Point defects ionic crystals ¾ Ionic bonding and ionic crystals: Brief review ¾ Defects in ionic crystal and oxides. Kröger-Vink Notation ¾ Site, mass, and charge balance ¾ Frenkel and Schottky defects ¾ Extrinsic point defects in ionic crystals - impurities ¾ Non-stoichiometry in ionic crystals

Electronic defects in semiconductors ¾ Electronic defects in intrinsic semiconductors

optional reading (not tested)

¾ Extrinsic electronic defects in semiconductors - doping

References: Allen & Thomas, Ch. 5, pp. 263-270 Swalin: Ch. 14, pp. 317-350

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Ionic bonding: Brief review

Electropositive elements: Readily give up electrons to become positive ions (cations)

inert

accept 1e-

Electronegativity - a measure of how willing atoms are to accept electrons

accept 2e-

give up 3e-

give up 2e-

give up 1e-

Ionic bonding - typical between elements from horizontal extremities of the periodic table

Electronegative elements: Readily acquire electrons to become negative ions (anions)

IA: Alkali metals (Li, Na, K…) - one electron in outermost occupied s subshell - eager to give up electron VIIA: Halogens (F, Br, Cl...) missing one electron in outermost occupied p subshell - want to gain electron Metals are electropositive – they can give up their few valence electrons to become positively charged ions University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Ionic bonding: Brief review Example: table salt (NaCl) Na has 11 electrons, 1 more than needed for a full outer shell (Neon) 11 Protons Na 1S2 2S2 2P6 3S1 11 Protons Na+ 1S2 2S2 2P6

donates 10 e- left

e-

Na

e-

Cl

Cl has 17 electron, 1 less than needed for a full outer shell (Argon) 17 Protons Cl 1S2 2S2 2P6 3S2 3P5 17 Protons Cl- 1S2 2S2 2P6 3S2 3P6

receives e18 e-

Na+

Cohesive energy of NaCl crystal (energy needed to convert NaCl crystal into individual Na and Cl atoms): Na (gas) + 5.14 eV (ionization energy) → Na+ + ee- + Cl (gas) → Cl- + 3.61 eV (electron affinity) energy of (long-range) interaction among the ions Na+ + Cl- → NaCl (crystal) + 7.9 eV balance: ΔE = 7.9 eV - 5.1 eV + 3.6 eV = 6.4 eV per NaCl formula unit < 0 → it costs energy to transfer e from Na to Cl University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Cl-

Ionic crystals: Brief review • Charge neutrality: the total charge in the base must be zero • There are no free electrons, ionic crystals are insulators • Interatomic bonding is mostly defined by long-range inter-ionic Coulomb interactions ±q2/r and is rather strong (Ec ~ 600-1000 kJ/mol ~ 6-10 eV/atom) and has no directionality NaCl structure fcc with 2 atoms in the base: at (0, 0, 0) and (½, 0, 0)

Na+ ions filling octahedral holes in the fcc structure

KCl, AgBr, KBr, PbS, MgO, FeO

fluorite structure fcc with 3 atoms in the base: cations at (0,0,0) and two anions at (¼, ¼, ¼), and (¼, ¾, ¼)

F- ions filling tetrahedral holes in the fcc structure

CaF2 or ZrO2

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Ionic crystals: Brief review zinc blende structure

CsCl structure

fcc with two atoms in the base at (0,0,0) and (¼, ¼, ¼) ZnS, CuF, CuCl GaAs, GaP, InP- semiconductors

simple cubic with two atoms in the base at (0,0,0) and (½, ½, ½) CsCl, TlI, TlCl AlNi, CuZn - intermetallic comp.

tetrahedral sites are preferred because of the relative sizes of the positive and negative ions, but not all of them are filled to maintain stoichiometry

spinel structure named after the mineral spinel (MgAl2O4) Fe3+( Fe2+ Fe3+)O4, Mg2+( Al23+)O4, Fe3+(Cr23+)O4 can contain vacancies as an integral part of the structure to satisfy the charge balance, Fe21,67Vac2,33O32 if all Fe converted to Fe+3

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Defects in ionic crystal and oxides. Kröger-Vink Notation Introduction of a concentration α of vacancies to Na+ sites (or the same concentration of Cl- interstitials) creates net charge of -eαN charge in a crystal with N lattice sites → very high energy → Na1-αCl cannot exist pure ionic crystals must be perfectly stoichiometric (?) introduction of impurities with different valence and electronegativity than the host ions can require additional point defects to charge balance The concentration of vacancies can be much higher than required by thermal equilibrium electrochemical equilibrium must be maintained. How to incorporate point defects into chemical reaction equations?

Kröger-Vink Notation:

X YZ

X – nature of species located on a site: element symbol for an atom, V for vacancy Y – type of the site occupied by X: (i for an interstitial, element symbol for site normally occupied by this element) Z – charge relative to the normal ion charge on the site ′ negative relative charge • positive relative charge x zero relative charge (x is often omitted) University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Kröger-Vink Notation X YZ Examples: interstitial Ag ion in AgCl:

Ag•i

vacancy on a Ag site in AgCl:

' VAg

Ca2+ ion on a Na site in NaCl:

Ca•Na

vacancy on an O site in Al2O3:

VO••

Cu+ on a Cu2+ site CuO:

Cu'Cu

Intrinsic point defects and Pu impurity in UO2 crystal

Y

In a generic discussion of defect reactions, M and X are often used: M - atom of electropositive element X - atom of electronegative element

X U4+ ion O2- ion

Vacancy

Pu4+ ion

U site

UxU

O'U'''''

VU''''

Pu xU

O site

U•O•••••

OOx

VO••

Pu•O•••••

i site

U•i •••

Oi''

Vix

Pu•i •••

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Site, mass, and charge balance Formation and annihilation of point defects in ionic crystals must satisfy the following 3 rules:

1. Site balance – Ratios of regular lattice sites must be conserved, i.e., fixed proportion of M and X sites must be created regardless of whether they are occupied or not. Total number of sites may change, but the ratio must remain constant. Example: Al2O3: by oxidation of aluminum create 3OO then 2AlAl must also be created, although they may be vacant.

2. Mass balance – Total number of atoms of each species on right and left side of defect formation reaction must be equal – Vacancies and electronic defects do not affect mass balance

3. Charge balance (electroneutrality) – Compounds are assumed to remain neutral Any charge inbalance, global or local, leads to high electrostatic energy that exceed any other contributions to the Gibbs free energy, making the charged state to be strongly nonequilibrium one. University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Frenkel defects Frenkel defect (Frenkel pair) = vacancy + interstitial in close proximity first discussed in 1926 by Frenkel for AgCl +

two types of Frenkel defects: • cation Frenkel pair: cation vacancy + cation interstitial • anion Frenkel pair: anion vacancy + anion interstitial Typically, the enthalpies of formation are very different for the two types and, in a given crystal, one type of Frenkel defect is prevalent. formation reaction for a cation Frenkel pair in AgCl:

' AgxAg ↔ Ag•i + VAg

this reaction satisfies the mass, charge, and site balance Analysis of the equilibrium concentration of Frenkel defects can be done similarly to our derivation for vacancies. We just have to keep in mind that both vacancy and self-interstitial are generated and ni = nv = nFP

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Frenkel defects FP n Equilibrium concentration of Frenkel defects: ΔG = n FP ( Δh FP f − T Δs f ) − T Δ S c

N! N i! S = k B ln v + k B ln i v n ! N −n ! n ! N i − ni ! n c

(

)

(

)

where Ni is the number of interstitial sites (may depend on configuration, e.g., dumbbell vs. octahedral)

n i = n v = n FP ∂Δ G ∂n FP

= Δh

FP f

− TΔs

FP v

FP n FP = neq

⎛ neq neq ⎞ ⎟=0 + k BT ln ⎜⎜ i ⎟ ⎝ N N ⎠

compound

reaction

Δh FP f , eV

AgBr

x ' AgAg ↔ Ag•i + VAg FFx ↔ VF• + Fi' x '' CaCa ↔ Ca•i • + VCa

1.1

CaF2 Li2O TiO2

x LiLi ↔ Lii• + VLi' OOx ↔ VO•• + Oi'' x TiTi ↔ Tii•••• + VTi'''' OOx ↔ VO•• + Oi''

2.3-2.8

⎛ Ni = ⎜⎜ N ⎝ N

neqFP

1/ 2

⎞ ⎟⎟ ⎠

⎛ ΔsvFP exp ⎜⎜ ⎝ 2k B

⎛ Δh FP ⎞ f ⎟⎟ exp ⎜ − ⎜ 2k T B ⎠ ⎝

⎞ ⎟ ⎟ ⎠

to measure the concentration of intrinsic point defects, ionic crystals of high purity have to be made, e.g., by zone refining

7 2.3 8.7

Δh FP includes energy of electrostatic interactions f

12 3.0

UO2

UxU ↔ U•i ••• + VU''''

9.5

ZnO

OOx ↔ VO•• + Oi''

2.5

many more anion Frenkel defects than cation ones

from Allen & Thomas

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Schottky defects Schottky defect = cation vacancy + anion vacancy in close proximity formation reaction for a Schottky defect in BeO: x '' x BeBe + OOx ↔ VBe + VO•• + BeBe + OOx

this reaction satisfies the mass, charge, and side balance

compound

Schottky defect

Δh SD f , eV

α-Al2O3

2VAl''' + 3Vo••

26

CaF2

'' VCa + 2VF•

5.5

BeO

'' VBe + VO••

6

TiO2 NaCl

V + 2V V + 2V V +V

2.2-2.4

KCl

VK' + V

2.6

UO2

'''' Ti '''' U ' Na

•• O •• O • Cl • Cl

5.2 6.4

electrostatic attraction between cation and anion vacancies → binding energy of the Schottky defect and temperature dependent degree of association equilibrium concentration: ⎛ ΔsvSD = exp ⎜⎜ N ⎝ 2k B

neqSD

⎛ Δh SD ⎞ f ⎟⎟ exp ⎜ − ⎜ 2k T B ⎠ ⎝

⎞ ⎟ ⎟ ⎠

relative low Δhf → Schottky defects dominate

from Allen & Thomas University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Extrinsic point defects in ionic crystals - impurities Introduction of impurities may require simultaneous introduction of additional defects, e.g., in addition to the thermally-induced (intrinsic) vacancies, some additional extrinsic vacancies may be induced by impurity ions with valence different from the one of the ions in the host crystal. Let’s consider incorporation of CaCl2 to KCl crystal as a substitutional impurity: x x CaCl2 (s) + 2K Kx + 2ClCl ↔ Ca •K + VK' + 2ClCl + 2KCl(g)

or

x CaCl2 ↔ Ca •K + VK' + 2ClCl

Site balance: the 1:1 ratio of K and Cl sites must be maintained. Two Cl anions occupy the existing Cl sites → two cation sites must be created. One of the cation sites is occupied by Ca2+ and one is left vacant. Mass balance: the numbers of atoms of each species on both sides of the equation are equal. Charge balance: placing Ca2+ on a K+ gives a net charge of +1 that has to be compensated by a vacancy. If Ca2+ occupies an interstitial site, the equation has to be modified: x x CaCl2 (s) + 2K Kx + 2ClCl ↔ Ca i•• + 2VK' + 2ClCl + 2KCl(g)

or

x CaCl2 ↔ Ca i•• + 2VK' + 2ClCl

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Extrinsic point defects in ionic crystals - impurities Different schemes of impurity incorporation can be sometimes distinguished from experimental measurements of the effect of impurity concentration on material density. Let’s consider incorporation of ZrO2 to Y2O3 crystal Two simplest options: (1) Zr4+ fully occupy Y sites and anion defects take care of the charge balance

2ZrO2 ↔ 2ZrY• + 3O Ox + Oi'' (2) O2- fully occupy O sites and cation defects take care of the charge balance

3ZrO2 ↔ 6O Ox + 3ZrY• + VY''' Experimental observation that density of Y2O3 increases with addition of ZrO2 is in favor of option (1), since appearance of vacancies would decrease density and Zr has slightly higher atomic mass and smaller ionic radius than Y.

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Extrinsic point defects in ionic crystals - impurities Let’s consider incorporation of CaO to ZrO2

Ca2+ cation, Zr4+ cation, O2- anion

We have two possible scenarios: (1) charge compensation by anion vacancy

CaO ↔ Ca 'Zr' + O Ox + VO••

(2) charge compensation by Zr interstitials 2CaO + ZrZrx ↔ 2Ca 'Zr' + 2O Ox + Zri•••• site balance: 1Zr = 2Zr + 2O null = ZrO2

need an extra Zr site

2CaO + ZrZrx + 2O Ox + ZrZrx ↔ 2Ca 'Zr' + 2O Ox + Zri•••• + ZrO2

Solid solutions produced by these different reactions will have different densities. For reaction (1), the defective formula unit is made up of one Ca, one O, and one vacancy. For reaction (2), the defective formula unit is made up of one Ca, two O, and one half of a Zr (in an interstitial site). Thus, the formulas for weight and density of each solid solution are:

(1) x(M CaO ) + (1 − x)M ZrO2

ρ=

Z × [ x(M CaO ) + (1 − x)M ZrO2 ] N aVcell

1 (2) x(M CaO2 + M Zr ) + (1 − x)M ZrO2 2

ρ=

Z × [ x(M CaO2 + 0.5M Zr ) + (1 − x)M ZrO2 ] N aVcell

where M is the molecular weights of the corresponding species, Vcell is the volume of the unit cell, and ZofisVirginia, the number of6020: formula unitsand perMicrostructure unit cell. University MSE Defects in Materials, Leonid Zhigilei

Extrinsic point defects in ionic crystals - impurities CaO ↔ Ca 'Zr' + O Ox + VO•• ρ=

Z × [ x(M CaO ) + (1 − x)M ZrO2 ] N aVcell

2CaO + ZrZrx ↔ 2Ca 'Zr' + 2O Ox + Zri•••• ρ=

Z × [ x(M CaO2 + 0.5M Zr ) + (1 − x)M ZrO2 ]

small amount of CaO stabilizes cubic fluorite structure (we are neglecting changes in the size of the unit cell with composition)

N aVcell 4×(ZnO2) per unit cell

Thus, the two hypothetical models predict ~8% difference in the density. When mass and volume measurements can be done with sufficient accuracy, it is possible to distinguish the models.

ρ, g/cm3

For x = 0.15, Z = 4, MCaO = 56.2 g/mole, MZrO2 = 123.2 g/mole, MCaO2 = 72.1 g/mole, MZr = 91.2 g/mole, and the cubic lattice constant a = 5.15 Å, we calculate densities of ρ = 5.5 g/cm3 and ρ = 5.95 g/cm3 for the vacancy and interstitial models, respectively.

Experiments by Diness and Roy [Solid State Commun. 3, 123, 1965]

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

a

Extrinsic point defects in ionic crystals - impurities concentration of point defects in KCl with 0.1 ppm CaCl2 intrinsic (Schottky) defects: x x K Kx + ClCl ↔ VK' + VCl• + K Kx + ClCl

⎛ ΔsvSD = exp ⎜⎜ N ⎝ 2k B

neqSD

ln(10-7)

= 16.12

⎛ Δh SD ⎞ f ⎟⎟ exp ⎜ − ⎜ 2k T B ⎠ ⎝

⎞ ⎟ ⎟ ⎠

Δh SD = 2.6 eV f extrinsic defects x CaCl2 ↔ Ca •K + VK' + 2ClCl

extrinsic defects dominate at low T from Allen & Thomas

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Non-stoichiometry While some of the compounds become unstable at small deviations from stoichiometric composition (e.g. NaCl), other compounds can exhibit large deviations from stoichiometric composition or even be unstable at the stoichiometric composition (e.g. FeO - wüstite phase).

M1−δO

MO

1 O2 ( g ) ↔ OOx + VM (transfer of neutral O) For a transfer of oxygen to a metal oxide MO: 2 1 But if the crystal is ionic, O will accept 2 e- that should O2 ( g ) ↔ OOx + 2h• + VNi'' come from metal that is already ionized, e.g., for NiO: 2 O2- Ni2+ O2- Ni2+ 2+

Ni

O

2-

2+

Ni

O

2-

O2- Ni2+ O2- Ni2+ Ni2+ O2- Ni2+ O2-

1 O2 ( g ) 2

O2- Ni2+ O2- Ni2+ O2Ni2+ • O2- VNi'' O2- Ni2+ h 2+ 2O Ni • O2- Ni2+ h Ni2+ O2- Ni2+ O2cation vacancy + 2 holes

Ni2+ can then transform into Ni3+ (2 Ni ions have to be transformed for each vacancy) (equivalent to solution of Ni2O3 in NiO) This scenario works for transition metals (e.g., Fe, Ti, Cu, Ni), where the ionization energy is low, but not for metals with high ionization energy (e.g. Na or K) University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Non-stoichiometry

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Electronic defects in semiconductors

¾ Electronic defects in intrinsic semiconductors ¾ Extrinsic electronic defects in semiconductors - doping

optional reading (not tested)

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

insulators - semiconductors - metals σ = n|e|μe + p|e|μh

σ (Ωcm)-1

n ≈ 1.4×1010 cm-3 (Si at 300 K)

n ~ 1 cm-3

n ≈ 1.8×1023 cm-3 (Al)

n - number of “free” or conduction electrons per unit volume partially filled band

filled valence band filled band

empty conduction band

?

Energy

Energy

empty band empty band

GAP

Eg < 2 eV filled valence band

partly filled band

filled band

filled band

filled states

filled states

Eg > 2 eV

Energy

filled states

empty conduction band

filled states

Energy

overlapping bands

Cu: 1s22s22p63s23p63d104s1

filled band

filled band

Mg: 1s22s22p63s2

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Electronic defects in intrinsic semiconductors ε

ε T=0K

conduction band

EC

Eg EV

Fermi level, EF

valence band

holes, h 0

1 f(ε)

EF represents probability of ½ that an available energy state is occupied by an electron EF = electrochemical potential of electrons material

T>0K

electrons, e

Eg, eV

Si

1.1

SiC

2.9

ZnO

3.3

Al2O3

9.5

0

1 f(ε)

Thermal generation of electron-hole pairs. Electrons excited to the conduction band leave holes in the valance band

The concentration of electrons, n, in the conduction band −1

⎡ ⎛ ε − EF ⎞⎤ ⎛ 2πme*kBT ⎞ ⎟⎟⎥ dε ≈ 2⎜⎜ ⎟⎟ n = ∫ g (ε) f (ε, T )dε = ∫ g (ε)⎢1 + exp⎜⎜ 2 ⎝ h ⎠ ⎝ kBT ⎠⎦ EC EC ⎣ ∞

∞

3/ 2

⎛ E − EF ⎞ ⎟⎟ exp⎜⎜ − C kBT ⎠ ⎝ 3/ 2

⎛ 2me* ⎞ We used the free electron gas model approximations: g (ε) = 4π⎜⎜ 2 ⎟⎟ (ε − EC )1/ 2 ⎝ h ⎠ ∞ ⎞ ⎞ ⎛ ⎛ as well as 1 + exp⎜⎜ ε − EF ⎟⎟ ≈ exp⎜⎜ ε − EF ⎟⎟ and ∫ x exp(− x) = π / 2 ⎝ kBT ⎠ ⎝ kBT ⎠ 0

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Electronic defects in intrinsic semiconductors ε

⎛ 2πme*kBT ⎞ ⎟⎟ n = 2⎜⎜ 2 ⎝ h ⎠

EC

3/ 2

Similarly, the concentration of holes, p, in the valence band 3/ 2

EV

⎛ E − EF ⎞ ⎛ E − EF ⎞ C ⎟⎟ = Neff ⎟⎟ exp⎜⎜ − C exp⎜⎜ − C kBT ⎠ kBT ⎠ ⎝ ⎝

g(ε)

⎛ 2πmh*kBT ⎞ ⎟⎟ p = 2⎜⎜ 2 ⎝ h ⎠

⎛ E − EV exp⎜⎜ − F kBT ⎝

⎞ ⎛ E − EV V ⎟⎟ = Neff exp⎜⎜ − F kBT ⎠ ⎝

⎞ ⎟⎟ ⎠

C V Neff and Neff - effective densities of state at the conduction and valence band edges

for an intrinsic semiconductor:

ni = n = p

mh* Eg 3 + kBT ln * ≈ EF = me 2 2 4 Eg

intrinsic concentration of charge carriers: ⎛ Eg ⎞ C V ⎟⎟ exp⎜⎜ − ni = np = Neff Neff ⎝ 2kBT ⎠

n = C T3/2 exp(-Eg/2kT)

depends only on T and Eg = EC -EV University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Electronic defects in intrinsic semiconductors ⎛ 2πme*kBT ⎞ ⎟⎟ n = 2⎜⎜ 2 h ⎝ ⎠

3/ 2

⎛ Eg ⎞ ⎛ Eg ⎞ C ⎟⎟ = Neff ⎟⎟ exp⎜⎜ − exp⎜⎜ − k T k T 2 B ⎠ ⎝ B ⎠ ⎝

for T = 1500 K h = 6.626×10−34 Js kB = 1.381×10−23 J/K me* ≈ 9.11×10−31 kg

C Neff

⎛ 2πme*kBT ⎞ ⎟⎟ = 2⎜⎜ 2 ⎝ h ⎠

3/ 2

= 2.8 ×1026 m-3

⎛ Eg ⎞ ⎟⎟ = 0.014 exp⎜⎜ − 2 k T B ⎠ ⎝ surprisingly good semiquantitative agreement, given that very rough approximations are used, e.g. g(ε) for free electron model, decrease of Eg with increasing T is neglected…

Eg = 0.67 eV for Ge Eg = 1.11 eV for Si

- can find Eg from the temperature dependence of intrinsic carrier concentration University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Electronic defects in extrinsic semiconductors Extrinsic semiconductors - electrical conductivity is defined by impurity atoms. Si is considered to be extrinsic at room T if impurity concentration is one impurity per 1012 lattice sites

Unlike intrinsic semiconductors, an extrinsic semiconductor may have different concentrations of holes and electrons. p-type if p > n and n-type if n > p

intrinsic: n ≈ 1.4×1016 m-3 (Si at 300 K) molar volume of Si ≈12 cm3/mol NA ≈ 6×1023 atoms/mol 6×1023 / 12×10-6 = 5×1028 atoms/m3 fraction of excited intrinsic electrons per atom ~10-13

One can engineer conductivity of extrinsic semiconductors by controlled addition of impurity atoms – doping (addition of a very small concentration of impurity atoms). Two common methods of doping are diffusion and ion implantation. n-type: excess electron carriers are produced by substitutional impurities that have more valence electron per atom than the semiconductor matrix (elements in columns V and VI of the periodic table are donors for semiconductors in the IV column, Si and Ge) Example: P (or As, Sb..) with 5 valence electrons, is an electron donor in Si since only 4 electrons are used to bond to the Si lattice when it substitutes for a Si atom. Fifth outer electron of P atom is weakly bound in a donor state (~ 0.01 eV) and can be easily promoted to the conduction band. p-type: excess holes are produced by substitutional impurities that have fewer valence electrons per atom than the matrix (elements in columns III of the periodic table (B, Al, Ga) are donors for semiconductors in the IV column, Si and Ge) University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Electronic defects in extrinsic semiconductors Ec EF

Ed

EV

Ec EF

Ed

Ec

EF EV

EV

T = 0K

Ec

T > 0K

donor dopant creates energy level near conduction band easy to promote electron from donor level to conduction band Fermi level moves toward conduction band

Ea

EF EV

T = 0K

Ea

T > 0K

acceptor dopant creates energy level near valence band easy to promote electrons from valence levels to acceptor band (create holes are in valence band) Fermi level moves toward valence band.

Out of the total number of dopants (substitutional extrinsic point defects), some will be neutral and some ionized. For example, for P in Si, the total concentration [P] = [P]0 + [P]+ ⎛ E − Ed [P+ ] = [P]{1 − f (Ed , EF , T)} ≈ [P] exp⎜⎜ F ⎝ kBT

⎞ ⎟⎟ ⎠

vacancies can also introduce energy levels within the band gap and can be ionized University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei