
Intrinsic Carrier Concentration Ec Ev •Intrinsic carrier concentration is the number of electron (=holes) per cubic centimeter populating the conduction band (or valence band) is called the intrinsic carrier concentration, ni •ni = f(T) that increases with increasing T (more thermal energy) At Room Temperature (T=300 K) ni~2e6 cm-3 for GaAs with Eg=1.42 eV, ni~1e10 cm-3 for Si with Eg=1.1 eV, ni~2e13 cm-3 for Ge with Eg=0.66 eV, ni~1e-14 cm-3 for GaN with Eg=3.4 eV
Temperature Dependence of Intrinsic Carrier Concentration
ni = N c N v e
!
Eg 2 kT
k = boltzmann constant = 1.38e-23 J/K
Definition of “effective mass” of a charge carrier
Carrier Movement in Free Space
Newton’s second law of motion! F=ma
dv F = "qE = mo dt F ! force, v ! velocity, t ! time, q ! electronic ch arg e, mo ! electron mass
Carrier Movement Within the Crystal
Electron sees a periodic potential due to the atomic cores
Carrier Movement Within the Crystal dv F = "qE = m dt F ! force, v ! velocity, t ! time, q ! electronic ch arg e,
dv dt F ! force, v ! velocity, t ! time, q ! electronic ch arg e,
mn* ! electron effective mass
m*p ! hole effective mass
F = qE = m*p
* n
Effective Mass used to estimate “mobility” of carrier in the lattice electron
holes
mn*= 0.33mo
mh*= 0.5mo
Germanium
mn*= 0.22mo
mh*= 0.31mo
GaAs
mn*= 0.063mo
mh*= 0.5mo
Silicon
A.K.A. Conductivity Effective Mass
Effective Mass for Different Estimations
Definition of “extrinsic semiconductor” of a charge carrier
Extrinsic, (or doped material): Concept of a Donor “adding extra” electrons
Example: P, As, Sb in Si
Concept of a Donor “adding extra” electrons: Band diagram equivalent view
Extrinsic, (or doped material): Concept of an acceptor “adding extra” holes
Example: B, Al, In in Si
Concept of an Acceptor“adding extra hole”: Band diagram equivalent view
Hole Movement
All regions of material are neutrally charged.
Empty state is located next to the Acceptor
Hole Movement
+
Another valence electron can fill the empty state located next to the Acceptor leaving behind a positively charged “hole”.
Hole Movement
+
The positively charged “hole” can move throughout the crystal (really it is the valance electrons jumping from atom to atom that creates the hole motion).
Hole Movement
+
The positively charged “hole” can move throughout the crystal (really it is the valance electrons jumping from atom to atom that creates the hole motion).
Hole Movement
+
The positively charged “hole” can move throughout the crystal (really it is the valance electrons jumping from atom to atom that creates the hole motion).
Hole Movement
Region around the acceptor has one extra electron and thus is negatively charged.
+
Region around the “hole” has one less electron and thus is positively charged.
The positively charged “hole” can move throughout the crystal (really it is the valance electrons jumping from atom to atom that creates the hole motion).
Summary of Important terms and symbols Bandgap Energy: Energy required to remove a valence electron and allow it to freely conduct. Intrinsic Semiconductor: A “native semiconductor” with no dopants. Electrons in the conduction band equal holes in the valence band. The concentration of electrons (=holes) is the intrinsic concentration, ni. Extrinsic Semiconductor: A doped semiconductor. Many electrical properties controlled by the dopants, not the intrinsic semiconductor. Donor: An impurity added to a semiconductor that adds an additional electron not found in the native semiconductor. Acceptor: An impurity added to a semiconductor that adds an additional hole not found in the native semiconductor. Dopant: Either an acceptor or donor. N-type material: When electron concentrations (n=number of electrons/cm3) exceed the hole concentration (normally through doping with donors). P-type material: When hole concentrations (p=number of holes/cm3) exceed the electron concentration (normally through doping with acceptors). Majority carrier: The carrier that exists in higher population (ie n if n>p, p if p>n) Minority carrier: The carrier that exists in lower population (ie n if n>NA and ND>>ni
n ! ND
and
2 i
n p! ND
If NA>>ND and NA>>ni
p ! NA
and
ni2 n! NA
Developing the mathematical model for electrons and holes Example: An intrinsic Silicon wafer has 1e10 cm-3 holes. When 1e18 cm-3 donors are added, what is the new hole concentration?
n ! ND = 1018 cm"3
(
10
)
2
10 n "3 "3 p= = cm = 100 cm 18 n 10 2 i
Developing the mathematical model for electrons and holes Example: An intrinsic Silicon wafer has 1e10 cm-3 holes. When 1e18 cm-3 acceptors and 8e17 cm-3 donors are added, what is the new hole concentration? 2
" 1x10 ! 8x10 % 1x10 ! 8x10 10 p= + $ + 1x10 ' 2 2 # & 18
17
18
17
p = 2x1017 cm!3 = NA ! ND
(
)
2
Developing the mathematical model for electrons and holes Example: An intrinsic Silicon wafer at 470K has 1e14 cm-3 holes. When 1e14 cm-3 acceptors are added, what is the new electron and hole concentrations? ND=0
2
! 1x10 $ 1x10 14 p= + # + 1x10 & 2 " 2 % 14
14
(
p = 1.62x1014 cm'3 ( NA ' ND
1x10 ) ( n= 14
2
1.62x1014
= 6.2x1013 cm'3
)
2
Developing the mathematical model for electrons and holes Example: An intrinsic Silicon wafer at 600K has 4e15 cm-3 holes. When 1e14 cm-3 acceptors are added, what is the new electron and hole concentrations? ND=0
2
! 1x10 $ 1x10 15 p= + # + 4x10 & 2 " 2 % 14
14
(
)
2
p = 4x1015 cm'3 = ni ( NA ' ND
4x10 ) ( n= 15
4x1015
2
= 4x1015 cm'3 = ni
) Intrinsic Material at High Temperature
Where is Ei? Since we started with descriptions of intrinsic materials then it makes sense to reference energies from the intrinsic energy, Ei. Intrinsic Material:
n = Nce
( E f ! Ec ) / kT
Nce
( Ei ! Ec ) / kT
= Nve
( Ev ! E f ) / kT
= Nve
( Ev ! Ei ) / kT
E c + E v kT & N v # !! Ei = + ln$$ 2 2 % Nc "
=p
Where is Ei? Intrinsic Material:
But,
N v &$ m #! = N c $% mn* !" * p
3/ 2
* & m E c + E v 3kT $ p Ei = + ln * $m 2 4 % n
Letting Ev=0, this is Eg / 2 or “Midgap”
# ! ! "
-0.007 eV for Si @ 300K ( 0.6% of EG )
Where is Ei? Extrinsic Material:
n = ni e
( E f ! Ei ) / kT
p = ni e
( Ei ! E f ) / kT
Solving for (Ef-Ei)
&n E f ' Ei = kT ln$$ % ni or for N D >> N A
# & p# !! = 'kT ln$$ !! " % ni " and N D >> ni
& ND # !! E f ' Ei = kT ln$$ % ni " or for N A >> N D and N A >> ni & NA # !! E f ' Ei = 'kT ln$$ % ni "
