carrier modeling. quantization concept plank constant core electrons valence electrons

Carrier Modeling

Quantization Concept

plank constant

Core electrons

Valence electrons

Periodic Table of the Elements

Group**

Period 1 IA 1A

18

VIIIA 8A

1 1 H

1.008

2

IIA 2A

13

IIIA 3A

14

IVA 4A

15

VA 5A

16

VIA 6A

17

VIIA 7A

2 He 4.003

2 3

Li 6.941

4 Be 9.012

5 B

10.81

6 C

12.01

7 N

14.01

8 O

16.00

9 F

19.00

10 Ne 20.18

8 9 10 3

11 Na 22.99

12 Mg 24.31

3

IIIB 3B

4

IVB 4B

5

VB 5B

6

VIB 6B

7

VIIB 7B ------- VIII -------

------- 8 -------

11

IB 1B

12

IIB 2B

13 Al 26.98

14 Si

28.09

15 P

30.97

16 S

32.07

17 Cl

35.45

18 Ar 39.95

4 19 K

39.10

20 Ca 40.08

21 Sc 44.96

22 Ti

47.88

23 V

50.94

24 Cr 52.00

25 Mn 54.94

26 Fe 55.85

27 Co 58.47

28 Ni 58.69

29 Cu 63.55

30 Zn 65.39

31 Ga 69.72

32 Ge 72.59

33 As 74.92

34 Se 78.96

35 Br 79.90

36 Kr 83.80

5 37

Rb 85.47

38 Sr

87.62

39 Y

88.91

40 Zr

91.22

41 Nb 92.91

42 Mo 95.94

43 Tc (98)

44 Ru 101.1

45 Rh 102.9

46 Pd 106.4

47 Ag 107.9

48 Cd 112.4

49 In

114.8

50 Sn 118.7

51 Sb 121.8

52 Te 127.6

53 I

126.9

54 Xe 131.3

6 55 Cs 132.9

56 Ba 137.3

57 La* 138.9

72 Hf 178.5

73 Ta 180.9

74 W

183.9

75 Re 186.2

76 Os 190.2

77 Ir

190.2

78 Pt

195.1

79 Au 197.0

80 Hg 200.5

81 Tl

204.4

82 Pb 207.2

83 Bi

209.0

84 Po (210)

85 At (210)

86 Rn (222)

7 87 Fr

(223)

88 Ra (226)

89 Ac~ (227)

104 Rf (257)

105 Db (260)

106 Sg (263)

107 Bh (262)

108 Hs (265)

109 Mt (266)

110 ---

()

111 ---

()

112 ---

()

114 ---

()

116 ---

()

118 ---

()

2s2p

1sK

L


The shell model of the atom in which the electrons are confined to live within certain shells and in sub shells within shells.

The Shell Model

1s22s22p2 or [He]2s22p2

L shell with two sub shells

Nucleus

ro

v+e

e


y

x

x

y

z

z

x

y

z

x

y

z

Stable orbit has radius r0

The planetary model of hydrogen atom in which the negatively charged electron orbits the positively charged nucleus.

Orbitals

1s orbital 2px orbital

2py orbital 2pz orbital (ml = 0)

Atomic Bonding

a. Ionic bonding (such as NaCl)

b. Metallic bonding (all metals)

c. Covalent bonding (typical Si)

d. Van der Waals bonding (water…)

e. Mixed bonding (GaAs, ZnSe…, ionic & covalent)

Bonding forces in Solids

Splitting of energy states into allowed bands separated by a forbidden energy gap as the atomic spacing decreases; the electrical properties of a crystalline material correspond to specific allowed and forbidden energies associated with an atomic separation related to the lattice constant of the crystal.

Allowed energy levels of an electron acted on by the Coulomb potential of an atomic nucleus.

Energy Band Formation

Broadening of allowed energy levels into allowed energy bands separated by forbidden-energy gaps as more atoms influence each electron in a solid.


One-dimensional representation

Two-dimensional diagram in which energy is plotted versus distance.


Strongly bonded materials: small interatomic distances.

Thus, the strongly bonded materials can have larger energy bandgaps than do weakly bonded materials.

Energy Bandgapwhere ‘no’ states exist

As atoms are brought closer towardsone another and begin to bond together, their energy levels mustsplit into bands of discrete levelsso closely spaced in energy, theycan be considered a continuum ofallowed energy.

Pauli Exclusion Principle

Only 2 electrons, of spin+/-1/2, can occupy the same energy state at the

same point in space.

The 2N electrons in the 3s sub-shell and the 2N electrons in the 3p sub-shell undergo sp3 hybridization.

Energy Band Formation (Si)

Energy levels in Si as a function of inter-atomic spacing

The core levels (n=1,2) in Si are completely filled with electrons.

conduction band(empty)

valence band(filled)

N electrons filling half of the 2N allowed states, as can occur in a metal.


Energy band diagrams.

A completely empty band separated by an energy gap Eg from a band whose 2N states are completely filled by 2N electrons, representative of an insulator.

Allowed electronic-energy-state systems for metal and semiconductors.

States marked with an “X” are filled; those unmarked are empty.

Metals, Semiconductors, and Insulators

Metal Semiconductor

Ef

Ef

2 s Band

Overlapping energy bands

Electrons2 s2 p

3 s3 p

1 s 1sSOLIDATOM

E = 0

Free electronElectron Energy, E

2 p Band

3s BandVacuum

level

In a metal the 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.

Typical band structures of Metal


Electron motion in an allowed band is analogous to fluid motion in a glass tube with sealed ends; the fluid can move in a half-filled tube just as electrons can move in a metal.

Electron Motion in Energy Band

E = 0 E 0

Current flowing


E = 0 E 0

No fluid motion can occur in a completely filled tube with sealed ends.

Energy-band diagram for a semiconductor showing the lower edge of the conduction band Ec, a donor level Ed within the forbidden gap, and Fermi level Ef, an acceptor level Ea, and the top edge of the valence band Ev.


Energy band diagrams.


No flow can occur in either the completely filled or completely empty tube.

Fluid can move in both tubes if some of it is transferred from the filled tube to the empty one, leaving unfilled volume in the lower tube.

Fluid analogy for a semiconductor


Insulator Semiconductor Metal

Typical band structures at 0 K.

Ease of achieving thermal population of conduction band determines whether a material is an insulator, metal, or semiconductor.

Material Classification based on Size of Bandgap

10610310010-310-610-910-1210-1510-18 109

Semiconductors Conductors

1012

AgGraphite NiCrTeIntrinsic Si

Degeneratelydoped Si

Insulators

Diamond

SiO2

Superconductors

PETPVDF

AmorphousAs2Se3

Mica

Alumina

Borosilicate Pure SnO2

Inorganic Glasses

Alloys

Intrinsic GaAs

Soda silica glass

Many ceramics

MetalsPolypropylene


Range of conductivities exhibited by various materials.

Conductivity (m)-1

r

PE(r)

x

V(x)

x = Lx = 0 a 2a 3a

0aa

Surface SurfaceCrystal

PE of the electron around an isolated atom

When N atoms are arranged to form the crystal then there is an overlap of individual electron PE functions.

PE of the electron, V(x), inside the crystal is periodic with a period a.

The electron potential energy [PE, V(x)], inside the crystal is periodic with the same periodicity as that of the crystal, a.

Far away outside the crystal, by choice, V = 0 (the electron is free and PE = 0).

Energy Band Diagram

E-k diagram, Bloch function.

Energy Band Diagram

E-k diagram, Bloch function.

0)(2

22

2

xVEm

dx

d e

Schrödinger equation

...3,2,1)()( mmaxVxV

Periodic Potential

xkikk exUx )()(

Periodic Wave function

Bloch Wavefunction

There are many Bloch wavefunction solutions to the one-dimensional crystal each identified with a particular k value, say kn which act as a kind of quantum number.

Each k (x) solution corresponds to a particular kn and represents a state with an energy Ek.

Ek

kš /a–š /a

Ec

Ev

ConductionBand (CB)

Ec

Ev

CB

The E-k D iagram The Energy BandD iagram

Em pty k

O ccupied kh+

e-

Eg

e-

h+

h

V B

h

V alenceBand (V B)

The E-k curve consists of many discrete points with each point corresponding to a possible state, wavefunction k (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 [k

(x), solutions].

Energy Band Diagram

E-k diagram of a direct bandgap semiconductor

Energy Band Diagram

The bottom axis describe different directions of the crystal.

Ge Si GaAs

The energy is plotted as a function of the wave number, k, along the main crystallographic directions in the crystal.

E

CB

k–k

Direct Bandgap

GaAs

E

CB

VB

Indirect Bandgap, Eg

k–k

kcb

Si

E

k–k

Phonon

Si with a recombination center

Eg

Ec

Ev

Ec

Ev

kvb VB

CB

ErEc

Ev

Photon

VB

In GaAs the minimum of the CB is directly above the maximum of the VB. direct bandgap semiconductor.

In Si, the minimum of the CB is displaced from the maximum of the VB.indirect bandgap semiconductor

Recombination of an electron and a hole in Si involves a recombination center.

Energy Band Diagram

E-k diagram

Direct and Indirect Energy Band Diagram

(a) Direct transition with accompanying photon emission.(b) Indirect transition via defect level.

Energy Band

A simplified energy band diagram with the highest almost-filled band and the lowest almost-empty band.

valence band edge

conduction band edge

vacuum level

: electron affinity

Only the work function is given for the metal.

Semiconductor is described by the work function qΦs, the electron affinity qs, and the band gap (Ec – Ev).

Metals vs. Semiconductors

Pertinent energy levels

Metal Semiconductor

Electron energy, E

Conduction Band (CB)Empty of electrons at 0 K.

Valence Band (VB)Full of electrons at 0 K.

Ec

Ev

0

Ec+

Covalent bondSi ion core (+4e)

A simplified two dimensional view of a region of the Si crystal showing covalent bonds.

The energy band diagram of electrons in the Si crystal at absolute zero of temperature.

Typical band structures of Semiconductor


Band gap = Eg

Electrons: Electrons in the conduction band that are free to move throughout the crystal.

Holes: Missing electrons normally found in the valence band (or empty states in the valence band that would normally be filled).

Electrons and Holes

These “particles” carry electricity. Thus, we call these “carriers”

e–hole

CB

VB

Ec

Ev

0

Ec+

Eg

Free e–h > Eg

Hole h+

Electron energy, E

h

Electrons and Holes

A photon with an energy greater then Eg can excite an electron from the VB to the CB.

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.

Generation of Electrons and Holes

Effective Mass (I)

An electron moving in respond to an applied electric field.

E

E

within a Vacuum within a semiconductor crystal

dt

dvmEqF 0

dt

dvmEqF n

It allow us to conceive of electrons and holes as quasi-classical particles and to employ classical particle relationships in semiconductor crystals or in most device analysis.

Carrier Movement Within the Crystal

Density of States Effective Masses at 300 K

Ge and GaAs have “lighter electrons” than Si which results in faster devices

Effective Mass (II)

Electrons are not free but interact with periodic potential of the lattice. Wave-particle motion is not as same as in free space.

Curvature of the band determine m*.m* is positive in CB min., negative in VB max.

The motion of electrons in a crystal can be visualized and described in a quasi-classical manner.

In most instances

The electron can be thought of as a particle. The electronic motion can be modeled using Newtonian

mechanics.

The effect of crystalline forces and quantum mechanical properties are incorporated into the effective mass factor.

m* > 0 : near the bottoms of all bands m* < 0 : near the tops of all bands

Carriers in a crystal with energies near the top or bottom of an energy band typically exhibit a constant (energy-independent) effective mass.

` : near band edge

Effective Mass Approximation

constant2

2

dk

Ed

Covalent Bonding

Band Occupation at Low Temperature (0 K)

Without “help” the total number of “carriers” (electrons and holes) is limited to 2ni.

For most materials, this is not that much, and leads to very high resistance and few useful applications.

We need to add carriers by modifying the crystal.

This process is known as “doping the crystal”.

Impurity Doping

The need for more control over carrier concentration

Concept of a Donor “Adding extra” Electrons

Binding energies in Si: 0.03 ~ 0.06 eV

Binding energies in Ge: ~ 0.01 eV

Binding Energies of Impurity

Hydrogen Like Impurity Potential (Binding Energies)

Effective mass should be used to account the influence of the periodic potential of crystal.

Relative dielectric constant of the semiconductor should be used (instead of the free space permittivity).

: Electrons in donor atoms

: Holes in acceptor atoms


Band diagram equivalent view

e–As+

x

As+ As+ As+ As+

Ec

Ed

CB

Ev

~0.05 eV

As atom sites every 106 Si atoms

Distance intocrystal

Electron Energy

The four valence electrons of As allow it to bond just like Si but the 5th electron is left orbiting the As site. The energy required to release to free fifth- electron into the CB is very small.

Energy band diagram for an n-type Si dopedwith 1 ppm As. There are donor energy levels

below Ec around As+ sites.


n-type Impurity Doping of Si

just

Energy band diagram of an n-type semiconductor connected to a voltage supply of V volts.

The whole energy diagram tilts because the electron now has an electrostatic potential energy as well.

Current flowingV

n-Type Semiconductor

Ec

EF eV

A

B

V(x), PE (x)

x

PE (x) = – eV

E

Electron Energy

Ec eV

Ev eV

V(x)

EF

Ev


Energy Band Diagram in an Applied Field

Concept of a Acceptor “Adding extra” Holes

All regions of

material are neutrally

charged

One less bond

means

the acceptor is

electrically

satisfied.

One less bond

means

the neighboring

Silicon is left with

an empty state.

Hole Movement

Empty state is located next to the Acceptor

Hole Movement

Another valence electron can fill the empty state located next tothe 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)

Regionaround the“hole” hasone lesselectronand thus ispositivelycharged.

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)


Band diagram equivalent view

B–h+

x

B–

Ev

Ea

B atom sites every 106 Si atoms

Distanceinto crystal

~0.05 eV

B– B– B–

h+

VB

Ec

Electron energy

p-type Impurity Doping of Si


Boron doped Si crystal. B has only three valence electrons. When it substitute for a Si atom one of its bond has an electron missing and therefore a hole.

Energy band diagram for a p-type Si crystal doped with 1 ppm B. There are acceptor energy levels just above Ev around B- site. These acceptor levels accept electrons from the VB and therefore create holes in the VB.

Ec

Ev

EFi

CB

EFp

EFn

Ec

Ev

Ec

Ev

VB

Intrinsic semiconductors

In all cases, np=ni2

Note that donor and acceptor energy levels are not shown.

Intrinsic, n-Type, p-Type Semiconductors

Energy band diagrams

n-type semiconductors

p-type semiconductors

CB

g(E)

E

EFp

Ev

Ec

EFn

Ev

Ec

CB

VB

Degenerate n-type semiconductor. Large number of donors form a band that overlaps the CB.

Heavily Doped n-Type, p-Type Semiconductors

Degenerate p-type semiconductor

Impurities forming a band

Impurity Doping

Impurity Doping

Valence Band

Valence Band

Impurity Doping

Position of energy levels within the bandgap of Si for common dopants.

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 (i.e. n if n>p, p if p>n)

Minority carrier: The carrier that exists in lower population (i.e. n if n<p, p if p<n)

Other important terms: Insulator, semiconductor, metal, amorphous, polycrystalline, crystalline (or single crystal), lattice, unit cell, primitive unit cell, zincblende, lattice constant, elemental semiconductor, compound semiconductor, binary, ternary, quaternary, atomic density, Miller indices

Summary of Important terms and symbols

carrier modeling. quantization concept plank constant core electrons valence electrons

Documents

energy bandgapwhere

energy gaps

allowed energy bands

energy levels mustsplit

forbidden energy gap

continuum ofallowed

larger energy bandgaps

atomic bondingionic