Point defects ionic crystals Electronic defects in semiconductors

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

Point defects ionic crystals

Ionic bonding and ionic crystals: Brief reviewDefects in ionic crystal and oxides. Kröger-Vink Notation Site, mass, and charge balanceFrenkel and Schottky defects Extrinsic point defects in ionic crystals - impuritiesNon-stoichiometry in ionic crystals

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

Electronic defects in semiconductorsElectronic defects in intrinsic semiconductorsExtrinsic electronic defects in semiconductors - doping

optional reading(not tested)


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

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

give

up

1e-

give

up

2e-

give

up

3e-

iner

t

acce

pt 1

e-

acce

pt 2

e-

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

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 electronVIIA: Halogens (F, Br, Cl...) missing one electron in outermost occupied p subshell - want to gain electronMetals are electropositive – they can give up their few valence electrons to become positively charged ions


Example: table salt (NaCl)

Na has 11 electrons, 1 more than needed for a full outer shell (Neon)

Cl has 17 electron, 1 less than needed for a full outer shell (Argon)

11 Protons Na 1S2 2S2 2P6 3S1

11 Protons Na+ 1S2 2S2 2P6

donates e-

10 e- left

17 Protons Cl 1S2 2S2 2P6 3S2 3P5

17 Protons Cl- 1S2 2S2 2P6 3S2 3P6

receives e-

18 e-

Na Cle-

Cl-Na+

Na (gas) + 5.14 eV (ionization energy) → Na+ + e-

e- + Cl (gas) → Cl- + 3.61 eV (electron affinity)

energy of (long-range) interaction among the ionsNa+ + Cl- → NaCl (crystal) + 7.9 eV

balance: ΔE = 7.9 eV - 5.1 eV + 3.6 eV = 6.4 eV per NaCl formula unit

Cohesive energy of NaCl crystal (energy needed to convert NaClcrystal into individual Na and Cl atoms):

< 0 → it costs energy to transfer e from Na to Cl

Ionic bonding: Brief review


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

fcc with 2 atoms in the base: at (0, 0, 0) and (½, 0, 0)

NaCl structure

KCl, AgBr, KBr, PbS, MgO, FeO

Na+ ions filling octahedral holes in the fcc structure

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

fluorite structure

F- ions filling tetrahedral holes in the fcc structure

CaF2 or ZrO2


Ionic crystals: Brief review

simple cubic with two atoms in the base at (0,0,0) and (½, ½, ½)

CsCl structure

CsCl, TlI, TlClAlNi, CuZn - intermetallic comp.

fcc with two atoms in the base at (0,0,0) and (¼, ¼, ¼)

zinc blende structure

ZnS, CuF, CuClGaAs, GaP, InP- semiconductors

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 structurenamed after the mineral spinel (MgAl2O4)

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

Fe3+( Fe2+ Fe3+)O4, Mg2+( Al23+)O4, Fe3+(Cr2

3+)O4


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 – 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 chargex zero relative charge (x is often omitted)

ZYX


Kröger-Vink Notation ZYX

Examples:

interstitial Ag ion in AgCl:

vacancy on a Ag site in AgCl:

Ca2+ ion on a Na site in NaCl:

vacancy on an O site in Al2O3:

Cu+ on a Cu2+ site CuO:

•iAg

'AgV•NaCa••

OV'CuCu

i siteO siteU site

Pu4+ ionVacancyO2- ionU4+ ionY X

''''UV••

OVxiV

xOO

''''''UO

''iO

xUU

••••••OU

••••iU

xUPu

••••••OPu

••••iPu

Intrinsic point defects and Puimpurity in UO2 crystal

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


Site, mass, and charge balance

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.

Formation and annihilation of point defects in ionic crystals must satisfy the following 3 rules:

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.


Frenkel defects

+

Frenkel defect (Frenkel pair) = vacancy + interstitial in close proximityfirst 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.

'VAgAg AgixAg +↔ •formation reaction for a cation Frenkel pair in AgCl:

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


Frenkel defects

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ⎟⎟⎠

⎞⎜⎜⎝

⎛=

Tkh

ks

NN

Nn

B

FPf

B

FPv

iFPeq

2exp

2exp

2/1

FPvi nnn ==

( ) ( )!!!ln

!!!ln iii

i

BvvBnc nNn

NknNn

NkS−

+−

=

nc

FPf

FPf

FP STsThnG Δ−Δ−Δ=Δ )(Equilibrium concentration of Frenkel defects:

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

0ln =⎟⎟⎠

⎞⎜⎜⎝

⎛+Δ−Δ=

∂Δ∂

=i

eqeqB

FPv

FPf

nnFP N

nNn

TksThn

GFPeq

FP

2.5ZnO9.53.0

UO2

128.7

TiO2

2.3Li2O7

2.3-2.8CaF2

1.1AgBrreactioncompound eV ,FP

fhΔ

''iO

xO OVO +↔ ••

''''Ui

xU VUU +↔ ••••

'Agi

xAg VAgAg +↔ •

'iF

xF FVF +↔ •

''Cai

xCa VCaCa +↔ ••

'Lii

xLi VLiLi +↔ •

''iO

xO OVO +↔ ••

''''Tii

xTi VTiTi +↔ ••••

''iO

xO OVO +↔ •• from Allen & Thomas

many more anion Frenkel defects than cation ones

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

includes energy of electrostatic interactionsFPfhΔ


Schottky defects

6.4UO2

2.2-2.4NaCl

5.2TiO2

5.5CaF2

2.6KCl

6BeO

26α-Al2O3

Schottky defectcompound eV ,SDfhΔ

••+ o'''

Al V32V

from Allen & Thomas

electrostatic attraction between cation and anion vacancies → binding energy of the Schottky defect and temperature dependent degree of association

Schottky defect = cation vacancy + anion vacancy in close proximity

xO

xBeO

''Be

xO

xBe OBeVVOBe +++↔+ ••

formation reaction for a Schottky defect in BeO:

this reaction satisfies the mass, charge, and side balance

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ=

Tkh

ks

Nn

B

SDf

B

SDv

SDeq

2exp

2exp

equilibrium concentration:•+ F

''Ca V2V

••+ O''

Be VV••+ O

''''Ti V2V

••+ O''''

U V2V•+ Cl

'Na VV

•+ Cl'K VV relative low Δhf → Schottky defects dominate


Extrinsic point defects in ionic crystals - impuritiesIntroduction 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:

2KCl(g)2ClVCa2Cl2K)s(CaCl xCl

'KK

xCl

xK2 +++↔++ •

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.

xCl

'Ki2 2ClV2CaCaCl ++↔ ••or

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:

2KCl(g)2ClV2Ca2Cl2K)s(CaCl xCl

'Ki

xCl

xK2 +++↔++ ••

xCl

'KK2 2ClVCaCaCl ++↔ •or


Extrinsic point defects in ionic crystals - impuritiesDifferent 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

''i

xOY2 OO32Zr2ZrO ++↔ •

(2) O2- fully occupy O sites and cation defects take care of the charge balance '''

YYxO2 V3ZrO63ZrO ++↔ •

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.


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

We have two possible scenarios:

(1) charge compensation by anion vacancy ••++↔ OxO

''Zr VOCaCaO

(2) charge compensation by Zr interstitials

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:

••••++↔+ ixO

''Zr

xZr ZrO2Ca2Zr2CaO

2ZrOCaO )M(1)(M (1) xx −+22 ZrOZrCaO )M(1)M

21(M (2) xx −++

where M is the molecular weights of the corresponding species, Vcell is the volume of the unit cell, and Z is the number of formula units per unit cell.

cellaVNxxZ ])M(1)(M[

2ZrOCaO −+×=ρ

cellaVNxxZ ])M(1)M5.0(M[

22 ZrOZrCaO −++×=ρ

need an extra Zr sitesite balance: 1Zr = 2Zr + 2Onull = ZrO2

2ixO

''Zr

xZr

xO

xZr ZrOZr2O2CaZr2OZr2CaO +++↔+++ ••••

Ca2+ cation, Zr4+ cation, O2- anion


Extrinsic point defects in ionic crystals - impurities

cellaVNxxZ ])M(1)(M[

2ZrOCaO −+×=ρ

cellaVNxxZ ])M(1)M5.0(M[

22 ZrOZrCaO −++×=ρ

••++↔ OxO

''Zr VOCaCaO ••••++↔+ i

xO

''Zr

xZr ZrO2Ca2Zr2CaO

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

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 themodels.

small amount of CaO stabilizes cubic fluorite structure a4×(ZnO2) per unit cell

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 latticeconstant 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.

ρ, g

/cm

3

(we are neglecting changes in the size of the unit cell with composition)


Extrinsic point defects in ionic crystals - impurities

from Allen & Thomas

extrinsic defects dominate at low T

concentration of point defects in KCl with 0.1 ppm CaCl2

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ=

Tkh

ks

Nn

B

SDf

B

SDv

SDeq

2exp

2exp

xCl

xKCl

'K

xCl

xK ClKVVClK +++↔+ •

intrinsic (Schottky) defects:

eV 6.2=Δ SDfh

xCl

'KK2 2ClVCaCaCl ++↔ •

extrinsic defects

ln(10-7) = 16.12


Non-stoichiometry

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)

Ni2+ can then transform into Ni3+ (2 Ni ions have to be transformed for each vacancy)

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 - wüstitephase).

MxO2 VO)(O

21

+↔gFor a transfer of oxygen to a metal oxide MO:

cation vacancy + 2 holes

(transfer of neutral O)

But if the crystal is ionic, O will accept 2 e- that should come from metal that is already ionized, e.g., for NiO:

''Ni

xO2 V2O)(O

21

++↔ •hg

-2O-2O-2O-2O

-2O-2O-2O-2O

•h

+2Ni+2Ni+2Ni+2Ni

+2Ni+2Ni+2Ni+2Ni

''NiV

-2O-2O-2O-2O

-2O-2O-2O-2O

+2Ni+2Ni+2Ni+2Ni

+2Ni+2Ni+2Ni+2Ni

•h

-2O)(O

21

2 g

(equivalent to solution of Ni2O3 in NiO)

OM δ1−MO


Non-stoichiometry


Electronic defects in semiconductors

Electronic defects in intrinsic semiconductorsExtrinsic electronic defects in semiconductors - doping

optional reading(not tested)


insulators - semiconductors - metals

σ (Ωcm)-1

n ≈ 1.4×1010 cm-3 (Si at 300 K)n - number of “free” or conduction electrons per unit volume

n ≈ 1.8×1023 cm-3 (Al)n ~ 1 cm-3

σ = n|e|μe + p|e|μh

filled band

Energy

partly filled band

empty band

GAP

fille

d st

ates

partially filled band

filled band

Energy

partly filled band

empty band

GAP

fille

d st

ates

filled band

Energy

partly filled band

empty band

GAP

fille

d st

ates

partially filled band

Energy

filled band

filled band

empty band

fille

d st

ates

overlapping bandsEnergy

filled band

filled band

empty band

fille

d st

ates

Energy

filled band

filled band

empty band

fille

d st

ates

overlapping bands

Energy

filled band

filled valence band

fille

d st

ates

empty

bandconductionempty

bandconduction

Energy

filled band

filled valence band

fille

d st

ates

?

empty

bandconductionempty

bandconduction

Mg: 1s22s22p63s2Cu: 1s22s22p63s23p63d104s1

Eg > 2 eV Eg < 2 eV


Electronic defects in intrinsic semiconductors

conduction band

Eg

T = 0 K

material Eg, eV

Si 1.1

SiC 2.9

ZnO 3.3

Al2O3 9.5

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

electrons, e

holes, h

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

valence band

Fermi level, EF

f(ε)

ε

T > 0 K

f(ε)

ε

10 0 1

The concentration of electrons, n, in the conduction band

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−⎟⎟

⎠

⎞⎜⎜⎝

⎛ π≈ε⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −ε+ε=εεε=

−∞∞

∫∫ TkEE

hTkmd

TkEgdTfg

B

FCBe

B

F

EE CC

exp22exp1)(),()(n2/3

2

*1

EV

EC

2/12/3

2

*

)(24)( Ce E

hmg −ε⎟⎟

⎠

⎞⎜⎜⎝

⎛π=ε

⎟⎟⎠

⎞⎜⎜⎝

⎛ −ε≈⎟⎟

⎠

⎞⎜⎜⎝

⎛ −ε+

TkE

TkE

B

F

B

F expexp1

We used the free electron gas model approximations:

( ) 2/exp0

π=−∫∞

xxas well as and



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

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−⎟⎟

⎠

⎞⎜⎜⎝

⎛ π=

TkEEN

TkEE

hTkmn

B

FCCeff

B

FCBe expexp222/3

2

*

Similarly, the concentration of holes, p, in the valence band

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−⎟⎟

⎠

⎞⎜⎜⎝

⎛ π=

TkEEN

TkEE

hTkmp

B

VFVeff

B

VFBh expexp222/3

2

*EV

g(ε)

ε

EC

for an intrinsic semiconductor:

pnni ==2

Eln

43

2E

E g*

*g ≈+=

e

hBF m

mTk

Veff

Ceff and NN

intrinsic concentration of charge carriers:

⎟⎟⎠

⎞⎜⎜⎝

⎛−==

TkE

NNnpnB

gVeff

Ceffi 2

exp

depends only on T and Eg = EC -EV

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



Js 10626.6 34−×=h

surprisingly good semi-quantitative 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 GeEg = 1.11 eV for Si

3-262/3

2

*

m 108.222 ×=⎟⎟⎠

⎞⎜⎜⎝

⎛ π=

hTkmN BeC

eff

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛ π=

TkE

NTk

Eh

TkmnB

gCeff

B

gBe

2expexp22

2/3

2

*

J/K 10381.1 23−×=Bkkg 1011.9 31* −×≈em

for T = 1500 K

014.02

exp =⎟⎟⎠

⎞⎜⎜⎝

⎛−

TkE

B

g

- can find Eg from the temperature dependence of intrinsic carrier concentration


Electronic defects in extrinsic semiconductorsExtrinsic 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

molar volume of Si ≈12 cm3/molNA ≈ 6×1023 atoms/mol6×1023 / 12×10-6 = 5×1028 atoms/m3

fraction of excited intrinsic electrons per atom ~10-13

intrinsic: n ≈ 1.4×1016 m-3 (Si at 300 K)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

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)


Electronic defects in extrinsic semiconductors

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

Ec

EV

T = 0K

EdEc

EV

T > 0K

EdEF EF

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.

Ec

EV

T = 0K

Ea

Ec

EV

T > 0K

EaEF EF

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]+

{ } ⎟⎟⎠

⎞⎜⎜⎝

⎛ −≈−=+

TkEEf

B

dFexp]P[)T,E,E(1]P[]P[ Fd

vacancies can also introduce energy levels within the band gap and can be ionized

Documents

Point defects ionic crystals Electronic defects in semiconductors