Download The quantum states of muons in fluorides

The quantum states of muons in fluorides Johannes Möller, Oxford University Fron=ers of Muon Spectroscopy St Hugh’s College, Oxford 17 September 2012 Acknowledgments Collabora'on Thanks also to Davide Ceresoli (Milano) Stephen Blundell (Oxford) Tom Lancaster (Durham) Nicola Marzari (Lausanne) Steve Cox (ISIS) Andrea dal Corso (SISSA) Emine Küçükbenli (Lausanne) Roberto de Renzi (Parma) Pietro Bonfà (Parma) Fabio Bernardini (Cagliari) 5.2), the presence of some additional relaxing fraction in the sample, or may be
n artefact resulting from possible distortions in the data at very early times.
What do muons measure? 28 considered, including
Alternative interpretations of the data below TN were
3
he possibility of incommensurate magnetic order, although these resulted in sig-
nificantly worse fits to the measured spectra.
•  The interac=on of the muon n PrO2 . The
magnetic
at a muon site B
(rµ )bise given
by the sum of dipolar
with its efield
nvironment cµan complicated: ontributions. Assuming that the local field is due entirely to the dipole field of
•  Dipolar interac=on with localized moments Pr ions:
rµ − ri
µ0 ! 3(µi · n̂i )n̂i − µi
n̂
=
Bµ (rµ ) =
,
(5.6)
i
3
4π
|
r
−
r
|
|rµ − ri |
µ
i
i
where r• 
µ is the position of the muon, µi is the ordered magnetic moment of the
For delocalized polarized spin density, there could be th
Pr ion and n̂i (= (rµ − ri )/ | rµ − ri |) is the unit vector from the muon to
a Fermi contact interac=on. he Pr ion at site ri . The positive muon’s position is usually in the vicinity of the
•  There may lectronegative
O2− ions
[55].be an external magne=c field. A perturbation
The• exact
structure
the 3.2
available
neutron
2 is unknown
In magnetic
systems with ofaPrO
non-­‐zero mand
agne=za=on also happroach
ave data are consistent
with either
1-k, 2-k (oreg. 3-k transverse
type-I magnetic
structures
finite-­‐size effects demagne=zing field). Our measurement of νi (0) allows us to attempt to determine the muon sites
Figure 3.1: Semiclassical view of the muon in a magnetic field.
about B0 with frequency ω0 .
arranged) we must average this result over all values of θ. Thi
1 2
+ cos ω0 t.
3 3
Pz (t) =
This corresponds to an ensemble with 1/3 of the spin componen
to the local field and 2/3 with components in a perpendicular
The general problem, where fluctuations are present, requires a
ory approach [36]. A magnetic field B
63], together with a second component with twice the period
along one axis
tot (t)
fluctuating dynami
B0 , may be written as
60]. Although we might assume the 3-k structure of UO2 [64], the dipole fieldB
tot (t)
= B0 + B(t).
We express the Hamiltonian of the muon in this B-field as
alculation yields consistent sites for 2-k and 1-k structures as
well. The best
Muon sites and the elusive role of the muon •  Muons probe the dynamic and sta=c behaviour of the local magne=c field. •  They are a very sensi=ve probe of magne=sm. •  However, the site of implanta'on is generally unknown so magne=c moments and spin structures cannot be measured directly. •  How much do muon charge and moment perturb the sample? •  Few systema=c microscopic studies to date of diamagne=c muon states in solids. What can DFT tells us about this? Brief introduc=on to DFT •  “Chemistry has come to an end” [Paul Dirac]. •  Full many-­‐body problem intractable with wavefunc=on-­‐based methods. 



�2 � 2 � Z l e 2
1�
e2
−
∇j −
+
− E Ψ(r1 , r2 , ..., rN ) = 0
�|
 2m

|r
−
R
|
2
|r
−
r
j
l
j
j
�
j
j,l
j�=j
•  Fundamental principle of DFT: any property of a correlated many-­‐body system can be expressed as a func.onal of the ground state density n 0 (r)
. •  This is the Hohenberg-­‐Kohn theorem. The Kohn-­‐Sham equa=ons �
1
− ∇2 + Vext (r) + VHartree + Vxc (r) − �j
2
VHartree
=
VXC (r)
=
n(r)
=
�
ψj (r) = 0
�
n(r� ) 3 �
d r
|r − r� |
δExc [n(r)]
� δn
σ
fji
ψi∗ (r)ψj (r),
i,j,σ
Phys. Rev. 140, A1133 (1965), 15,000 cita=ons Some technical details Pseudopoten'als •  Replace full poten=al of the nucleus and core electrons by an effec=ve pseudopoten.al. •  We use the GGA flavour of Perdew, Burke, and Ernzerhof (PBE) �
GGA
Exc
[n] = d3 r f (n, ∇n)
Periodic boundary condi'ons •  We’re dealing with a defect so we need a supercell approach. We use a supercell consis=ng of 2x2x2 conven=onal unit cells. How do we model the muon? •  It’s a proton with the mass of a muon. The quantum states of muons in LiF, NaF, CaF2, and BaF2 Why fluorides? •  In fluorinated compounds diamagne=c muons omen dipole couple strongly to a small number of fluorine nuclear spins. � µ0 γ i γ j
�
H=
[Si · Sj − 3(Si · r̂)(Sj · r̂)] −
γ i S i · Bi
3|
4π|r
i<j
i
�
�
1 �
Dz (t) =
|�m|σq |n�|2 exp(iωmn t)
N m,n
q
•  Diagonaliza=on of this Hamiltonian allows the local geometry of the muon-­‐fluorine entangled state to be determined experimentally. •  Usually this informa=on is not experimentally accessible hence fluorides provide a promising tes=ng ground for any theore=cal predic=ons about muon states in solids. The fluorides LiF and NaF CaF2 and BaF2 •  Rock-­‐salt structure •  Wide-­‐gap, non-­‐magne=c insulators •  a=4.028 A (LiF) •  a=4.78 A (NaF) •  Fluorite structure •  Wide-­‐gap, non-­‐magne=c insulators •  a=5.46 A (CaF2) •  a=6.2 A (BaF2) Experimental diamagne=c muon site geometries •  Brewer et al., PRB 33, 7813 (1986). Predic=ng the diamagne=c sites •  We add a single muon (hydrogen) to the system. •  Supercell has charge +1, i.e. is deficient of one electron. •  Then allow all atoms to relax their posi=on based on forces calculated with Hellmann-­‐Feynman theorem F = −∇R ER
∂Eλ
=
∂λ
�
�
� �
� ∂Hλ �
� ψλ
ψλ ��
∂λ �
DFT predic=ons for diamagne=c muon sites: LiF and NaF •  Muon-­‐fluorine bound states predicted correctly. •  Fμ state unstable. •  Bond angle 180° as observed experimentally. •  Bond lengths extremely accurate, even trends are correct. F-­‐F separa'on LiF NaF Experimental 2.36(2) A 2.38(1) A DFT 2.35 A 2.34 A How much does the muon perturb the crystal and magne=c structure? Change in Lowdin projection (Bohr)
LiF F F state
Li
F
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0.0
0
1
2
3
4
5
6
Perturbed distance from defect (A)
Change in distance from defect (A)
•  Perturba=on due to muon is very short ranged. •  The nearest neighbour ions, including the ca=on, are perturbed considerably. •  If the ca=on was magne=c, quan=ta=ve es=mates of the magne=c moment would have to take this into account. •  The conjecture that the perturba=on mainly affects the fluorine ions involved in the FμF state appears incorrect. 0.5
Li
F
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
0
1
2
3
4
Original distance from defect (A)
5
6
DFT predic=ons for diamagne=c muon sites: CaF2 and BaF2 •  FμF bound states predicted correctly. •  Bond angle 180° as observed experimentally. •  Bond lengths extremely accurate, even trends are correct. F-­‐F separa'on CaF2 BaF2 Experimental 2.34(2) A 2.37(2) A DFT 2.33 A 2.305 A CaF2 F F state
Ca
F
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0.0
0
1
2
3
4
5
6
7
8
9
Perturbed distance from defect (A)
Change in distance from defect (A)
•  General picture the same as for LiF and NaF, albeit with a slightly more long-­‐ranged perturba=on. •  Again the nearest neighbour ca=on posi=on is more perturbed than the posi=on of the fluorines involved in the muon-­‐
fluorine state. Change in Lowdin projection (Bohr)
How much does the muon perturb the crystal and magne=c structure? 0.5
Ca
F
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
0
1
2
3
4
5
6
7
Original distance from defect (A)
8
9
The paramagne=c state: muonium •  The muon atracts an electron, forming the muonium hydrogen-­‐like atom. •  Do same as above, but now with a neutral supercell: i.e. add muon and one electron. How to calculate the Fermi contact coupling •  Hyperfine coupling consists of Fermi contact and dipolar terms. H = Se · (Ai I + Aa ) · Sµ
•  Here, because of the tetrahedral and octahedral symmetry of the muon site (see below), all dipolar terms cancel. •  Contact term given by spin density at muon site. µ0 �
Ai =
γe γµ |ρ(rµ )|2
3π
How to calculate the Fermi contact coupling •  However, when using a pseudopoten=al, the wavefunc=on inside the core region is pseudized. •  Can either use full Coulomb poten=al for the muon (very slow convergence). •  Or reconstruct wavefunc=on near nuclei using the Gauge-­‐Including Projector Augmented Wave method. Pickard and Mauri PRB 63, 245101 (2001). Implemented in GIPAW code, www.gipaw.net •  All results below use the GIPAW method. DFT predic=ons for muonium sites and hyperfine couplings: LiF and NaF •  Muonium predicted to sit in tetrahedral cage. •  Hyperfine couplings quite good, but a bit too high. •  Avac=3919 MHz (calculated). •  Avac=4463.3 MHz (exp.). A/Avac LiF NaF Experimental ≈ 1.027 ≈ 1.04 DFT (classical) 1.114 1.12 How much does the muonium perturb the crystal and magne=c structure? Change in Lowdin projection (Bohr)
LiF muonium state
Li
F
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0.0
0
1
2
3
4
5
Perturbed distance from defect (A)
Change in distance from defect (A)
•  Less crystallographic distor=on than for bare muon FμF states. •  Plausible since the electron screens the muon’s charge. •  However, some of the unpaired spin density “leaks” onto the neighbouring ions. 0.5
Li
F
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
0
1
2
3
4
Original distance from defect (A)
5
DFT predic=ons for muonium sites and hyperfine couplings: CaF2 and BaF2 •  Muonium predicted to sit in octahedral cage. •  F-­‐F BC site 1.27 eV (CaF2) 0.76 eV (BaF2) higher in energy even w/o ZPE. •  Hyperfine couplings quite good, but a bit too high. •  Avac=3919 MHz (calculated). •  Avac=4463.3 MHz (exp.). A/Avac Experimental CaF2 BaF2 1.0036 (10) ? DFT (classical) 1.176 1.175 CaF2 muonium state
Ca
F
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0.0
0
1
2
3
4
5
6
7
8
9
Perturbed distance from defect (A)
Change in distance from defect (A)
•  There is a short-­‐ranged effect on the Löwdin projec=on but essen=ally no distor=on even of the nearest neighbour ions. •  The octahedral cavity seems to be large enough to encompass the full muonium wavefunc=on without much effect on its neighbours. •  Similar results for BaF2. Change in Lowdin projection (Bohr)
How much does the muonium perturb the crystal and magne=c structure? 0.5
Ca
F
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
0
1
2
3
4
5
6
7
Original distance from defect (A)
8
9
Quantum correc=ons: what are we missing? •  The muon is a par=cle in a box! Muon zero point mo=on and “quantum spread” of muon wavefunc=on so far completely ignored. �2 π 2 2
En =
n
2
2mL
•  There is a compe''on between ZP energy and poten'al energy. But what’s the effect on: •  the rela=ve stability of diamagne=c and paramagne=c sites, •  the hyperfine coupling parameters? Double-­‐adiaba=c approxima=on •  AR Porter, MD Towler, RJ Needs, PRB 60 13534 (1999), looked at tetrahedral muonium in silicon. •  Si, muon, electron decouple: mSi >> mµ >> me
•  Parameterized the poten=al calculated from DFT and solved Schrödinger equa=on for muon. VT (x, y, z) =VT (0, 0, 0) + a2 (x2 + y 2 + z 2 ) + a3 xyz
+ a4 (x4 + y 4 + z 4 ) + b4 (x2 y 2 + x2 z 2 + y 2 z 2 )
+ a5 xyz(x2 + y 2 + z 2 ) + a6 (x6 + y 6 + z 6 )
+ b6 (x2 y 4 + x2 z 4 + x4 y 2 + x4 z 2 + y 2 z 4 + y 4 z 2 )
+ c 6 x2 y 2 z 2
A harmonic approxima=on •  Parameterizing the poten=al and solving for the muon wavefunc=on allows an accurate quantum correc=on of the hyperfine couplings. •  However, it requires considerable effort, e.g. the tetrahedral wavefunc=on consisted of > 2000 basis func=ons. Poten=al had to be expanded to 8th order. •  Let’s ignore details of the poten=al, model as harmonic oscillator: 1
2 2
V
=
mω
r
2
•  In the harmonic approxima=on, frequencies are given by phonon eigenfrequencies of the muon from DFPT. Eigenmodes of muonium in tetrahedral cage •  Eigenfrequencies are very high because of the low muon mass (mμ = 1/9 mp). ω=
�
ν (1/cm) k
m
2702.774480 2706.395958 2706.405884 ZPE: 0.503 eV •  Rigid la|ce approxima=on: amplitude of ionic vibra=ons < 1% of amplitude of muonium eigenmodes. Muonium: quantum effects in LiF •  In the harmonic approxima=on, the wavefunc=on is known analy=cally =
�
1
2π
Rx
1
2π
Ry
�1/2 �
1
2π
Ry
Ri =
1
Rz2 π
�1/2 �
�
1
Rz2 π
e−(x
�1/2
�2
A (GHz)
�2
+y +z )
H
0
0.35
e−r
�2
H
0.3
0.25
0.2
0.15
Muon
0.1
�
mωi
0.05
0.0
2
| | (a.u.)
25
•  Quantum average of hyperfine coupling given by H
20
15
10
Muon
5
0
2
r dr A(r)|Ψ0 (r)|
�
r2 dr |Ψ0 (r)|2
2
Energy above ground state (eV)
< A >quantum =
3
1
x� = x/Rx ...
�
4
2
�2
2
1
2π
Rx
�1/2
| | (a.u.)
=
�1/2 �
2
2
�1/2 �
Muon
5
r
|Ψ0 (r)|
�
F
Li
6
Calculated energy
Harmonic approx. (fit)
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Distance from equilibrium (A)
0.8
1.0
Muonium: quantum effects in LiF Quantum correc=on -­‐2.5 % Experiments ≈ 1.027 -­‐0.7 % ? F
Li
6
Muon
A (GHz)
5
4
3
H
2
1
0
0.35
H
0.25
2
| | (a.u.)
0.3
0.2
Muon
r
2
0.15
0.1
0.05
0.0
2
| | (a.u.)
25
H
20
15
10
Muon
5
0
Energy above ground state (eV)
•  There is a quantum correc=on to the hyperfine coupling even though the muonium equilibrium posi=on in LiF is at a saddle point of spin density. •  Correc=on likely much larger in CaF2 and BaF2. •  Direct analogy between muonium and hydrogen limited by ZP effects. Muonium Hydrogen A/Avac 1.114 1.114 Classical With quantum 1.086 1.106 correc=on Calculated energy
Harmonic approx. (fit)
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Distance from equilibrium (A)
0.8
1.0
What about the diamagne=c state? •  The linear FμF complex, looked like a molecule-­‐in-­‐a-­‐crystal defect, not unlike the Vk centre in the alkali halides. •  Molecules vibrate, modes for linear CO2 in vacuum: 1333 1/cm 667 1/cm 2349 1/cm What about the diamagne=c state? •  The linear FμF complex, looked like a molecule-­‐in-­‐a-­‐crystal defect, not unlike the Vk centre in the alkali halides. •  Molecules vibrate, modes for linear CO2 and (FμF)-­‐ in vacuum: 1333 1/cm 1299 1/cm 583 1/cm 667 1/cm 648 1/cm 3501 1/cm 2349 1/cm 2339 1/cm 4761 1/cm CO2 (exp.) CO2 (calc.) (FμF)-­‐ (calc.) What about the diamagne=c state? •  The low mass of the muon implies very large vibra=onal frequencies and zero point energies. •  The μF and (FμF)-­‐ molecules have larger ZPEs than any naturally occurring di-­‐/triatomic molecules. ZPE (eV) ZPE/number of modes (eV) CO2 0.311 (exp.) 0.290 (calc.) 0.0778 (exp.) 0.0725 (calc.) (FμF)-­‐ 0.765 0.191 H2O 0.558 0.186 μF 0.693 0.693 HF 0.257 (exp.) 0.249 (calc.) 0.257 (exp.) 0.249 (calc.) H2 0.273 0.273 What about the diamagne=c state? •  In a solid, the muon eigenmodes are modified depending on the local geometry of the site. Vacuum In LiF 2819 1/cm 3501 1/cm 4601 1/cm 4879 1/cm 4879 1/cm ZPE in LiF 3 main modes Total Muonium 0.503 0.535 eV FμF 0.866 eV 0.763 Which charge state is more favourable? •  Combining total and ZP energies, we can compare the forma=on energies of the diamagne=c and muonium states Ef (µq ) = Etot (µq ) − Etot (bulk) + EZPE + qEF
Formation energies in LiF
-4
Formation energy (eV)
-6
-8
-10
Mu
-12
-14
+
-16
-20
g
g
E
-18
0
2
4
6
E
calc
8
Fermi level (eV)
10
exp
12
14
Which charge state is more favourable? •  Combining total and ZP energies, we can compare the forma=on energies of the diamagne=c and muonium states Ef (µq ) = Etot (µq ) − Etot (bulk) + EZPE + qEF
Formation energies in LiF
-4
Formation energy (eV)
-6
-8
-10
Mu
-12
-14
+
-16
-20
g
g
E
-18
0
2
4
6
E
calc
8
10
exp
12
14
Fermi level (eV)
ZnO: Cox et al PRL 86 2601 (2001) Van de Walle PRL 85 1012 (2000) •  Neglects kinema=c effects of the muon stopping process Prokscha et al PRL 98 227401 (2007) & Storchak et al PRL 78 2835 (1997). Conclusions and Outlook •  DFT can provide a comprehensive quan=ta=ve descrip=on of the quantum states of muons in solids. •  Muons cause a significant short-­‐ranged perturba=on to their host, that should be taken into account for an accurate quan=ta=ve analysis. •  Muons are the “most quantum” defect available. •  Muon states in a range of interes=ng materials amenable to characteriza=on with DFT. •  Details on the arxiv soon. Dependence of HF coupling on band gap Double-­‐adiaba=c approxima=on •  ZP mo=on stabilizes T site. Path-­‐integral molecular dynamics •  Miyake et al PRL 81 1873 (1998) and PRB 60 14197 (1999): (i) poten=als from DFT, (ii) Feynman path-­‐integral method allows quantum effects to be included, (iii) molecular dynamics to treat mo=on of pseudo-­‐classical par=cles. •  Find substan=al differences between Mu and H. perly in the
rgy differquite accucupation of
ues. As an
on !an apy adding a
es for inteture of the
al occupahe kind of
rical value
f the LDA
' DFT+U EKS#$qI%& = min EKS#n!r"& +
n!r",!I
(I !IKS!nI ! qI"
)
, !14"
we would have obtained nonvanishing results as well because by varying the site occupation a rehybridization of the
localized orbitals with the other degrees of freedom is induced that gives rise to a nonlinear change in the energy of
the system. This curvature coming from rehybridization,
originating from the noninteracting band structure but
present also in the interacting case, has clearly nothing to do
with the Hubbard U of the interacting
system and should be
U�
subtracted
from the
curvature
EDFT+U
= total
EDFT
[ρ] +
Tr[nIσ (1 − nIσ )]
•  Plain DFT is known to underes=mate the gap in transi=on metal oxides. Plain DFT will not work for a Mot insulator. arameter U
as a funcrious selfhe system.
value of U
preceding
on matrices
correlation
the exact
d in solids
2
Iσ
!2E#$qI%& !2EKS#$qI%&
!
.
U=
!qI2
!qI2
!15"
In Ref. 8 Anisimov and Gunnarsson, in order to avoid
dealing with the abovementioned noninteracting curvature,
exploited the peculiarities of the LMTO method, used in
their calculation, and decoupled the chosen localized orbitals
from the remainder of the crystal by suppressing in the
LMTO Hamiltonian the corresponding hopping terms. This
reduced the problem to the one of an isolated atom embed-