Positrons in Homogeneous Electron Gases

Neil D. Drummond∗, Chris J. Pickard† and Richard J. Needs∗

∗ TCM Group, Cavendish Laboratory, University of Cambridge

† University of St. Andrews

QMC in the Apuan Alps II, TTI, Vallico Sotto, Italy

23rd July, 2006

Positrons

• Positrons are anti-electrons (solutions of Dirac equation).

• Positrons are produced in β+ decays of proton-rich nuclei,e.g. 25

13Al12 → 2512Mg13 + β+ + ν.

• A positron may bind with an electron to form a positronium atom.

• Ground-state energy of positronium is −1/4 a.u. (Like hydrogen atom, but reducedmass of electron is 1/2.)

• Annihilation of a parallel-spin electron-positron pair is a 3rd order process in quantumelectrodynamics, producing 3 photons.

• Annihilation of an antiparallel-spin electron-positron pair is a 2nd order process,producing 2 photons.

• We consider only 2-photon annihilation events (experimentally relevant).

• Two-photon annihilation cross-section is σ = π/(vc3), where v is the positron velocityand c is the speed of light (in a.u.).1

• Positron annihilation is widely used to study material properties. Usual source ofpositrons in experiments: 22

11Na11 → 2210Ne12 + β+ + ν + γ.

1P. A. M. Dirac, Proc. Cam. Phil. Soc. 26, 361 (1930).

Positron Lifetime Spectroscopy (POLIS)

• Suppose a positron is injected into a sample of material.

• Positron rapidly thermalises and diffuses through material, before ending up in itsground state. (Often settles in negatively charged defects.)

• Positron remains in ground state for some time before annihilating an electron.

• Measure time difference between positron birth (one 1.274 MeV photon emitted) andannihilation (two 0.511 MeV photons emitted).

• Annihilation rate is characteristic of the defects at which positrons settle.

• Sensitive, nondestructive technique allowing simultaneous measurement of type andquantity of defects in metals and semiconductors.

Sensitivity of different experimental methods to defect concentration and size as afunction of depth. Green box indicates depth, size and concentration relevent to studies

of defects in electronic interconnects on semiconductor chips.

Annihilating-Pair Momentum Spectroscopies

• Conservation of momentum: total momentum of γ rays equals momentum of electronwith which positron annihilates.

• Measure distribution of momenta of annihilation radiation to find out distribution ofelectron momenta.

• Angular correlation of annihilation radiation (ACAR) spectroscopy and Doppler-broadening spectroscopy (DOBS) are powerful methods for identifying defects andmeasuring Fermi surfaces.

POLIS, DOBS and ACAR spectroscopy.

Magnetically confined positron beam for DOBS experiments at the University of Bath

Applications of Positron Annihilation Spectroscopy

Recent PAS studies at the University of Bath include:

• Surface modification of polymer films by laser or plasma treatment;

• Defects caused by the implantation of Ge ions into SiC;

• Transition region between SiO2 and Si;

• Fluorine diffusion and agglomeration in Si;

• Interfaces between nanocrystals of Si and a silica matrix;

• Defects in ferroelectric films.

Challenges for Theory

• Positron modifies electron charge density and momentum distribution.

• Annihilation rate by a naıve calculation from cross-section:

λ = πc−3n,

where c is speed of light and n is electron number density.

• Actual annihilation rate is higher, because positron attracts electrons to it: contactdensity enhancement.

• Positron causes increase in momentum distribution near Fermi edge: Kahanaenhancement.2

2S. Kahana, Phys. Rev. 129, 1622 (1963).

Positron Immersed in a Homogeneous Electron Gas

• Want to calculate annihilation rates and annihilating-pair momentum distributionsfor positrons in real materials.

• First step: calculate annihilation rate and momentum distribution for a positron in ahomogeneous electron gas.

• Use energy data to construct electron-positron correlation functionals, enabling DFTsimulations of positrons in real materials.

Electron-Positron Hamiltonian

• Hamiltonian for positron in homogeneous electron gas:

H =∑

i

−12∂2

∂r2i

− 12∂2

∂s2−

∑

i

1|ri − s| +

∑

i

∑

j>i

1|ri − rj|.

• Make coordinate transformation suggested by Leung et al.3

X =1

N + 1

(s +

N∑

i=1

ri

)

xi = ri − s.

• Then Hamiltonian is

H =−1

2(N + 1)∂2

∂X2−

∑

i

(∂2

∂x2i

+1|xi|

)+

∑

i

∑

j>i

(− ∂2

∂xi∂xj+

1|xi − xj|

).

3C. H. Leung, et al., Phys. Lett. 57A, 26 (1976).

• First term on RHS is CoM KE operator. May neglect in ground state.

• Left with Hamiltonian for N interacting particles of mass 1/2 a.u. and charge −1 a.u.in the presence of a fixed positive charge of magnitude 1 a.u. at the origin.

• There is an additional interaction, resembling mass-polarisation. This was neglectedby Leung et al., who argued that it is small.

Additional Interaction

• Suppose Ψ is a product of determinants of orthonormal orbitals ψi for spin-up andspin-down electrons. Then

⟨Ψ

∣∣∣∣∣∣

N−1∑

i=1

N∑

j=i+1

∇i · ∇j Ψ

⟩=

12

N∑

i=1

N∑

j=1

(〈ψi|∇ψi〉 · 〈ψj|∇ψj〉

− δmsi,msj〈ψi|∇ψj〉 · 〈ψj|∇ψi〉

),

where msiis the spin of particle i.

• First term vanishes in a closed-shell ground state. We consider only closed-shellground states.

• Require expectation of transformed Hamiltonian to be stationary with respect tovariations in each ψi; obtain one-electron DFT equations:

[−∇2 + VH(x) + Vext(x) + Vxc(x)]ψi(x)

+N∑

j=1

fjδmsi,msj〈ψj|∇ψi〉 · ∇ψj(x) = εiψi(x),

where VH, Vext and Vxc are the Hartree, external and exchange-correlation potentials,and fj and εj are occupation no. and eigenvalue of state j.

• Equations are solved by a modified version of the CASTEP plane-wave DFT code.4

4M. D. Segall et al., J. Phys. Cond. Matt. 14, 2717 (2002).

Positron Pseudopotential

• We use an ultrasoft positron pseudopotential.

• All-electron orbitals recovered using projector augmented-wave method.

• Have checked that results are converged w.r.t. cutoff radius, number of projectors,plane-wave cutoff energy, etc.

• We have used the bare Coulomb potential in some of our calculations.

• Bare Coulomb and pseudopotential calculations are in agreement.

Overlap Integral Theorem for Determinants

Theorem5: Let

Ψ(R) =

∣∣∣∣∣∣

ψ1(r1) · · · ψN(r1)... ...

ψ1(rN) · · · ψN(rN)

∣∣∣∣∣∣and

Φ(R) =

∣∣∣∣∣∣

φ1(r1) · · · φN(r1)... ...

φ1(rN) · · · φN(rN)

∣∣∣∣∣∣.

Then

〈Ψ|Φ〉 = N !

∣∣∣∣∣∣

〈ψ1|φ1〉 · · · 〈ψ1|φN〉... ...

〈ψN |φ1〉 · · · 〈ψN |φN〉

∣∣∣∣∣∣.

5P. O. Lowdin, Phys. Rev. 97, 1474 (1955).

Annihilating-Pair Momentum Distribution

• CoM and difference coordinates: ri ≡ (ri + s)/2 and δri ≡ ri − s.

• Positron-electron 1 centre-of-mass momentum wave function:

Ψ(p1, δr1; r2, . . . , rN) =1V

∫exp(−ip1 · r1)Ψ(r1, δr1; r2, . . . , rN) dr1,

• Assumption: distribution of annihilating-pair momenta same as distribution of CoMmomenta when positron coincides with an electron of opposite spin.

• Unnormalised distribution of the CoM momentum for positron annihilating withelectron 1: ∫

· · ·∫|Ψ(p1,0; r2, . . . , rN)|2 dr2 . . . drN .

• Normalise and use antisymmetry of wave function to get momentum distribution:

ρ↑(p) =∫ · · · ∫ |Ψ(p,0; r2, . . . , rN)|2 dr2 . . . drN∑p

∫ · · · ∫ |Ψ(p,0; r2, . . . , rN)|2 dr2 . . . drN

=

∫ · · · ∫ ∣∣∫ exp(−ip · r1)Ψ(r1; r1, . . . , rN) dr1

∣∣2 dr2 . . . drN

V∫ · · · ∫ |Ψ(r1; r1, . . . , rN)|2 dr1 . . . drN

.

• Now suppose

Ψ(s; r1, . . . , rN) =1√

N↑!N↓!

∣∣∣∣∣∣∣

φ↑1(r1 − s) · · · φ↑N↑(r1 − s)... ...

φ↑1(rN↑ − s) · · · φ↑N↑(rN↑ − s)

∣∣∣∣∣∣∣

×

∣∣∣∣∣∣∣

φ↓1(rN↑+1 − s) · · · φ↓N↓(rN↑+1 − s)... ...

φ↓1(rN − s) · · · φ↓N↓(rN − s)

∣∣∣∣∣∣∣.

• Then numerator of momentum distribution is

ρ↑u(p) =1

N↑!N↓!

∫ ∫exp (ip · (r′1 − r1))

×∫· · ·

∫∣∣∣∣∣∣∣∣∣∣

φ↑∗1 (0) · · · φ↑∗N↑(0)

φ↑∗1 (r2′′ − r′1) · · · φ↑∗N↑(r

′′2 − r′1)

... ...

φ↑∗1 (r′′N↑ − r′1) · · · φ↑∗N↑(r′′N↑ − r′1)

∣∣∣∣∣∣∣∣∣∣

×

∣∣∣∣∣∣∣∣∣∣

φ↑1(0) · · · φ↑N↑(0)

φ↑1(r2′′ − r1) · · · φ↑N↑(r

′′2 − r1)

... ...

φ↑1(r′′N↑ − r1) · · · φ↑N↑(r

′′N↑ − r1)

∣∣∣∣∣∣∣∣∣∣

dr′′2 . . . dr′′N↑

×∫· · ·

∫∣∣∣∣∣∣∣

φ↓∗1 (r′′N↑+1 − r′1) · · · φ↓∗N↓(r′′N↑+1 − r′1)

... ...

φ↓∗1 (r′′N − r′1) · · · φ↓∗N↓(r′′N − r′1)

∣∣∣∣∣∣∣

×

∣∣∣∣∣∣∣

φ↓1(r′′N↑+1 − r1) · · · φ↓N↓(r

′′N↑+1 − r1)

... ...

φ↓1(r′′N − r1) · · · φ↓N↓(r

′′N − r1)

∣∣∣∣∣∣∣dr′′N↑+1 . . . dr

′′N dr′1 dr1

=∫e−ip·R

∫· · ·

∫ N↑∑

i=1

N↑∑

j=1

(−1)i+jφ↑∗i (0)φ↑j(0)M1iN1j dr2 . . . drN↑

× V

N↑!

∣∣∣∣∣∣∣

∫φ↓∗1 (r)φ↓1(r−R) dr · · · ∫

φ↓∗N↓(r)φ↓1(r−R) dr

... ...∫φ↓∗1 (r)φ↓N↓(r−R) dr · · · ∫

φ↓∗N↓(r)φ↓N↓(r−R) dr

∣∣∣∣∣∣∣dR,

• We have: (i) substituted R = r1− r′1; (ii) substituted ri = r′′i − r′1 ∀i ∈ 2, . . . , N,allowing us to perform the integral over r′1; (iii) made use of the overlap integraltheorem; (iv) defined Mij to be the (i, j)th minor of

∣∣∣∣∣∣∣∣∣∣

φ↑∗1 (0) · · · φ↑∗N↑(0)

φ↑∗1 (r2) · · · φ↑∗N↑(r2)... ...

φ↑∗1 (rN↑) · · · φ↑∗N↑(rN↑)

∣∣∣∣∣∣∣∣∣∣

;

and (v) defined Nij to be the (i, j)th minor of

∣∣∣∣∣∣∣∣∣∣

φ↑1(0) · · · φ↑N↑(0)

φ↑1(r2 −R) · · · φ↑N↑(r2 −R)... ...

φ↑1(rN↑ −R) · · · φ↑N↑(rN↑ −R)

∣∣∣∣∣∣∣∣∣∣

.

• For each i, j ∈ 1, . . . , N↑, we can use the overlap integral theorem to determine a(N↑ − 1)× (N↑ − 1) matrix BR(i, j) such that

det(BR(i, j)

)=

1(N↑ − 1)!

∫· · ·

∫M1iN1j dr2 . . . drN↑.

• So unnormalised annihilating-pair momentum distribution is

ρ↑u(p) =∫

exp(−ip ·R)N↑∑

i=1

N↑∑

j=1

(−1)i+jφ↑∗i (0)φ↑j(0) det(BR(i, j)

)

×

∣∣∣∣∣∣∣

∫φ↓∗1 (r)φ↓1(r−R) dr · · · ∫

φ↓∗N↓(r)φ↓1(r−R) dr

... ...∫φ↓∗1 (r)φ↓N↓(r−R) dr · · · ∫

φ↓∗N↓(r)φ↓N↓(r−R) dr

∣∣∣∣∣∣∣dR.

Electron-Positron Pair-Correlation Function

• Spin-down positron–spin-up electron pair-correlation function:

g↑(r, s) =ρ↑1p(r, s)

ρ↑1(r)ρp(s)= V 2

∫ · · · ∫ |Ψ(s; r, r2, . . . , rN)|2 dr2 . . . drN∫ · · · ∫ ∫ |Ψ(s; r1, . . . , rN)|2 dr1 . . . drN ds.

• Denominator:

V

∣∣∣∣∣∣∣

〈φ↑1|φ↑1〉 · · · 〈φ↑1|φ↑N↑〉... ...

〈φ↑N↑|φ↑1〉 · · · 〈φ↑N↑|φ

↑N↑〉

∣∣∣∣∣∣∣×

∣∣∣∣∣∣∣

〈φ↓1|φ↓1〉 · · · 〈φ↓1|φ↓N↓〉... ...

〈φ↓N↓|φ↓1〉 · · · 〈φ↓N↓|φ

↓N↓〉

∣∣∣∣∣∣∣.

• Numerator:

V 2

N↑!

∫· · ·

∫∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣

φ↑1(r− s) · · · φ↑N↑(r− s)

φ↑1(r2) · · · φ↑N↑(r2)... ...

φ↑1(rN↑) · · · φ↑N↑(rN↑)

∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣

2

dr2 . . . drN↑

×

∣∣∣∣∣∣∣

〈φ↓1|φ↓1〉 · · · 〈φ↓1|φ↓N↓〉... ...

〈φ↓N↓|φ↓1〉 · · · 〈φ↓N↓|φ

↓N↓〉

∣∣∣∣∣∣∣.

• But

∫· · ·

∫∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣

φ↑1(r− s) · · · φ↑N↑(r− s)

φ↑1(r2) · · · φ↑N↑(r2)... ...

φ↑1(rN↑) · · · φ↑N↑(rN↑)

∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣

2

dr2 . . . drN↑

=N↑∑

i=1

N↑∑

j=1

(−1)i+jφ↑∗i (r− s)φ↑j(r− s)∫· · ·

∫N∗

1iN1j dr2 . . . drN

= (N↑ − 1)!N↑∑

i=1

N↑∑

j=1

(−1)i+jφ↑∗i (r− s)φ↑j(r− s) det(B(i, j)),

where Nij is the (i, j)th minor of

∣∣∣∣∣∣∣∣∣∣

φ↑1(r− s) · · · φ↑N↑(r− s)

φ↑1(r2) · · · φ↑N↑(r2)... ...

φ↑1(rN↑) · · · φ↑N↑(rN↑)

∣∣∣∣∣∣∣∣∣∣

and the overlap integral theorem is used to define the matrix B(i, j) for each i and j.

• So

g↑(r− s) =V

∑N↑i=1

∑N↑j=1(−1)i+jφ↑∗i (r− s)φ↑j(r− s) det(B(i, j))

N↑

∣∣∣∣∣∣∣

〈φ↑1|φ↑1〉 · · · 〈φ↑1|φ↑N↑〉... ...

〈φ↑N↑|φ↑1〉 · · · 〈φ↑N↑|φ

↑N↑〉

∣∣∣∣∣∣∣

.

• Have written a code to evaluate the momentum distribution and PCF.

• NB, augmented orbitals (pseudo-orbitals in plane waves plus augmentation functionson radial grid) must be used.

Contact Density Enhancement and Annihilation Rate

• Probability density that a spin-up electron coincides with the positron:

n↑eff =N↑Vg↑(0).

• This contact density should be used to calculate annihilation rate using the 2-photonannihilation cross-section.

• Annihilation rate for spin-up electrons with a spin-down positron:

λ =3g(0)4c3r3s

.

Immersion and Relaxation Energies

• Let E(N,M, V ) be the energy of a uniform plasma of N electrons and M positronsin fixed volume V .

• Immersion energy of a positron:

∆E ≡ E(N + 1, 1, V )− E(N, 0, V )− E(1, 1, V ).

• Relaxation energy of a positron:

∆Ω ≡ E(N + 1, 1, V )− E(N + 1, 0, V )

= ∆E + E(N, 0, V )− E(N + 1, 0, V ) + E(1, 1, V )

= ∆E − µ(N,V ) + E(1, 1, V ) +O(N−1),

where

µ(N,V ) ≡(∂E

∂N

)

N,0,V

is the zero-temperature chemical potential of the N -electron HEG.

• In the infinite-system limit,

∆Ω(rs) = ∆E(rs)− µ(rs) + Epos,

• Zero-temperature chemical potential of a HEG is given by

µ =d

dn(nE) = E − 1

3rsdEdrs

,

where n is number density and E is total energy per electron.

• Chemical potential is easily calculated within HF theory.

• Correlation chemical potential has been parametrised by Perdew and Zunger6 (usingGell-Mann & Brueckner’s analytical result for high-density HEG and Ceperley-AlderDMC data for rs ≥ 1).

6J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981).

5 10 15 20 25r

s (a.u.)

-0.1

-0.05

0

0.05

0.1

µ (a.u.)

HF + corr. pot. of Perdew & ZungerHartree-Fock

• Hence we can obtain the immersion energy from relaxation-energy calculations andvice versa.

Electron-Positron Correlation Functionals

• If there is no electron-positron correlation then the positron occupies its zero-momentum ground state.

• In this case, energy of the HEG+positron is same as energy of the HEG.

• So the electron-positron correlation energy is the difference of the energy of theHEG+positron and the HEG.

• This is just the relaxation energy; can calculate it as a function of rs.

• Hence we can construct an electron-positron correlation functional, for use in DFTstudies of positrons in real materials.

Preliminary results

Pair correlation function

0 1 2 3 4 5 6Distance from positron (a.u.)

0

10

20

30

40

50

60

70

Pair

-cor

rela

tion

func

tion r

s = 1 a.u.

rs = 2 a.u.

rs = 3 a.u.

rs = 4 a.u.

rs = 5 a.u.

rs = 6 a.u.

rs = 7 a.u.

rs = 8 a.u.

Contact PCF is of greatest interest.

Contact density enhancement factor7

0 2 4 6 8 10r

s (a.u.)

0

20

40

60

80

100

120

140

g(0)

Present workBoronski & NieminenSterne & KaiserArponen & PajanneDMC data (Fraser & Foulkes)VMC data (Ortiz)FHNC data (Apaja et al.)FHNC data (Stachowiak & Lach)FHNC data (Lantto)Present work, w/o extra int.

• Our results agree with the VMC calculations of Ortiz.

• Extra interaction is important, especially at high densities.

7P. A. Sterne & J. H. Kaiser, Phys. Rev. B 43, 13892 (1991); E. Boronski & R. M. Nieminen, Phys. Rev. B 34, 3820(1986); J. Arponen & E. Pajanne, J. Phys. F 9, 2359 (1979); G. Ortiz, PhD thesis, Lausanne (1992); L. Fraser, PhD thesis,London (1995); V. Apaja et al., Phys. Rev. B 68, 195118 (2003); H. Stachowiak & J. Lach, Phys. Rev. B 48, 9828 (1993);L. J. Lantto, Phys. Rev. B 36, 5160 (1987).

Annihilating-pair momentum distribution

0 0.5 1 1.5 2 p / p

F

0

0.2

0.4

0.6

0.8

1

ρ / ρF

rs = 1.0 a.u.

rs = 2.0 a.u.

rs = 3.0 a.u.

rs = 4.0 a.u.

rs = 5.0 a.u.

rs = 6.0 a.u.

rs = 7.0 a.u.

rs = 8.0 a.u.

Maximum Kahana enhancement at rs ≈ 2 a.u.

Relaxation energy8

2 4 6 8Density parameter r

s (a.u.)

-0.5

-0.4

-0.3

-0.2

-0.1

Rel

axat

ion

ener

gy (

a.u.

)

37 electrons53 electrons65 electrons113 electrons161 electrons185 electrons245 electrons53 electrons, no extra int.65 electrons, no extra int.113 electrons, no extra int.DFT energy of positroniumBoronski-Nieminen formula

• Well behaved for small systems, badly behaved for large ones.

• Perhaps CASTEP is converging to the wrong self-consistent solution?

8E. Boronski & R. M. Nieminen, Phys. Rev. B 34, 3820 (1986)

QMC Studies of Positrons in Electron Gases

• Previous QMC studies of positrons in HEGs have used plane-wave orbitals for theelectrons and positron.

• At low densities positron binds with a single electron to form a positronium atom.

• Plane-wave orbitals are inappropriate for describing positronium.

• We can use modified CASTEP to generate orbitals for CASINO; no difficulty describingpositronium.

• Will either have to (i) modify DMC Green’s function etc. in accordance with thetransformation to the Hamiltonian or (ii) use the electron coordinates relative to thepositron in the Slater wave function.

• Option (ii) is probably the easiest way to proceed.

• DMC should be able to provide definitive answers to these technologicallyimportant questions about the behaviour of positrons in materials.

Acknowledgements

We acknowledge financial support from Jesus College, Cambridge and the Engineeringand Physical Sciences Research Council.