Intrinsic Defects and Dopability of Zinc Phosphide

Steven Demers∗ and Axel van de WalleDepartment of Applied Physics and Materials Science,

California Institute of Technology, Pasadena, California 91125, USA(Dated: March 6, 2012)

Zinc Phosphide (Zn3P2) could be the basis for cheap and highly efficient solar cells. Its use inthis regard is limited by the difficulty in n-type doping the material. In an effort to understandthe mechanism behind this, the energetics and electronic structure of intrinsic point defects in zincphosphide are studied using generalized Kohn-Sham theory and utilizing the Heyd, Scuseria, andErnzerhof (HSE) hybrid functional for exchange and correlation. Novel ’perturbation extrapolation’is utilized to extend the use of the computationally expensive HSE functional to this large-scaledefect system. According to calculations, the formation energy of charged phosphorus interstitialdefects are very low in n-type Zn3P2 and act as ’electron sinks’, nullifying the desired dopingand lowering the fermi-level back towards the p-type regime. This is consistent with experimentalobservations of both the tendency of conductivity to rise with phosphorus partial pressure, and withcurrent partial successes in n-type doping in very zinc-rich growth conditions.

I. INTRODUCTION

Zinc Phosphide has great potential as a photovoltaicmaterial; it absorbs strongly in the visible spectrum(> 104 − 105cm−1), has long minority carrier diffusionlengths (5-10 µm), and a direct, almost ideal bandgap(∼1.5eV)1. Both zinc and phosphorus are Earth-abundant elements, greatly aiding their widespread use.However, no practical means of creating Zn3P2 crystalswith n-type doping has been found. This has preventedthe typical p-n homojunction solar cells and currentimplementations instead rely upon metal-semiconductorjunctions or p-n semiconductor heterojunctions2. Thebest results are currently with p-Zn3P2/Mg Schottkydiode where the maximal open-circuit voltage (∼0.5eV)is the limiting factor on the efficiency of these devices(currently 6%)3.

Differences in growth environment has been shown toaffect the electrical properties significantly4. Specifically,resistivity ranging over 100 to 105Ω-cm have been mea-sured in single and polycrystalline samples. Such a largerange of values may point to an intrinsic defect mech-anism dominating carrier concentrations. Despite manyyears of research, the exact nature of the various intrinsicdefects as well as their small-scale structure and proper-ties are only partially known. So, as a step towards un-derstanding the n-type doping difficulties, it is natural tofirst explore the role of intrinsic defects in this system.

Density Functional Theory (DFT) is the method ofchoice for studying the electronic structure of defects insemiconductors. It is the only ab initio approach cur-rently tractable for calculation of the energetics of cellsof sufficient size to explore isolated defects (at least onthe order of 100’s of atoms). For computational sim-plicity, the most common exchange-correlation function-als utilized are the local-density approximation (LDA)and generalized-gradient approximation (GGA). How-ever, while the defect-cell total energies are typicallywell-described by LDA or GGA, the well-known bandgapproblem of these methods poses problems in the de-

termining the precise electronic structure of the defectsthemselves. Subsequently, this can lead to different pre-dictions of defect stability amongst calculations even onidentical systems. For example, recent work of variousgroups on ZnO has lead to predictions that the oxygen va-cancy acts as both a shallow and deep defect5,6. Recently,more accurate techniques such as hybrid functionals7 andGW excited-state calculations8 yield greatly improvedbandgap prediction. However, these techniques are stilltoo expensive for the large scales required of defect sys-tems.

Regardless of the exact ab initio procedure chosen,there is a fundamental problem in studying defects witha necessarily small amount of atoms (even the most ef-ficient approach of LDA-based DFT is limited to on theorder of hundreds of atoms); that is, we are imposing adegenerate doping condition, which is seldom the regimeof interest (typical defect or dopant concentrations rarelyexceed parts-per-million). As such, care must be takento correct for interactions that are artifacts of this incor-rectly high defect concentration such as image chargesand spurious hybridization. In the following section wedetail the methods that allow us to compute defect levelsand energetics separate from these effects.

Zn3P2 exists in two phases, tetragonal and cubic.We have investigated the intrinsic point defects of thetetragonal phase, which is of primary concern becauseit is the room-temperature phase. These defects includezinc vacancy (VZn), phosphorus vacancy (VP ), zincinterstitial (Zni), and phosporus interstitial (Pi). Theantisite defects were also studied, but their formationenergies are such that they would be unstable to dis-sociating into a vacancy-interstitial pair, thus they areexcluded from the following discussion. What we findis that the formation energy of the charged acceptordefects, especially Pi, is small enough at even moderaten-type conditions to form enough compensating defectsas to completely neutralize the desired extrinsic doping.

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II. METHODOLOGY

Of central concern is the stability of the intrinsic de-fects in Zn3P2 and their effect on the fermi level. Thestability of a defect is largely determined by its formationenergy; in the supercell formalism put forth by Lany andZunger9 the formation energy of a defect with charge qis given by a sum of three terms:

∆Hf = [ED −EH ] + q(EV + ∆Ef ) +∑α

nα(µoα + ∆µα)

(1)EH and ED are the total energy of the perfect host andhost+defect supercells respectively. The first term on theright is the difference in bond energy brought about bythe defect.

The second term in (1) represents the energy cost ofexchanging electrons with the ’electron reservoir’. EVis the reference energy of the reservoir and is the pricewe pay for removing an electron from the top of the va-lence band (ie. the energy of a hole at the Valence BandMaximum)10. Consequently, equation (1) describes theenergetics of forming a defect while conserving charge9.The calculated total energy (EH) of the system followsJanak’s Theorem11:

dEH(ni)

dni= ei (2)

where ni is the occupation of the highest occupied state iwith eigenvalue ei. For an infinite system ei is identical toEV . Thus, EV can be calculated as the energy differencebetween a host and a host+hole cell in the limit that thenumber of electrons (N) tends to infinity:

limN→∞

[EH(N)− EH(N − 1)] = EV (3)

As a practical matter, a good approximation can be at-tained for relatively small systems - in the present workthe difference between a neutral perfect supercell and asupercell and a hole (EH(0) - EH(+)) is used. We nowhave a good approximation for the electron chemical po-tential at the VBM. Finally, ∆Ef is the additional en-ergy of electrons in our system above the VBM and isthe proxy for specifying the doping regime (p or n-type)of the bulk.

The crystal growth environment affects the formationenergy via the chemical potentials (µα = µoα + ∆µα) inequation (1). These represent the energy cost of exchang-ing atoms with the chemical reservoir. By convention,the formation energies are defined in relation to the stan-dard states of the constituents of a system. Thus, theelemental chemical potentials are broken into two parts:µoα is the chemical potential of the standard state of theelement and ∆µα is the chemical potential of the elementin relation to the standard state. The growth conditions

FIG. 1. Zn3P2 allowed chemical potentials. The red andgreen lines are the stability limits for Zn3P2 and ZnP2 re-spectively. The dark purple region corresponds to the rangeof chemical potentials where we are assured of forming zincphosphide crystals.

-1.0 -0.8 -0.6 -0.4 -0.2 0.0-1.0

-0.8

-0.6

-0.4

-0.2

0.0Zn3P2

P Chemical Poten.al

Zn3P2 ZnP2

Zn-­‐rich/P-­‐deficient

P-­‐rich/Zn-­‐deficient

X

X

are reflected in ∆µα (ie. a maximally rich growth en-vironment of a certain element would have ∆µα = 0),it becomes more negative for lower concentrations of anelement during crystal formation. The chemical poten-tials are added or subtracted from the formation energyaccording to the number of atoms of a certain species isdeposited or withdrawn from the growth reservoir (nα);it is +1 if an atom is added (ie. vacancy defects), if weremove an atom it is -1 (ie. interstitial defects).

Thus, the formation energy is determined as a func-tion of fermi-level (∆Ef ) for a specific growth condition(dictated through ∆µα). Assuming equilibrium growthconditions, the chemical potentials are restricted to val-ues that maintain a stable compound and don’t permitcompeting phases to exist. For Zn3P2 we have the fol-lowing constraints:

In order not to precipitate the elemental form of Zn orP, we must have:

µZn ≤ µoZn; µP ≤ µoP (4)

or, equivalently:

∆µZn ≤ 0; ∆µP ≤ 0 (5)

where ∆µZn=0 would define the maximumly-rich Zngrowth condition.

The stability condition for Zn3P2 is:

3∆µZn + 2∆µP ≤ ∆Hf (Zn3P2) (6)

The only other competing phase is ZnP2 (phosphorus-rich phase) and avoiding its formation yields the followingrestriction:

∆µZn + 2∆µP ≥ ∆Hf (ZnP2) (7)

There is a the further issue of incomplete error cancella-tion in DFT when energy differences are taken between

3

chemically dissimilar systems12 (such as between com-pounds and their elemental constituents). Here we cor-rect the chemical potentials according to [ 12] which re-sults in better agreement between predicted and exper-imental formation energies (1.04 eV non-corrected, 1.79eV corrected, 1.5 eV experimental).

Combining all the conditions described by Eq 5-7, wecan determine an allowed region of chemical potentialswhere we are assured of forming only Zn3P2 - seeFigure 1. Later, when we discuss formation energiesfor a specific growth environment it will be this figurewhich defines the chemical potentials to use for theZn-rich/P-poor versus the Zn-poor/P-rich regimes.

All these calculations are performed within the Kohn-Sham framework (DFT) and utilize the projector aug-mented pseudopotentials as implemented in VASP13–15.We use valence configurations of 3d4s and 3s3p for zincand phosphorus respectively with the Perdew, Burke,and Ernzerhof (PBE) potential16. GGA as well as hybridfunctionals (HSE) for exchange-correlation were used,the specifics of which will be discussed later. For super-cells of 2x2x2 unit cells (320 atoms) energies were calcu-lated with two k-points (0,0,0) and (1/2,1/2,1/2) and aplane-wave basis with a cutoff of 300eV. These settingsresult in a numerical error on the order of 0.1 eV in thedefect formation energy which is on par with the othersources of error (e.g. uncertainty in the bandgap andfinite-size corrections). All calculations were performedconsidering spin polarization.

Although the use of a supercell geometry within a DFTframework is a common approach for defect calculations,care must be taken to avoid spurious finite-size effectsas well as known deficiencies in DFT’s ability to modelexcited state energies.

A. Finite Size Effects

With current limitations on computing power, DFTcalculations are typically restricted to cells on the or-der of hundreds of atoms. Even the addition of a sin-gle point defect would thereby result in defect concen-trations on the order of tenths of a percent, normallydescribing degenerate conditions. Usual semiconductordefect concentrations are much more dilute (on the orderof parts-per-million), and often result in far different ma-terial properties than within the degenerate regime. Dueto the low dielectric strength (and hence poor screening)of Zn3P2 these effects are especially troublesome in thisstudy. However, if we are careful about correcting our re-sults, we can still make accurate predictions about whatthe dilute environment should look like. The three mainfinite-size corrections are discussed below.

1. Image Charge Correction

The drawback of the use of a standard supercell geome-try is that defects are periodically and infinitely repeatedspatially. The defect, instead of being surrounded by alarge region of perfect bulk crystal as it would be undernon-degenerate conditions, is now surrounded by mirrorimages of itself. This will result in somewhat frustratedionic relaxation, though these elastic energy effects tendto be short range and is rarely a problem for even mod-estly sized cells (there is very little difference in relaxationenergies for even a 2x2x2 supercell versus a unit cell ofZn3P2). However, when dealing with charged defectswe form ’image charges’ leading to spurious electrostaticinteractions. These coulombic interactions between thedefect and its mirror charges are long-ranged and signif-icant even for large cells.

Corrections for this ’image charge’ effect have been thesubject of much research, though the most common ap-proach is based on the work of Makov and Payne17. Theyconsidered the charge density to be the contribution ofthe periodic charge of the underlying crystal structureand the charge density of the aperiodic defect (which issimply the electron density difference between the hostand host+defect cells). The multipole correction to theformation energy is:

EIC =q2αM2εL

+2πqQr3εL3

+O(L−5) (8)

where αM is the supercell lattice-dependant Madelungconstant, L is the length of the supercell, ε is the staticdielectric constant, andQr is the second radial moment ofthe aperiodic charge density. The first two terms are themonopole and quadrupole corrections respectively; thequadrupole correction typically ∼ 30% of the monopoleterm11.

2. Potential Alignment Correction

In the case of charged defects with periodic bound-ary conditions there is a violation of charge neutrality,which causes the Coulomb potential to diverge18. Inmomentum-space formalism, one usually sets the G=0term of the electrostatic and ionic potential (VH(G=0)and VI(G=0)) to zero. The Kohn-Sham eigenvalues arethus only defined with respect to the average electrostaticpotential of the cell. For neutral systems this arbitraryoffset still leads to a well-defined total energy since theelectron-electron and ion-ion contributions exactly can-cel. In a charged system, ignoring the G=0 term canbe viewed as equivalent to a uniform background charge(jellium) compensating for the net charge - though it isimportant to note that this only occurs for the potential.In a charged cell there is now an arbitrary offset to thetotal energy. The charged cell energies ED and EH(+)in Eq. (1) have to be compensated for in order to treat

4

FIG. 2. Potential alignment correction. Differences in elec-trostatic potential between perfect and defect cells far fromdefect area are used to correct charged cell energies.

VZn

0 2 4 6 8 10 12-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Distance from Defect HangL

PotentialDifferenceHevL

POTENTIAL ALIGNMENT - VZn

0 2 4 6 8 10 12-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Distance from Defect HangL

PotentialDifferenceHevL

POTENTIAL ALIGNMENT - VPVP

4 6 8 10 12-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Distance from Defect HangL

PotentialDifferenceHevL

POTENTIAL ALIGNMENT - IZn0.3

0.0

-­‐0.3

0.3

0.0

-­‐0.3 2 4 6 8 10 12 0 Distance from Defect (ang)

2 4 6 8 10 12-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Distance from Defect HangL

PotentialDifferenceHevL

POTENTIAL ALIGNMENT - IP

2 4 6 8 10 12 0 Distance from Defect (ang)

Zni

Pi

them on an equal footing with the neutral cell19. Thepotential-alignment correction is:

EPA = q(V rD,q − V rH) (9)

where the charged defect (V rD,q) and host reference (V rH)potentials are atomic sphere-averaged electrostatic po-tentials far from the defect site, q is the charge of thedefect- see Figure 2.

3. Band-filling Correction

The artificially degenerate doping regime can cause de-fects to form bands rather than isolated states withinthe bandgap20,21. For shallow defect states, this incor-rect dispersion can cause abnormally large hybridizationwith the extended band states (either conduction or va-lence) and subsequently partially populate these bands.In order to obtain accurate defect cell energies (ED) wehave to correct for the extra energy in the system due toelectron populations at these higher (or lower) energies.For shallow donors, the correction is:

EBF = −∑n,k

Θ(en,k − eC)(wkfn,ken,k − eC) (10)

Where en,k is the k-dependent energy of state at bandindex n, eC is the conduction band minimum energy ofthe defect-free bulk after potential alignment, fn,k is theband occupation, and wk is the k-point weight. Θ is theHeaviside step function.

B. Band-gap Error

The most common exchange-correlation potentials,LDA and GGA, severely underestimates the bandgap ofmost semiconductors. This has two damaging effects ondefect calculations. The defect-induced states may lie

artificially close in energy to some band states causingexcessive hybridization and ambiguity between shallowor deep defect behaviour. Secondly, the range of electronchemical potentials used to calculate defect formation en-ergies will be too small, possibly incorrectly predictingunstable charged defect states.

Fundamentally, the bandgap error in LDA or GGA isthe result of a lack of continuity with respect to the num-ber of electrons in the exchange-correlation potential22.This in turn leads to self-interaction error (SIE) associ-ated with a bias towards delocalized wavefunctions. Themost common means to correct this situation is to adda Hubbard-like potential to penalize partial state occu-pancies as in GGA+U. Unfortunately, there are manyequivalent ways to apply this correction (ie. which choiceof ’orbitals’ to apply this to) yielding the same bandgap.Since defect levels are sensitive not to the bandgap itself,but to their position relative to the host band states, thisambiguity can result in many different predictions for de-fect ground states5,6,23. Furthermore, corrections for SIEare especially important for charged defect calculationsas reducing interaction error tends to increase the ion-icity of the crystal24 resulting in more ionic relaxationas a defect state is populated and hence greater energybenefit for a charged defect.

Functionals which attempt to correct for SIE have re-cently emerged. One of the most robust and computa-tionally tractable is the HSE functional7. The exchange-correlation potential is divided into short and long-rangecomponents via a screening length parameter. For thelong-range portion, things are unchanged from a typicalGGA calculation. Short-range interactions have a por-tion of exact Hartree-Fock exchange mixed into the ex-change potential, which partially corrects for SIE. Thisapproach has been shown to greatly increase the accu-racy of the bandgap as well as relative band positionsfor a wide variety of materials25. In this work, theamount of short-range HF-exchange mixing is set to 25%which is the amount suggested by the adiabatic connec-tion theorem26. We use a screening length of .1 A−1

instead of the more typical .2 A−1 to account for the lowdielectric strength, and hence poor screening of Zn3P2.This results in close agreement between our calculatedbandgap and experiment (see Table I).

However, the added memory requirements associatedwith the use of the HSE functional for a 2x2x2 supercell ofZn3P2 (320 atoms) make it computationally intractablewith our existing resources (even smaller supercells rep-resent a challenge). Here we have decided to use the’perturbation extrapolation’ method put forth by Lany,et al.9 This is based on the idea of expressing the defect-influenced states in the basis of the states of the perfectbulk (assuming these form a complete basis):

ΨD(r) =∑n,k

An,kΨn,k(r) (11)

If we model the bandgap correction of the HSE func-

5

FIG. 3. Perturbation Extrapolation Workflow - assuming thatthe perfect unit cell GGA forms a complete basis for thesupercell defect wavefunctions, we project the GGA defectstates onto the GGA perfect states and then use the offsetsbetween the GGA and HSE unit cells to extrapolate HSEsupercell behavior.

2/22/12 - 1

Bandgap Error - Workflow

Unit cell HSE

Unit cell GGA

Calculate Band offsets

Defect Supercell GGA

Project States (An,k)

Apply perturbation get new bandstructure

Steven Demers, EAS, Caltech

tional as a perturbation (Hp) of the perfect bulk systemHamiltonian (Hbulk) via a multiplier λ, the band energiesshift:

en,k(λ) = 〈Ψn,k|Hbulk|Ψn,k〉+ λ〈Ψn,k|Hp|Ψn,k〉 (12)

en,k(λ) = en,k(0) + λ∂en,k(λ)

∂λ(13)

If we apply this same perturbation to the defect system(with Hamiltonian Hbulk +HD) then we have:

eD(λ) = 〈ΨD|Hbulk +HD|ΨD〉+ λ〈ΨD|Hp|ΨD〉 (14)

eD(λ) = eD(0) + λ∂eD(λ)

∂λ(15)

Under the assumption of first-order perturbation the-ory and via Eq. (11) we have the final result:

eD(λ) = eD(0) + λ∑n,k

A2n,k

∂en,k(λ)

∂λ(16)

As long as the assumptions of first-order perturbationtheory are justified (ie. unchanged wavefunctions uponapplication of the band-correcting perturbation Hp) wewould predict the defect states to track the correctionsto the perfect bulk states in proportion to the square ofthe coefficients in the expansion (A2

n,k) of Eq. (11). AsHSE doesn’t significantly affect the dispersion of bands somuch as their relative positions to each other, we expectthat this approximation should be justified.

There is still a question as to whether the perfect bulkstates form a reasonably complete basis to describe thedefects. We are helped here by the fact that the staticdielectric strength of Zn3P2 is low (∼ 3) which leads to

FIG. 4. Tetragonal Zn3P2 has a 40-atom unit cell withP42/nmc symmetry.

[010]

[001]

[100]

the tendency of the defects to not be very localized. Thenumber of bands we need to calculate in the expansionof Eq. (11) is thus fairly limited.

The general workflow is shown in Figure 3. Perfectbulk unit cell wavefunctions are determined with GGAand HSE functionals. Then a 2x2x2 defect supercell iscalculated only with the GGA functional. The defectsupercell GGA wavefunctions are then projected onto theunit cell GGA wavefunctions obtaining An,k. Each bandis then offset by the amounts prescribed by comparing theGGA and HSE unit cells. We can then make a predictionfor what the supercell defect bandstructure would be ifwe had been able to utilize the HSE functional.

III. RESULTS

At room temperature and atmospheric pressure Zn3P2

forms a tetragonal phase with a 40-atom unit cell pos-sessing symmetry P42/nmc. It is derived from the cubicfluorite structure with zinc at the center of a distortedphosphorus tetrahedra and the phosphorus surroundedby eight zinc sites lying roughly at the corners of a cube,only six of which are occupied28 - see Figure 4. Thecalculated structural and basic thermodynamic param-eters are summarized in Table I. Results for the latticeconstant and formation enthalpy are in good agreementwith experiment with both GGA and HSE functionals.However, the bandgap is severely underestimated withGGA, which at 0.32eV is only about 1/5 of the exper-imental result. Using the HSE functional, we find the

TABLE I. Calculated lattice constants, bandgap, and heatof formation of tetragonal Zn3P2 using both GGA-PBE andHSE functionals (using .1 A−1 screening length and 25% HF-exchange mixing).

functional a (angs) c/a Eg (eV) ∆Hf (eV)

GGA-PBE 8.108 1.408 .32 -1.79

HSE 8.160 1.390 1.42 -1.32

Experiment1,27 8.097 1.286 1.49 -1.53

6

FIG. 5. DFT Bandstructure - calculate with HSE functional.Partial density of states shows makeup of VBM mainly ofphosphorus p-character, while the CBM is mixed p and s-character from phosphorus and zinc states.

G A M X G R Z G

-5

0

5

EnergyHevL

p s

d

bandgap to within 5% of experiment, highlighting thevalue of using HSE for this system.

In order to accurately describe the defect levels weneed to determine the effect of the HSE functional onthe bandstructure (see Figure 5) relative to GGA. Cal-culating the band shifts from GGA to HSE requires morethan simple bulk calculations, as the band energies givenby DFT are only referenced to the average electrostaticpotential of the simulation cell. For a periodically re-peated solid this is an ill-defined quantity which makesit impossible to directly compare band energies betweencells with different constituents or for calculations per-formed with different functionals. However, if we have aregion of equivalent potential (ie. vacuum) in both GGAand HSE systems we can compare the bulk potentialsin each system to this region, and subsequently computethe HSE and GGA band positions relative to each other.Care must be taken that the solid and vacuum regionsare large enough that the electron wavefunctions becomenegligible in the vacuum and the effects of the surfacestates are not felt in the bulk. In this work, we use a 4unit-cell slab along the non-polar [100] direction and anequal amount of vacuum (resulting in more than 35 Aof vacuum) - see Figure 6. Non-polar, non-reconstructedsurfaces were used to avoid creating surface dipoles withthe accompanying undesirable step in potential acrossthe solid-vacuum interface.

The alignment of GGA and HSE bandstructures isshown in Figure 7. The lowering of the VBM with HSEis due to the reduced self-interaction of the phosphorusp-orbitals which primarily form the highest-energy va-lence bands. The upward shift of the CBM is due tothe reduced hybridization of the Zn-s and P-s orbitalsthat make up the lower part of the conduction band withthe valence band states. With this alignment calculatedwe can compute the GGA-to-HSE band offsets neededfor the ’perturbation extrapolation’ method discussed inSection II B.

We are now in the position to apply all the correctionsdetailed in Section II. We turn our attention to the pre-

FIG. 6. Slab Geometry for band alignment determination.Red regions show areas where the potential was integrated inthe bulk and vacuum regions.

0 100 200 300 400 500

-10

-5

0

5

Distance HangL

Electrostatic

PotentialHevL

0 12 24 36 48 60 Distance (ang)

-­‐10

-­‐5

0

5

Unit cell

Vacuum Region

FIG. 7. GGA to HSE Band Alignment. Bandgaps for GGA(left) and HSE (right) are labeled at the gamma point. Cal-culated offsets for the VBM and CBM are shown.

ΔEC=.60 ev

ΔEV=-­‐.50 ev

EGAP=.32 ev EGAP=1.42 ev

dicted stability and doping effects of the various intrin-sic defects. Here it is useful to make a few notes aboutthe following discussion; first, we may find that a defectlevel lies outside of the bandgap region predicting thatthis charged state is unstable as a defect localized state.However, it may still be possible to bind electrons to thedefect through electrostatic interation and form hydro-genic effective-mass states just within the band edge. Adiscussion of these states is not in the scope of this work.Furthermore, we ignore the effects of both formation vol-ume and formation entropy in computing the defect for-mation energies. Formation volume is directly related tothe change in volume when a defect is created, but is typ-ically only important for degenerate defect regime or forvery high pressures. Formation entropy for point defectsis typically on the order of a few kB so are only importantfor very high temperatures - additionally, the entropy ofpoint defects in the same system tend to be similar andso largely cancel when we compare the likelihood of onedefect over another.

7

A. Zinc Vacancy

Zinc is surrounded by four nearly equidistant, tetrag-onally coordinated phosphorus atoms. Removing a zincatom from the lattice leaves four dangling bonds from theneighboring phosphorus atoms with mainly p-character -see Figure 8. These bonds are occupied with only six elec-trons, forming three low energy and one empty higher-energy defect state. This empty bonding state can cap-ture electrons and form an acceptor defect with a -1 or -2charge. As the defect states have a strong valence bandcharacter (e.g. in terms of symmetry), we would expectshallow defect behavior.

It is instructive to study the structural relaxationsaround the VZn site as this is intimately related to theoccupancy and energetics of the defect state. Removing azinc atom to form a neutral defect causes the four neigh-bouring phosphorus atoms to relax away from the defectsite as they seek to maximize their bond overlap with theremaining zinc atoms in the lattice. However, this move-ment also shifts the dangling bonds in the vacancy regionhigher in energy, fighting this relaxation. The phospho-rus atom closest to the defect site (the P-atom to the’north’ of the vacancy site in Figure 8) has more of itsdensity in the vacancy region and subsequently relaxesthe least. As the defect becomes occupied (as in the -1or -2 charged states) the electrons are chiefly populatingaround the phosphorus ions which cause a further relax-ation away from the defect due to increased coulombicrepulsion. As the relaxation is relatively slight in all de-fect charge states, we would expect that the defect itselfwould be delocalized. This is born out as the charge den-sity is poorly screened and well dispersed throughout thelattice.

Getting the correct energetics and occupancy of thedefect states is primarily important for their effect onthe formation energy of the defects, which is our centralgoal. Looking at the formation energy plots in Figure 9,zinc vacancies exist as charged defects for all but veryp-type regimes (fermi level close to VBM). The smallenergy difference between the different charged statesfor a fermi level at the VBM can be expected from thesmall difference in lattice relaxations associated with thecharged states of VZn. The formation energies in then-type regime are low enough in the Zn-deficient growthenvironment for this to be an important defect.

B. Phosphorus Vacancy

Phosphorus is surrounded by six nearest neighbor zincsites, four being almost equidistant and two zinc atomsabout 15% further away. Phosphorus normally exists ina -3 oxidation state and when we remove a phosphorusatom the electrons from the surrounding zinc atoms haveonly high energy bonding states to move into with mainlyZn-p and Zn-d character - see Figure 10. We would ex-

FIG. 8. VZn Electronic Structure - Partial Density of Statesof the neutral defect for the four nearest neighboring phospho-rus atoms. Defect states highlighted in yellow, show mainlyp-character. Bond lengths are given relative to distance todefect site in perfect bulk cell.

1.012

1.036

1.034

1.036

Energy(ev) -­‐15 -­‐10 -­‐5 0 5

0.4

0

0.2

P_p P_s

VZn0 VZn

-­‐2

0.997

1.053

1.035

1.053

[100]

[010]

FIG. 9. VZn Formation Energy. Acceptor-type defect, plottedfor the 0,-1,-2 charged states as the fermi level varies from thetop of the VBM (p-type) to the bottom of the CBM (n-type).Favorable growth condition on the left.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-1

0

1

2

3

4

5

Fermi LevelHevL

FormationEnergyHevL

Zinc Vacancy - Zn deficient

Fermi Level (ev)

0.0 1.5 .75 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-1

0

1

2

3

4

5

Fermi LevelHevLFormationEnergyHevL

Zinc Vacancy - Zn deficient

Fermi Level (ev)

0.0 1.5 .75

Zn Deficient Zn Rich

0

1

2

3

4

5

0

1

2

3

4

5

VZn0

VZn-­‐1

VZn-­‐2

VZn-­‐2

VZn-­‐1

VZn0

n-­‐type p-­‐type n-­‐type p-­‐type

pect that these states would want to depopulate andform +1 and +2 charged defects as these electrons areno longer needed for bonding a phosphorus atom to thelattice. The symmetry of the defect states is not similarto the conduction band symmetry so we would expectdeeper defect behavior than for VZn.

Removing a phosphorus atom results in a defect statethat is effectively screened by the neighboring zinc atomsand highly localized between the nearest zinc sites. Thehigh degree of localization results in a large lattice re-laxation, especially for the two zinc atoms closest to thedefect site. Since the majority of the electron density isconcentrated in the defect region, the four closest zincatoms relax closer to each other in order to lower the en-ergy of the occupied defect localized states. The two fur-thest zinc atoms relax away from the defect region due toincreased coulombic repulsion from the similarly chargedzinc atoms closer to the defect. As the defect becomesdepopulated there is a slight outward relaxation since thebenefit in lowering the energy of the defect states is re-

8

FIG. 10. VP Electronic Structure - Partial Density of States ofthe +2 charged defect for the nearest neighboring zinc atoms.Defect states highlighted in yellow, show mainly Zn-p andZn-d character. Bond lengths are given relative to distanceto defect site in perfect bulk cell.

0.807

0.963 0.963

1.085

Energy(ev) -­‐15 -­‐10 -­‐5 0 5

0.4

0

0.2

Zn_p Zn_s

VP0 1.085

0.807

0.807

0.969 0.969

1.089

VP+2

1.089

0.805

Zn_d

[100]

[010]

[001]

FIG. 11. VP Formation Energy. Donor-type defect, plottedfor the 0,+1,+2 charged states as the fermi level varies fromthe top of the VBM (p-type) to the bottom of the CBM (n-type). Favorable growth condition on the left.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0

2

4

6

8

Fermi LevelHevL

FormationEnergyHevL

Phosphorus Vacancy - P deficient

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0

2

4

6

8

Fermi LevelHevL

FormationEnergyHevL

Phosphorus Vacancy - P deficient

Fermi Level (ev)

0.0 1.5 .75 0.0 1.5

Fermi Level (ev)

.75

P Deficient P Rich

0

2

4

6

8

VP0

VP+1

VP+2

VP+2

VP+1

VP0

n-­‐type p-­‐type n-­‐type p-­‐type

0

2

4

6

8

duced and the zinc atoms want to increase their bondingwith the rest of the lattice. The high degree of localiza-tion would also suggest deep defect behavior.

Looking at the formation energies in Figure 11, wesee deep defect behavior where the defect only becomescharged for fermi levels in the neutral to p-type regime.Even for favorable growth conditions these defects aretoo high in energy to play a significant role in thedopability of zinc phosphide.

C. Zinc Interstitial

For both zinc and phosphorus interstitials we voxelizedthe Zn3P2 unit cell and tested the sites in order of fur-thest distance from neighboring atoms. Zinc interstitialsare predicted to form in the voids on the zinc plane of

FIG. 12. Zni Electronic Structure - Partial Density of Statesof the -2 charged defect for the interstitial Zn site. De-fect states highlighted in yellow, show mainly Zn-s character.Bond lengths are given relative to distance to defect site inperfect bulk cell.

Energy(ev) -­‐15 -­‐10 -­‐5 0 5

0.4

0

0.2 Zn_p Zn_s

1.073

VP+2

0.974

1.048

1.083

0.998

1.060

1.009

1.079

0.973

1.054

1.083

0.995

1.057

1.008

Zni0 Zni+2

Zn_d

[010]

[001]

atoms tetrahedrally coordinated with four phosphorusatoms and in-line with zinc atoms in adjacent planes.Incorporating an extra zinc atom into the lattice forms alocalized defect state of mainly ionic Zn-s character sincethe neighboring phosphorus atoms have closed-shell con-figurations. As for zinc atoms in the perfect bulk, theinterstitial zinc has a tendency to depopulate the s-shellstates. We would expect donor defect behavior with a+1 or +2 charged state and this is what our calculationsshow. The symmetry of the defect states are similar tothe conduction band character, consequently we wouldexpect shallow behavior.

The neighboring zinc atoms are affected the most bythe incorporation of the Zni defect as they relax awayfrom the similarly charged interstitial ion. The phospho-rus atoms relax closer to the interstitial site in generalas there has been a partial charge transfer from the Znidefect. There is only minor differences in relaxation be-tween the various charged states and the defect is poorlyscreened with a delocalized wavefunction centered on themajority of the zinc atoms in the lattice. All of whichwould suggest shallow defect formation.

In Figure 13, the formation energy plots show shallowbehavior where the defect becomes charged for all butfermi levels high in the n-type regime. As the fermi leveldrops to the VBM the formation energy of the defectsbecomes very small for the favorable growth conditions(Zn-rich regime). These defects should be important toconsider.

D. Phosphorus Interstitial

The most energetically favorable position for the phos-phorus interstitials are in the voids in the zinc plane ofatoms nearly equidistant from three zinc atoms and di-rectly above a phosphorus atom displaced from the planebelow. The additional electrons form states of mainly

9

FIG. 13. Zni Formation Energy. Donor-type defect, plottedfor the 0,+1,+2 charged states as the fermi level varies fromthe top of the VBM (p-type) to the bottom of the CBM (n-type). Favorable growth condition on the left.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-1

0

1

2

3

4

5

Fermi LevelHevL

FormationEnergyHevL

Zinc Interstitial - Zn rich

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-1

0

1

2

3

4

5

Fermi LevelHevL

FormationEnergyHevL

Zinc Interstitial - Zn rich

Fermi Level (ev)

0.0 1.5 .75

Fermi Level (ev)

0.0 1.5 .75

Zn Rich Zn Deficient

0

1

2

3

4

5

0

1

2

3

4

5

Zni0 Zni+1

Zni+2 Zni+2

Zni+1 Zni0

n-­‐type p-­‐type n-­‐type p-­‐type

covalent p-character between the Pi site and neighbor-ing zinc and phosphorus atoms - see Figure 14. Due tothe natural oxidation state of phosphorus (-3) we expectthat the defect should be an acceptor type, which is con-firmed by our calculations. Since the symmetry of thedefect states is similar to the VBM, we would predictshallow defect behavior.

The neutral defect state is highly localized, so Pi causesrelatively large ionic relaxation away from the defect site,though for the two zinc atoms most involved with thecovalent bonding this relaxation is reduced because thismovement also increases the energy of the populated de-fect states. The charged state is poorly screened andvery delocalized across the lattice. Consequently, there isvery little relaxation as the defect captures electrons fromthe conduction band. Apart from energetics, the severedelocalization of the charged state and good symmetrymatch between the defect and valence states causes al-most band-like behavior where we should readily captureelectrons from the conduction band.

The neutral defect has relatively low formation energyfor P-rich growth conditions. Due to the small differ-ences in relaxation between the charged defect states,there is only a small difference in formation energybetween them at the VBM. Thus, for fermi levels evenmodestly into the n-type regime the formation energyof the charged defects becomes very small making thesecritical defects to consider.

E. CONCLUSION

From our calculations, the likely candidate for the lackof n-type dopability is the Pi defects. Both of the accep-tor defects (VZn and Pi) have low formation energies aswe move the system into the n-type regime. While there

FIG. 14. Pi Electronic Structure - Partial Density of States ofthe neutral defect for the interstitial phosphorus site. Defectstates highlighted in yellow, show mainly P-p character. Bondlengths are given relative to distance to defect site in perfectbulk cell.

Energy(ev) -­‐15 -­‐10 -­‐5 0 5

0.4

0

0.2 Zn_p Zn_s

1.160

VP+2

Zn_d

Pi0 Pi-­‐2 1.128

1.996

1.109

1.159

1.128

1.998

1.109

[100]

[010] [001]

P_p P_s

FIG. 15. Pi Formation Energy. Acceptor-type defect, plottedfor the 0,-1,-2 charged states as the fermi level varies from thetop of the VBM (p-type) to the bottom of the CBM (n-type).Favorable growth condition on the left.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-1

0

1

2

3

4

5

Fermi LevelHevL

FormationEnergyHevL

Phosphorus Interstitial - P rich

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-1

0

1

2

3

4

5

Fermi LevelHevL

FormationEnergyHevL

Phosphorus Interstitial - P rich

Fermi Level (ev)

0.0 1.5 .75

Fermi Level (ev)

0.0 1.5 .75

P Rich P Deficient

0

1

2

3

4

5

0

1

2

3

4

5

Pi0 Pi-­‐1

Pi-­‐2

Pi-­‐2

Pi-­‐1 Pi0

n-­‐type p-­‐type n-­‐type p-­‐type

are more suitable VZn sites, the Pi defect are significantlyless costly and should be the vastly more prevalent defect.The Zni defects would tend to aid n-type doping as theyare electron donors, though they have too high a forma-tion energy for fermi levels even moderately n-typed tosignificantly compensate for the Pi defects.

Thus, as we dope our system with electrons we createa large amount of acceptor defects which act as ’electron-sinks’ and capture mobile electrons from the conductionband and neutralize the doping. In fact, the Pi defectrequires zero formation energy for fermi-levels midwaytowards the conduction band. This would ’pin’ the fermi-level - that is, as we try to approach this level of doping amassive amount of Pi defects would form in the crystal.Since the creation of Pi defects fight n-type doping, reach-ing this level would not be expected. There is some hopein that we would anticipate the Pi defects to repel eachother (especially the highly charged state), thus limiting

10

their concentration, and some n-type doping would thensurvive. Indeed, initial results show a significant penaltyin having two fully charged Pi defects within the same2x2x2 supercell. However, this still corresponds to a largedefect concentration and there is low likelihood that thedesireable properties of Zn3P2 such as high minority car-rier diffusion lengths would survive in the regime wherePi defects saturate.

Solving for the fermi level self-consistently using theabove formation energy functions of the various defects,we predict an intrinsic fermi-level of 0.55 eV, which ismildly p-type. Pushing the fermi level beyond roughly1.05 eV becomes impossible as the number of defects ex-ceeds the numer of sites in the cell at this point.

Since the Pi defects are the problematic defects, anymeans to suppress them should help the n-type dopingissue. For extreme Zn-rich growth conditions, where theacceptor Pi defects are suppressed and the donor Zni areenhanced there may be some hope of weakly n-type ma-

terials being formed. A suppression of interstitial defectsin general, such as straining the crystal as it is grownmay prove fruitful. Cluster doping29 donor atoms withones that suppress Pi formation would be another avenueto explore.

ACKNOWLEDGMENTS

Thanks to Qijun Hong for help in processing VASPPAW wavefunction files. Fruitful discussions with Prof.Harry Atwater, Jeff Bosco, and Greg Kimball are grate-fully acknowledged. This work made use of the Ter-aGrid/Xsede computing resources at the University ofTexas at Austin. This material is based upon work sup-ported by the Department of Energy National NuclearSecurity Administration under Award No. DE-FC52-08NA28613 and by the National Science Foundation un-der Grant No. DMR-0907669.

∗ steved@caltech.edu1 G. M. Kimball, A. M. Muller, N. S. Lewis, and H. A.

Atwater, Applied Physics Letters 95, 112103 (2009).2 F.-C. Wang, A. L. Fahrenbruch, and R. H. Bube, Journal

of Applied Physics 95, 112103 (1982).3 M. Bhushan and A. Catalano, Appl Phys Lett 38, 39

(1981).4 W. Zdanowicz and Z. Henkie, Bull. Acad. Polon. Sci. Ser.

Sci. Chim. 12, 729 (1964).5 S. Lany and A. Zunger, Physical Review Letters 98, 045501

(2007).6 A. Janotti and C. G. V. de Walle, Phys Rev B 76, 165202

(2007).7 J. Heyd, G. E. Scuseria, and M. Ernzerhof, The Journal

of Chemical Physics 118, 8207 (2003).8 L. Hedin, Phys. Rev. 139, A796 (1965).9 S. Lany and A. Zunger, Physical Review B 78, 235104

(2008).10 C. Persson, Y.-J. Zhao, S. Lany, and A. Zunger, Physical

Review B 72, 035211 (2005).11 S. Lany and A. Zunger, Model Simul Mater Sc 17, 084002

(2009).12 S. Lany, Phys Rev B 78, 245207 (2008).13 P. E. Blochl, Phys. Rev. B 50, 17953 (1994).14 G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).15 G. Kresse and J. Furthmller, Computational Materials Sci-

ence 6, 15 (1996).16 J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.

Lett. 77, 3865 (1996).17 G. Makov and M. Payne, Physical Review B 51, 4014

(1995).18 J. IHM, A. Zunger, and M. COHEN, J Phys C Solid State

12, 4409 (1979).19 S. Lany and A. Zunger, J Appl Phys 100, 113725 (2006).20 T. S. Moss, Proc. Phys. Soc. London 67, 775 (1954).21 E. Burstein, Phys. Rev. 93, 632 (1954).22 J. P. Perdew and M. Levy, Physical Review Letters 51,

1884 (1983).23 A. Janotti, J. B. Varley, P. Rinke, N. Umezawa, G. Kresse,

and C. G. V. de Walle, Physical Review B 81, 085212(2010).

24 P. Agoston and K. Albe, Physical Review Letters 103,245501 (2009).

25 J. Heyd, J. Peralta, G. Scuseria, and R. Martin, J ChemPhys 123, 174101 (2005).

26 J. P. Perdew, M. Ernzerhof, and K. Burke, 00219606 105,9982 (1996).

27 von Stackelberg M and P. R, Phys. Chem. B 28, 427(1935).

28 A. Catalano and R. Hall, J. Phys. Chem. Solids 41, 635(1979).

29 L. Wang and A. Zunger, Phys Rev B 68, 125211 (2003).