Lecture 08 Review of Solid State Physics · Crystal lattice, reciprocal lattice, symmetry Crystal directions and planes Energy bands, bandgap Direct bandgap vs indirect bandgap Crystal

Lecture 08 Review of Solid State Physics

ECE440 Nanoelectronics

Crystal lattice, reciprocal lattice, symmetry Crystal directions and planes Energy bands, bandgap

Direct bandgap vs indirect bandgap Crystal bonds

Basic concepts of semiconductors

A Brief review of Solid State Physics




Contents

(a) Crystalline (b) Amorphous (c) Polycrystalline

Three types of solids: (a) ordered crystalline and (b) amorphous materials are illustrated bymicroscopic views of the atoms, whereas (c) polycrystalline structure is illustrated by a moremacroscopic view of adjacent single-crystalline regions, each of which has a crystallinestructure as in (a).

A crystal is a solid, where the constituent atoms are arranged in acertain periodic fashion.A solid without any periodicity is called amorphous solid.A solid where only small regions are of a single‐crystal material iscalled polycrystalline solid.

Crystalline, polycrystalline, and amorphous

Crystal Structure

7 Crystal Systems 14 Bravais LatticeTriclinic 1Monoclinic 2Orthorhomic 4Tetragonal 2Cubic 3 (sc, fcc, bcc)Trigonal 1Hexagonal 1

(32 point groups) (230 space groups)

Symmetry = “rotation only” “rotation + translation”

(i.e., 230 crystal structures)

SC, BCC, FCC Lattices

(a) (b) (c)

Three types of cubic lattices:

(a) simple, (b) body‐centered, and (c) face‐centered.

Lattice and reciprocal lattice

( , , )

( , , )

Lattice Reciprocal lattice

2ᴨΩ ⨯ ,

2ᴨΩ ⨯ ,

2ᴨΩ ⨯

with Ω ≡ . ⨯

. δ . 2ᴨ

a2

a1

a3

SC Lattice Basis Vectors

Reciprocal Lattice of Simple Cubic (S.C.)

, , ,

Ω ⨯ . ⨯ . .

2ᴨΩ ⨯

2ᴨ⨯

2ᴨ

2ᴨΩ ⨯

2ᴨ ⨯

2ᴨ

2ᴨΩ ⨯

2ᴨ⨯

2ᴨ

The reciprocal lattice of SC is still SC!

BCC Lattice Basis Vectors

a1a2

a3x

y

z

Body-Centered Cubic (B.C.C.)

2

2

2

Ω ⨯ . 2 . 2

4 1 0 0 0 1 0

2

⨯ 4 ⨯ 4 0 0 0

4 2 2 2

⨯ 4 ⨯ 4 0 0 0

4 2 2 2

⨯ 4 ⨯ 4 0 0 0

4 2 2 2

Reciprocal Lattice of

2ᴨΩ ⨯

2ᴨ

22

2ᴨ

2ᴨΩ ⨯

2ᴨ

22

2ᴨ

2ᴨΩ ⨯

2ᴨ

22

2ᴨ

The reciprocal lattice of BCC is FCC!

a2

a1

a3

FCC Lattice Basis Vectors

x

y

z

Face-Centered Cubic (F.C.C.)

2

2

2

⨯ 4 ⨯ 4 0 4

⨯ 4 ⨯ 4 0 4

⨯ 4 ⨯ 4 0 4

Ω ⨯ . 4 . 2 8 1 0 0 1 0 0 4

Reciprocal Lattice of

2ᴨΩ ⨯

2ᴨ

44

2ᴨ

2ᴨΩ ⨯

2ᴨ

44

2ᴨ

2ᴨΩ ⨯

2ᴨ

44

2ᴨ

The reciprocal lattice of FCC is BCC!

Review of Semiconductor PhysicsLattice Reciprocal Lattice

, , ;S.C.

2ᴨ,

2ᴨ,

2ᴨ

S.C.

2

2

2

B.C.C.2ᴨ

2ᴨ

2ᴨ

F.C.C.

2

2

2

F.C.C.2ᴨ

2ᴨ

2ᴨ

B.C.C.

Diamond structure is of face-centered type of cubic lattices. Thetetrahedral bonding arrangement ofneighboring atoms is clear.

a

The basic lattice structure for diamond, silicon, and germanium is the diamond structure.The diamond structure is a face‐centered cubic structure with an extra atom placeda distance a1/4 + a2/4 + a3/4 from each of the original face‐centered atoms, as shownbelow. Thus, a diamond lattice contains twice as many atoms per unit volume as a face‐centered cubic lattice. Four nearest neighbor atoms to each atom are shown bycomplimentary color for easier visualization. The lattice of volume V = a3 consists of 8atoms.

Diamond Structure

Besides the translations, the crystal symmetry contains other symmetry elements, for example, specific rotations around high‐symmetry axes. In cubic crystals, axes directed along the basis vectors are equivalent and they are the symmetry axes.

Rotation symmetry in cubic crystals.

z

x

y

C3

C3

C3

C3

C4

C4

C4

Rotation Symmetry

Two-dimensional lattice. Threeunit cells are illustrated by A, B,and C. Two basis vectors areillustrated by a1 and a2 .

a1

a2

a1

a1

a2

B

C

A

a2

Translational symmetry is the property of the crystal to be “carried" into itself underparallel translation in certain directions and certain distances. For any three‐dimensionallattice it is possible to define three fundamental noncomplanar primitive translationvectors (basis vectors) a1, a2, and a3, such that the position of any lattice site can bedefined by the vector R = n1a1 + n2a2 + n3a3, where n1, n2, and n3 are arbitrary integers. Ifwe construct the parallelepiped using the basis vectors, a1, a2, and a3, we obtain just theprimitive cell.

Translational Symmetry

Crystal Structure

7 Crystal Systems 14 Bravais LatticeTriclinic 1Monoclinic 2Orthorhomic 4Tetragonal 2Cubic 3 (sc, fcc, bcc)Trigonal 1Hexagonal 1

(32 point groups) (230 space groups)

Symmetry = “rotation only” “rotation + translation”

Si, Ge = Diamond

GaAs = Zinc blende structures

GaN = Wurtzite

(i.e., 230 crystal structures)

Crystal structure = Bravais lattice + Basis

Crystal

= +

Lattice

Basis




Contents

Crystal directions in cubic crystals.

y[010]

[001][001]

[010]0

x

z

Crystal directions and planes

Miller Index

Planes:

Direction:

,

A plane All equivalent planes

A direction All equivalent directions ,

Brillouin Zone Wigner Seitz Cell

Constructed by drawing perpendicular bisector planes in thereciprocal lattice from the chosen center to the nearest equivalentreciprocal lattice sites.




Contents

Ener

gy b

and

Formation of energy band. Ground energy level E1 of a single atom evolves into energy bandwhen large number of single atoms are interacting with each other.

……

Large number of atoms

E1

E1

E1

E1

E1

Energy levels• Electrons can only have discrete values of energy

• Electrons in the outermost shell, called Valence Electrons.

• With large number of atoms in solid, energy levels form bands

• Important bands are the valence band, conduction band and energy gap

Schematics of energy bands 1, 2, and 3 that correspond to single atom energy levels E1 , E2,and E3 , respectively.

Energy level E3

1

2Overlap

3

Energy level E2

Energy level E1

Energy bands E1, E2, and E3

Case of dielectric: filled valence band and empty conduction band; Eg > 5 eV.

Case of a semiconductor: filled valence bandand empty conduction band (at lowtemperature); Eg < 5 eV.

Bandgap of Insulator, Semiconductor, and Conductor




Contents

Light wavelengths λ are much greater than electron de‐Broglie wavelengthsand the photon wavevectors q = 2π / λ, in turn, are much smaller than theelectron wavevectors ( |k1|, |k2| >> |q| ).Under light absorption and emission the electrons transferred between thevalence and conduction bands practically preserve their wavevectors.

1 2 and ( ) ( )k k k k kc vE E

GaAs

-1

1

2

3

E(e

V)

Eg

E(e

V)

-1

1

2

3Si

Eg

Γ ΓL X XL[111] [100] [111] [100]

Direct‐bandgap and indirect‐bandgap semiconductors

Direct vs. Indirect Bandgap Semiconductors

Figure 2 Simplified energy diagramof Si. The most relevant transitionsare shown. Black: absorption of aphoton via a phonon-assistedindirect transition; red: emission of aphoton via a phonon-assistedindirect transition; orange: freecarrier absorption; green: Augerrecombination process.

L. Pavesi, Routes toward silicon-based lasers. Materials Today 8, 18 (2005)

Figure 1 Energy band diagrams and major carrier transition processes in InPand silicon crystals. In a direct band structure (such as InP, left), electron-hole recombination almost always results in photon emission, whereas in an indirect band structure (such as Si, right), free-carrier absorption, Auger recombination and indirect recombination exist simultaneously, resulting in little photon emission.

D. Liang and J. E. Bowers, Nature Photonics 4, 511 (2010)

nonradrad

nonrad

nonradrad

radi

11

1

)()()(

Internal quantum efficiency ηi (defined as the ratio of the probability that an excited e-h pair recombine radiatively and the probability of e-h pair combination.)




Contents

Ionic crystals are made up of positive and negative ions. The ionic bond resultsprimarily from attractive electrostatic interaction of neighboring ions withopposite charges. There is also a repulsive interaction with other neighbors ofthe same charge. Attraction and repulsion together result in a balancing offorces that leads to the atoms being in stable equilibrium positions in such anionic crystal. As for the electronic configuration in a crystal, it corresponds to aclosed (completely filled) outer electronic shell. A good example of an ioniccrystal is NaCl (salt). Neutral sodium, Na, and chlorine, Cl, atoms have theconfigurations Na11 (1s22s22p63s1) and Cl17 (1s22s22p63s23p5), respectively.That is, the Na atom has only one valence electron, while one electron isnecessary to complete the shell in the Cl atom. It turns out that the stableelectronic configuration develops when the Na atom gives one valenceelectron to the Cl atom. Both of them become ions with opposite charges andthe pair has the closed outer shell configuration (like inert gases such ashelium, He, and neon, Ne). The inner shells are, of course, completely filledboth before and after binding of the two atoms. In general, for all elementswith almost closed shells, there is a tendency to form ionic bonds and ioniccrystals. These crystals are usually dielectrics (insulators).

Ionic crystals

Fig. 4.4. Four sp3-hybrid bonding orbitals in Si crystal.

Si

Si

Si

Si

Si

Covalent bonding is typical for atoms with a low level of outer shell filling up.An excellent example is provided by a Si crystal. As we discussed in theprevious Chapter, the electron configuration of Si can be presented as: core +3s23p2. To complete the outer 3s23p2 shell, a silicon atom in crystal forms fourbonds with four other neighboring silicon atoms. The central Si atom andeach of its nearest‐neighbor Si atoms share two electrons. This provides so‐called covalent (chemical) bonds (symmetric combinations of the sp3 orbitals)in the Si crystal. The four bonding orbitals form an energy band completelyfilled by the valence electrons. This band is called the valence band.

Covalent crystals

Type of crystal coupling Crystal Energy per atom (eV)

Ionic NaClLiF

7.910.4

CovalentDiamond, C

SiGe

7.43.73.7

MetallicNaFeAl

1.14.12.4

Molecular and Inert gas crystals

CH4Ar

0.10.8

Binding energies for different types of crystals.

The ionic and covalent crystals typically have binding energies in the range of1 ‐ 10 eV, while molecular and inert gas crystals are weakly‐coupled systems.Metals have intermediate coupling.The binding energy of a crystal is an important parameter, since it determinesthe stability of the crystal, its aging time, applicability of different treatmentprocesses, etc.




Contents

Semiconductor materials

Covalence Bonds

• Atoms of solid materials form crystals, which are 3D structures held together by strong bonds Between atoms—covalence Bonds like Silicon.

• Silicon forms a covalent crystal. The shared electrons are not mobile. Therefore there is a energy gap between the valence band and the conduction band.

Holes• An electron hole is the conceptual and mathematical opposite of an electron

• Holes do not travel like electrons• Only electrons can move from atom to atom

Doping in Silicon• Pure silicon wafers are called intrinsic silicon• Intrinsic silicon doesn’t have enough free electrons for current conduction.

• So impurities are introduced to increase conductivity• The introduction of these impurities is called doping

Donors and Acceptors• Donors are dopant atoms that added to a semiconductor to provide electrons

• Acceptors are dopant atoms that provide holes

Type of doping• N (negative) type: doped with donors and has extra free electrons.

• P (positive) type: doped with acceptors and has extra holes.

Electronic properties of doped silicon – qualitative picture.

Methods for doping

• Diffusion• impurities move by a difference in concentration gradients

• Conducted at very high temperatures

• Gas sources are the most common but liquids and solids are also used

• The sources react with silicon to form dopant oxide which then diffuses into the rest of the substrate by the increase of temperature

Methods for doping• Ion implantation: Alternative to high temperature diffusion

• A beam of highly energetic dopant ions is aimed at the semiconductor surface

• Collision with ion distorts the crystal structure

• Annealing has to be performed to correct the damage

Electronic properties: Silicon in general.

EG = 1.12 eVBoltzman constant: k = 8.62 10–5 eV/K

kTEEc

FceNn /)( Fundamental materials property:

Where n = concentration of negative (electron) carriers (typically in cm-3)Ec is the energy level of the conduction bandEF is the Fermi level.Nc is the intrinsic density of states in the conduction band (cm-3).

EF

Similarly, kTEE

VVFeNp /)(

Where p = concentration of positive (hole) carriers (typically in cm-3)EV is the energy level of the valence bandNV is the intrinsic density of states in the valence band (cm-3).

EC

EV

Density of states in conduction band, NC (cm-3) 3.22E+19

Density of states in valence band, NV (cm-3) 1.83E19

Note: without doping, n = p ni where ni is the intrinsic carrier concentration.For pure silicon, then

)/exp(2 kTENNn GVci

Thus ni = 9.6 x 109 cm-3

Similarly the Fermi level for the intrinsic silicon is,

)/ln()2/1(2/)( CVVCVi NNkTEEEE

Where we have used Ei to indicate intrinsic Fermi level for Si.

Electronic properties: intrinsic (undoped) silicon.

Energy Bands

Bloch function ɸ ħ ,

Valance band, Conduction band, Bandgap

Insulator, Semiconductor, Conductor

Direct bandgap vs. Indirect bandgap

1.42 ~3.4

1.12 0.66

Effective mass, 1∗

1ħ

Carrier Concentration

Donor

Acceptor

n-type

p-type

Fermi level

exp

exp

≡ 22ᴨ

ħ

≡ 22ᴨ

ħ

when ,2 2 ln

exp

exp2

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Lecture 08 Review of Solid State Physics · Crystal lattice, reciprocal lattice, symmetry Crystal directions and planes Energy bands, bandgap Direct bandgap vs indirect bandgap Crystal