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Dynamic Jahn-Teller effect of the
ππβ center in diamond by DFT
1
Abtew, Sun, Shih, Dev, S. Zhang, and P. Zhang
Phys. Rev. Lett. 107, 146403 (2011)
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
Ground 3π΄2 and excited 3πΈ states of ππβ
2
512-atom cell; negligible cell-cell interactions
Excited state: 1 electron from π1,β to an πβ level.
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
PBE calc.
PBE vs. HSE for ππβ
3
As the gap opens, majority-spin states shift down relative to the VBM
but only slightly; the occupied minority-spin π1 state moves up
somewhat, while empty minority-spin π states move up considerably.
Red: majority spin
Blue: minority spin
Fewer bulk bands
in the HSE results
due to the smaller
supercell (i.e.,
fewer band folding)
1
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
4
Eβe Jahn-Teller vibronic model for excited 3πΈ
π»ππ ππ₯, ππ¦ = π»ππ ππ₯ = ππ¦ = 0
+πΎ
2ππ₯2 + ππ¦
2 ππ§ + πΉ ππ₯ππ§ β ππ¦ππ₯ + πΊ[ ππ₯2 β ππ¦
2 ππ§ + 2ππ₯ππ¦ππ₯]
Bersuker, The Jahn-Teller Effect (Cambridge Univ. Press, 2006)
The electronic part:
where ππ₯ and ππ¦ are defined by
no net y comp.
no net x comp.
a breathing mode
and πΎ, πΉ, and πΊ are
the phenomenological
parameters and ππ are
the Pauli matrices.
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
5
Solutions of the model
Bersuker, The Jahn-Teller Effect (Cambridge Univ. Press, 2006)
Define polar variables π, π such that π = ππ₯2 + ππ¦
2 and π =
tanβ1ππ¦
ππ₯; the solutions for π»ππ ππ₯ , ππ¦ are π π, π =
1
2πΎπ2 Β±
π πΉ2 + πΊ2π2 + 2πΉπΊπ cos 3π
If πΊ = 0, π π, π =1
2πΎπ2 Β± πΉπ.
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
6
Solutions of the model
The lower-energy branch would be
ππΏπΈ π, π =1
2πΎπ2 β πΉπ, which has
the shape of a Mexican hat.
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
The adiabatic potential energy surface (APES)
7
Of course, πΊ β 0,
which leads to the
APES shown to the
right
There are 3 local
minima as a result of
Jahn-Teller distortion.
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
Three key parameters
8
1st: displacement from the origin to a minimum, π0 = πΉ/(πΎ β 2πΊ)
2nd: the corresponding energy lowering, πΈπ½π =πΉ2
2 πΎβ2πΊ=
πΉ
2π0
3rd: energy barriers between local minima, πΏ = 4πΈπ½πG/ πΎ + 2πΊ
PBE results: π0 =
0.068 β«, πΈπ½π = 25
meV, and πΏ = 10
meV
HSE+SOC: πΈπ½π = 42
meV, πΏ = 9 meV. (Thiering & Gali, 2017)
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
9
An example of the ππ¦ modes
Red arrows are
displacement
vectors, which sum
to zeros in the x-
and z-directions
Moving away from
the vacancy,
displacements
decay but do not
quickly diminish.
C
CN
N
V
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
10
Dynamic Jahn-Teller effect
From the PBE results for π0, πΈπ½π, and πΏ, we obtain πΉ = β0.74 eV/β«,
πΊ = 1.76 eV/β«2, and πΎ = 14.5 eV/β«2
Using the πΎ, we estimated the energy (βπ) for the local vibrational
mode (LVM) to be 71 meV
For a Mexican hat potential (πΊ = 0), the system dynamics involves a
radial vibration and a free rotation along the bottom of the trough in
the Mexican hat
However, since πΊ β 0 and βπ β« πΏ (= 10 meV), the dynamics is
better described as a hindered internal rotation, namely, we have a
dynamic Jahn-Teller system [Fu, et al., PRL103, 256404 (2009)].
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
11
A complete Eβe vibronic model for 3πΈ
π»π½π = π»ππ ππ₯, ππ¦ ββ2
2π
π2
πππ₯2 +
π2
πππ¦2 = 7.25 (eV/β«
2) ππ₯2 + ππ¦
2 ππ§ β
0.74 (eV/β«) ππ₯ππ§ β ππ¦ππ₯ + 1.76 (eV/β«2)[ ππ₯
2 β ππ¦2 ππ§ + 2ππ₯ππ¦ππ₯]
ββ2
2π
π2
πππ₯2 +
π2
πππ¦2
By diagonalizing
π»π½π, we obtain
vibronic levels
and transition
energies:
Symmetry Energy (meV) βE (meV) Expt. (meV)*
πΈ 36.7
π΄1 71.7 πΈ β π΄1 (3Ξ): 35
π΄2 103.8 πΈ β π΄2: 67 60
πΈ 114.8 π΄2 β πΈ: 11 10
*Davies and Hamer, Proc. R. Soc. A 348, 285 (1976)
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
12
Zero phonon line (ZPL) dephasing rate
Fu, et al. [PRL103, 256404 (2009)] showed that the dephasing of the
ZPL at low temperature (π) is dominated by a coupling between the
electronic πΈ doublet and e phonons
A π5-dependence of the linewidth was explained by a two-phonon
Raman process, which results in a dephasing rate:
π =2π
β 0
βππ· ππΈπ π + 1 ππ·ππ2 πΈ
π4 πΈ
πΈ2
where π is related to πΉ(DFT) by π(πΈ) = πΉ β2/2ππΈ
We simplified π to 8π
β
Ξ©ππππ
2π2β3π£π ππ’ππ3
2
πΆ4 ππ΅π5πΌ4
βππ·
ππ΅πwhere πΌ4 is
a Debye integral.
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
13
Comparison with experiment
Our approximation
for π can cover a
wider range of
temperatures
Agreement with
experiment, which
were carried out on
several different NV
centers, is rather
good.
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
14
Current status of the model
Huxter, et al., βVibrational and
electronic dynamics β¦ revealed
by 2D ultrafast spectroscopyβ,
Nature Physics 9, 744 (2013)
Ulbricht, et al., βJahn-Teller-induced fs electronic depolarization
dynamics β¦β, Nature Comm. 7: 13510, 2016
βinterestingly, (the measured) ππ closely approaches the calculated
tunneling splitting (here) of 35 meV, suggesting possible electronic
depolarization β¦β
β¦
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
15
Issues with ππβ centers in diamond
Exact positioning is difficult;
near-surface NV suffers from surface-charge fluctuations: blinking
due to jumps between ππβ and ππ0 / spectral diffusion due to
jumps within NV sublevels; and
in an ambient condition, water adsorption converts ππβ to ππ0
Low light-collection efficiency due to high refractive index
Low qubit yield. This is because (a) the N-to-NV conversion
efficiency is low, typically a few % and (b) the ZPL of the NV center is
relatively weak compared to other photonic transitions.
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
In addition,
NV-like center in cubic boron nitride
16
Abtew, Gao, Gao, Sun, S. Zhang, and P. Zhang
Phys. Rev. Lett. 113, 136401 (2014)
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
not h-
17
Electron counting and the ππ΅βππ0 center
To make an ππβ requires the removal
of 4 els with C and the addition of 2 els:
one by ππΆ and one by the (-) charge.
Effectively, the ππβ center is a 2 e-
deficient system.
In ππ΅-ππ, 3 els are removed with B, 1 el is added by ππ. This
amounts to a 2-el removal. Hence, the center should be ππ΅βππ0
In contrast in πΆπ΅-ππ, 5 els are removed with N, 1 el is added by πΆπ΅.
One needs 2 additional els to maintain 2e-deficiency [ πΆπ΅βππ2β].
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
18
Level shifts relative to ππβ center
NV in C OVB in BN
In both cases, the Fermi level is between the minority-spin π and π1
states, so the electron counting model really works
Good similarity but in cBN all defect levels move down considerably
If we have CVN, instead, these levels would move up considerably.
216-atom, HSE
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
19
Optical transitions within Franck-Condon
Good optical similarities
between NV in diamond and
OVB in cBN
ZPL of 1.60 eV for ππ΅βππ0
matches the experimental GC-2
center (1.61 eV) in cBN.
Constrained DFT
Ground-state DFT
Defect π¨ β π© (eV) Cβ π« (eV) ZPL (eV) βS (meV) βAS (meV)
ππβ 2.21 1.74 1.96 258 217
πππ΅0 1.75 1.47 1.60 150 130
HSE calculations for NV and OVB:
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
20
Origin of the optical similarities
Nearly identical
βatomicβ-like states
between NV and OVB
Small differences: (a) the
OVB states are more p-
like and (b) BN has less
donor contribution to
the π1 state than
diamond does.
π1 π
BN
Dia
mond
B
N
O
C
OVB offers a small variation to NV β good for studying NV physics.
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
21
Anything else that supports the GC-2 assignment?
Define βπΉπ = πΉπ ππ β πΉπ ππ the displacement vector, ππΌπ the
polarization vector of the πΌth phonon mode on the ith atom, and
ππΌ = π1
ππππΌπ β βπΉπ the projection of the displacement vector
Calculated local vibration modes (LVM) and projections: The first three
LVMs, weighted
by π 2, is 55
meV, which is in
good agreement
with the
experimental
result of 56 meV
for GC-2.
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
22
Scalable quantum information in a flatland?
Electron counting: While one can ignore the bonding along z, it
takes away 0 and 2 electrons for B and N, respectively. As such, B
and N become βequivalentβ in the 2D counting
Primary point defects to consider: ππ΅(-3), ππ(-3), ππ(+1), πΆπ΅(+1), as
well as ππ΅(0) and π΅π(0), where electron deficiency (excess) from
perfect crystal is indicated. Note that, due to symmetry difference,
the 2-electron-deficiency rule for diamond and cBN needs not apply
here
This yields possible 1st nn pairs: OVB(-2), CVN(-2), BNVB(-3), and
NBVN(-3) and 2nd nn pairs: CVB(-2), OVN(-2), NBVB(-3), and BNVN(-3).
Shengbai Zhang, RPIDynamic Jahn-Teller Effect
23Shengbai Zhang, RPIDynamic Jahn-Teller Effect
Summary
Reviewed our earlier dynamic Jahn-Teller calculation
Proposed πππ΅0 in cBN for its great similarity to ππβ
and suggested it as the experimental GC-2 center.