Computational Physics(Lecture 24)

PHY4370

DFT calculations in action: Strain Tuned Doping and Defects

Can Strain enhance doping?• A few theoretical studies suggested it.

– (Sadigh et al. suggested that the solubility of B in Si can be enhanced by a compressive biaxial strain; Ahn et al. proposed a general theory of strain effects on the solid solubility of impurities in Si)

• Bennett et al. suggested that Sb in Si is enhanced to 10 21/cm 3 under tensile strain.

• Strain enhanced doping in III-V semiconductors. – Junyi Zhu, Feng Liu, G. B. Stringfellow, Su-huai Wei, Phys. Rev. Lett.

105, 195503 (2010)

How does strain enhance doping? Under hydrostatic strain, the impurity

formation energy might first decrease and it may reach a minimum ; then it will increase with the change of strain?

Dop

ing

Ene

rgy

Strain0

i

Simulation Setup

• 64 zincblende atom cell of GaP– VASP, GGA PAW_PBE.– 4x4x4 k-point sampling– forces on all atoms are converged to be less

than 0.01 eV/Å– Plane wave cutoff 400 eV.

• Dopants: Zn, Cd, Al, In, Be, Si, Ge, Sn.

Dopant induced volume change• ∆V= ∆V(instrinsic) + ∆V(electronic)• ∆V(instrinsic) = 16/3 SQRT(3) [(R(dopant)+R(P)) 3 -(R(Ga)+R(P)) 3]

-10

-5

0

5

10

Intr

insi

c V

olu

me

Ch

an

ge

(Å

3)

Be

Sn

Zn+GeGe

InCd

AlZn

Intrinsic dopant induced volume change

Total dopant induced volume change

-10

-5

0

5

10

15

1.225Å

1.405Å

0.975Å

1.405Å 1.405Å

1.225Å

(1.23Å)

Zn+Ge

Sn

Be

Ge

InCd

Al

Zn(1.225Å)

Vol

um

e(Å3

)Dopant Induced Volume Change

Electronic environment induced volume change

-1

0

1

2

3

4

5

Ele

ctro

nic

En

viro

nm

en

t Vo

lum

e C

ha

ng

e (

Å3)

Zn

Al

Cd

In

Ge

Sn

Be

Zn+Ge

Doping energy vs. hydrostatic strain

Dismiss the speculation

E(host)= α(V –V(host)) 2

E(host+dopant)= α’(V –V(host+dopant)) 2

E(doping) = E(host+dopant)-E(host)~V(V(host)-V(host+dopant)) if α=α’

Bi-axial strain enhanced doping

• In epitaxial growth, bi-axial strain can be conveniently applied.

• Apply the strain along x and y, relax the z direction to achieve E minimum.

Red: biaxial Black: hydrostatic.

Strain tuned doping sites and type

• Interstitial doping and substitutional doping may induce different volume changes

• Strain provides a promising way to tune the doping sites and type. – Junyi Zhu, Su-huai Wei, Solid State

Communication 151, 1437 (2011)

Substitutional vs. Interstitial

• Enhance p-type substitutional doping and reduce

interstitial doping.

• Substitional dopants provide free carriers

• Interstitial dopants: Small dopants, sometimes deep

levels, passivating p-type dopants, introduce n-type

dopants.

• One example, Li in ZnO.

• Widely used in Energy applications: transparent electrode,

smart windows and LEDs.

Another Type of Problem

• Enhance interstitial doping and reduce substitutional doping.

• Li battery electrodes.

• Interstitial Li

• Good diffusitivities

• Good Reversibilities.

• Substitutional Li,

• Less Mobile

• Difficult to charge and discharge.

Simulation Setup• VASP

• PAW_PBE

• 72 atoms supercell.

• Plane wave cutoff energy: 600 eV.

• 4x4x4 k-points mesh.

• Lattice constant: 3.287 Å.

• c/a ration: 1.6137

Volume Change

Relax all three dimensions Fix x, y

Interstitial: 8.56 Å3 6.31Å3

Substitutional: -4.91 Å3 -2.711 Å3

Doping energy difference vs. Hydrostatic strain

• Doping energy: E(doping) = E(doped) − E(reference) + μ(Zn) − μ(Li);

Linear relationship.

1% strain enhance about 3-5 times

concentration of Substitutional

dopants at 900K.

1% strain reduces about one order

of magnitude of interstitial doping

at 900K.

Doping energy difference vs. biaxial strain

Formation Energy of Li at interstitial and substitutional sites vs. Fermi Energy.

VBM

Form

ati

on e

nerg

y

0.35eV 0.8 eV

Schematic illustration of Formation energy of Li at interstitial and substitutional sites in ZnO vs. Fermi Energy. Dashed(Blue): under 2% compressive strain. Solid(black): strain free.

Strain Tuned Defects• CZTS(Se)

– Important PV absorber.

• VCu : Important p-type dopant. Passivation of deep levels.

• CuZn: Deeper acceptor, lower formation energy than VCu .

• External strain: effective to tune their formation energies and enhance VCu.

Results

Junyi Zhu, Feng Liu and Mike Scarpulla, In preparation.

Summary• Dopant induced Volume Change:

– Intrinsic– Electronic environment

• Positive for n type• Negative for p type

• The sign of the dopant induced volume change for unstrained host lattice determines how strain affects doping. – volume expansion favors tensile strain– volume shrink favors compressive strain

• Doping energy change is super linear with strain.– No minimum at particular volume.

• Also an interesting general strategy to tune doping site and intrinsic defects.• Can be extended to other material systems.

Surfactant Tuned Doping and Defects

Tuning the electronic environment

• Codoping– Change the local electronic environment.

• Surfactant enhanced doping– Surface Active Agent– Surface metallic elements to modify the electronic

structure of thin films.

Revisit of Doping

• Either one electron more or one electron less• Suppose the host lattice is stable

– After doping, either electron shortage or extra electrons.

– Unstable– Electron counting rule.

Electron Counting Rule

• Metallic or nonmetallic surfaces ?– With a given distribution of dangling bonds

• Chadi, 1987, PRL, 43, 43• Pashley, 1989, PRB, 40, 10481

• The basic assumptions of ECR to apply to III-V(001) surface– to achieve Lowest-energy surface

• Filling dangling bonds on the electronegative element • empty dangling bonds on the electropositive element

Intrinsic difficulty of Doping

• The ECR can’t be satisfied during the doping.• One way to improve the doping

– to help the system satisfy ECR– Atomic H can serve that purpose

Surfactant enhanced doping in GaP/InGaP

• Sb/Bi are good surfactants for GaP– Low incorporation and low volatility.

• Zn doping is improved by the use of Sb as surfactant in InGaP and GaP.– Zn doping improved by an Order of magnitude

• D.C. Chapman, A.D. Howard and G.B. Stringfellow, Jour. of Crys. Growth, 287, Issue 2, 647 (2006).

• D. Howard, D. C. Chapman, and G. B. Stringfellow, J. Appl. Phys. 100, 44904 (2006).

Lack of physical understanding• Lack of in situ. observations.• Difficult to observe possible H as a codopant or a

surfactant.• DFT calculation can be a good tool.

J. Y. Zhu, F. Liu, G. B. Stringfellow, Phys. Rev. Lett. 101, 196103 (2008)

Simulation setup• GaP (001) films by a supercell slab

consisting of 4 layers of Ga atoms and 5 layers of P atoms, plus a 12.8 Å vacuum layer.

• 5.4 Å as the lattice parameter• plane wave cut-off energy:348 eV • 4x4x1 k-point mesh for Brillouin zone

sampling• energy minimization was performed by

relaxing atomic positions until the forces converged to less than 0.1 meV/Å

Calculation of the doping energy of Zn

• Replace a Ga

• Replace P dimer with Sb• Different concentration of H.

ZnGadoping undopeddoped EEE

Dual Surfactant Effect

• The two Surfactants work together to lower the Zn doping energy.– they do not lower the Zn doping energy

individually.

The role of H and Sb• The Effect of Sb:

– realized only when H is incorporated– Lower Electronegativity– Electron reservoir (High p orbital).

• H maintains ECR, filling the high 5 p orbital and

charge transfer to ZnGa to lower the doping energy

Codoping: One H goes into bulk

ZnGaHbulk 12H-Sb

between4th, ＋1.40eV EZnGaHbulk 12H-Sb

below4th, ＋eV 64.1 E

A possible doping process

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

E (

eV)

a

b cd

-0.93 eV

-0.12eV-0.23 eV

a b c d

Summary

• Dual-surfactant effect of Sb and H for Zn doping enhancement in GaP

• Greatly broaden the scope and application of the conventional surfactant effect of single element.

• The role of Sb• The role of H

Discussion• Surfactants also change the strain

distribution in the thin film.• A combination of surfactant, codoping and

strain enhanced doping.• Surfactants may also lower the vacancy

formation energy of host atoms to enhance the kinetic process of doping.

Functionals for exchange and correlation

• The exchange and correlation functional can be reasonably approximated– As a local or nearly local functional of the density.– The exact functional must be very complex!

The local spin density approximation (LSDA)

• Kohn and Sham showed in their seminal paper that the exchange and correlation function is generally local for solids, because solids can often be viewed as close to the limit of homogeneous electron gas.

• Thus they proposed the local spin density approximation, so that the exchange correlation energy is an integral over all space with the exchange correlation energy density at the point assumed to be the same as in a homogeneous electron gas with the same density

• Exc(LSDA) = • The LSDA is the most general local approximation

and is given explicitly for exchange (proportional to n1/3)and by approximate (or fitted) expressions for correlation.

• In PZ, the exchange and correlation follow a similar form.

• Read chapter 5 of Richard Martin’s book for detailed description.

• The rationale for the local approximation is that for the densities typical of those found in solids– The range of effects of exchange and correlation is

rather short.• This is not justified by a formal expansion in

some small parameter. • It will be the best for solids close to a

homogeneous gas (like a nearly free electron metal) and worst for very inhomogeneous cases.

• The self interaction term can be cancelled by the non-local exchange interaction in Hartree-Fock.

• However, in LDA, the cancellation is approximate and there remain self-interaction terms.

Generalized gradient approximations (GGA)

• The success of the LSDA has led to the development of various generalized-gradient approximations– With improvement over LSDA.

• In the chemistry community, GGA can provide the accuracy that has been accepted.

• The first step beyond the local approximation is a functional of the magnitude of the gradient of the density as well as the value n at each point. – Which was suggested in K-S’s original paper. – Gradient expansion approximation doesn’t have to

be better because it violates the sum rule and other relevant conditions.

• The term GGA denotes a variety of ways proposed for functions that modify the behavior at large gradients in such a way as to preserve desired properties.

• ExcGGA=

• Perdew and Wang (PW91), Perdew, Burke and Enzerhof (PBE) all proposed forms of the expansion of GGA.

• Many GGA functionals that are used in quantitative calculations in chemistry. – Correlation is often treated using Lee Yang Parr (LYP).– Krieger and coworkers have constructed a functional

KCIS based upon many-body calculations of an artificial jellium with a gap problem.