Dynamic Jahn-Teller effect of the

𝑁𝑉− center in diamond by DFT

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Abtew, Sun, Shih, Dev, S. Zhang, and P. Zhang

Phys. Rev. Lett. 107, 146403 (2011)

Shengbai Zhang, RPIDynamic Jahn-Teller Effect

Ground 3𝐴2 and excited 3𝐸 states of 𝑁𝑉−

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512-atom cell; negligible cell-cell interactions

Excited state: 1 electron from 𝑎1,↓ to an 𝑒↓ level.

Shengbai Zhang, RPIDynamic Jahn-Teller Effect

PBE calc.

PBE vs. HSE for 𝑁𝑉−

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As the gap opens, majority-spin states shift down relative to the VBM

but only slightly; the occupied minority-spin 𝑎1 state moves up

somewhat, while empty minority-spin 𝑒 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)

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Shengbai Zhang, RPIDynamic Jahn-Teller Effect

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E⊗e Jahn-Teller vibronic model for excited 3𝐸

𝐻𝑒𝑙 𝑄𝑥, 𝑄𝑦 = 𝐻𝑒𝑙 𝑄𝑥 = 𝑄𝑦 = 0

+𝐾

2𝑄𝑥2 + 𝑄𝑦

2 𝜎𝑧 + 𝐹 𝑄𝑥𝜎𝑧 − 𝑄𝑦𝜎𝑥 + 𝐺[ 𝑄𝑥2 − 𝑄𝑦

2 𝜎𝑧 + 2𝑄𝑥𝑄𝑦𝜎𝑥]

Bersuker, The Jahn-Teller Effect (Cambridge Univ. Press, 2006)

The electronic part:

where 𝑄𝑥 and 𝑄𝑦 are defined by

no net y comp.

no net x comp.

a breathing mode

and 𝐾, 𝐹, and 𝐺 are

the phenomenological

parameters and 𝜎𝑖 are

the Pauli matrices.

Shengbai Zhang, RPIDynamic Jahn-Teller Effect

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Solutions of the model

Bersuker, The Jahn-Teller Effect (Cambridge Univ. Press, 2006)

Define polar variables 𝜌, 𝜙 such that 𝜌 = 𝑄𝑥2 + 𝑄𝑦

2 and 𝜙 =

tan−1𝑄𝑦

𝑄𝑥; the solutions for 𝐻𝑒𝑙 𝑄𝑥 , 𝑄𝑦 are 𝜖 𝜌, 𝜙 =

1

2𝐾𝜌2 ±

𝜌 𝐹2 + 𝐺2𝜌2 + 2𝐹𝐺𝜌 cos 3𝜙

If 𝐺 = 0, 𝜖 𝜌, 𝜙 =1

2𝐾𝜌2 ± 𝐹𝜌.

Shengbai Zhang, RPIDynamic Jahn-Teller Effect

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Solutions of the model

The lower-energy branch would be

𝜖𝐿𝐸 𝜌, 𝜙 =1

2𝐾𝜌2 − 𝐹𝜌, which has

the shape of a Mexican hat.

Shengbai Zhang, RPIDynamic Jahn-Teller Effect

The adiabatic potential energy surface (APES)

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Of course, 𝐺 ≠ 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

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1st: displacement from the origin to a minimum, 𝜌0 = 𝐹/(𝐾 − 2𝐺)

2nd: the corresponding energy lowering, 𝐸𝐽𝑇 =𝐹2

2 𝐾−2𝐺=

𝐹

2𝜌0

3rd: energy barriers between local minima, 𝛿 = 4𝐸𝐽𝑇G/ 𝐾 + 2𝐺

PBE results: 𝜌0 =

0.068 Å, 𝐸𝐽𝑇 = 25

meV, and 𝛿 = 10

meV

HSE+SOC: 𝐸𝐽𝑇 = 42

meV, 𝛿 = 9 meV. (Thiering & Gali, 2017)

Shengbai Zhang, RPIDynamic Jahn-Teller Effect

9

An example of the 𝑄𝑦 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

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Dynamic Jahn-Teller effect

From the PBE results for 𝜌0, 𝐸𝐽𝑇, and 𝛿, we obtain 𝐹 = −0.74 eV/Å,

𝐺 = 1.76 eV/Å2, and 𝐾 = 14.5 eV/Å2

Using the 𝐾, we estimated the energy (ℏ𝜔) for the local vibrational

mode (LVM) to be 71 meV

For a Mexican hat potential (𝐺 = 0), the system dynamics involves a

radial vibration and a free rotation along the bottom of the trough in

the Mexican hat

However, since 𝐺 ≠ 0 and ℏ𝜔 ≫ 𝛿 (= 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

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A complete E⊗e vibronic model for 3𝐸

𝐻𝐽𝑇 = 𝐻𝑒𝑙 𝑄𝑥, 𝑄𝑦 −ℏ2

2𝑀

𝜕2

𝜕𝑄𝑥2 +

𝜕2

𝜕𝑄𝑦2 = 7.25 (eV/Å

2) 𝑄𝑥2 + 𝑄𝑦

2 𝜎𝑧 −

0.74 (eV/Å) 𝑄𝑥𝜎𝑧 − 𝑄𝑦𝜎𝑥 + 1.76 (eV/Å2)[ 𝑄𝑥

2 − 𝑄𝑦2 𝜎𝑧 + 2𝑄𝑥𝑄𝑦𝜎𝑥]

−ℏ2

2𝑀

𝜕2

𝜕𝑄𝑥2 +

𝜕2

𝜕𝑄𝑦2

By diagonalizing

𝐻𝐽𝑇, we obtain

vibronic levels

and transition

energies:

Symmetry Energy (meV) ∆E (meV) Expt. (meV)*

𝐸 36.7

𝐴1 71.7 𝐸 → 𝐴1 (3Γ): 35

𝐴2 103.8 𝐸 → 𝐴2: 67 60

𝐸 114.8 𝐴2 → 𝐸: 11 10

*Davies and Hamer, Proc. R. Soc. A 348, 285 (1976)

Shengbai Zhang, RPIDynamic Jahn-Teller Effect

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Zero phonon line (ZPL) dephasing rate

Fu, et al. [PRL103, 256404 (2009)] showed that the dephasing of the

ZPL at low temperature (𝑇) is dominated by a coupling between the

electronic 𝐸 doublet and e phonons

A 𝑇5-dependence of the linewidth was explained by a two-phonon

Raman process, which results in a dephasing rate:

𝑊 =2𝜋

ℏ 0

ℏ𝜔𝐷 𝑑𝐸𝑛 𝑛 + 1 𝜌𝐷𝑂𝑆2 𝐸

𝑉4 𝐸

𝐸2

where 𝑉 is related to 𝐹(DFT) by 𝑉(𝐸) = 𝐹 ℏ2/2𝑀𝐸

We simplified 𝑊 to 8𝜋

ℏ

Ω𝑐𝑒𝑙𝑙

2𝜋2ℏ3𝑣𝑠𝑜𝑢𝑛𝑑3

2

𝐶4 𝑘𝐵𝑇5𝐼4

ℏ𝜔𝐷

𝑘𝐵𝑇where 𝐼4 is

a Debye integral.

Shengbai Zhang, RPIDynamic Jahn-Teller Effect

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Comparison with experiment

Our approximation

for 𝑊 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

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Current status of the model

Huxter, et al., “Vibrational and

electronic dynamics … revealed

by 2D ultrafast spectroscopy”,

Nature Physics 9, 744 (2013)

Ulbricht, et al., “Jahn-Teller-induced fs electronic depolarization

dynamics …”, Nature Comm. 7: 13510, 2016

“interestingly, (the measured) 𝜔𝑐 closely approaches the calculated

tunneling splitting (here) of 35 meV, suggesting possible electronic

depolarization …”

…

Shengbai Zhang, RPIDynamic Jahn-Teller Effect

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Issues with 𝑁𝑉− centers in diamond

Exact positioning is difficult;

near-surface NV suffers from surface-charge fluctuations: blinking

due to jumps between 𝑁𝑉− and 𝑁𝑉0 / spectral diffusion due to

jumps within NV sublevels; and

in an ambient condition, water adsorption converts 𝑁𝑉− to 𝑁𝑉0

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

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Abtew, Gao, Gao, Sun, S. Zhang, and P. Zhang

Phys. Rev. Lett. 113, 136401 (2014)

Shengbai Zhang, RPIDynamic Jahn-Teller Effect

not h-

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Electron counting and the 𝑉𝐵−𝑂𝑁0 center

To make an 𝑁𝑉− requires the removal

of 4 els with C and the addition of 2 els:

one by 𝑁𝐶 and one by the (-) charge.

Effectively, the 𝑁𝑉− center is a 2 e-

deficient system.

In 𝑉𝐵-𝑂𝑁, 3 els are removed with B, 1 el is added by 𝑂𝑁. This

amounts to a 2-el removal. Hence, the center should be 𝑉𝐵−𝑂𝑁0

In contrast in 𝐶𝐵-𝑉𝑁, 5 els are removed with N, 1 el is added by 𝐶𝐵.

One needs 2 additional els to maintain 2e-deficiency [ 𝐶𝐵−𝑉𝑁2−].

Shengbai Zhang, RPIDynamic Jahn-Teller Effect

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Level shifts relative to 𝑁𝑉− center

NV in C OVB in BN

In both cases, the Fermi level is between the minority-spin 𝑒 and 𝑎1

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

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Optical transitions within Franck-Condon

Good optical similarities

between NV in diamond and

OVB in cBN

ZPL of 1.60 eV for 𝑉𝐵−𝑂𝑁0

matches the experimental GC-2

center (1.61 eV) in cBN.

Constrained DFT

Ground-state DFT

Defect 𝑨 → 𝑩 (eV) C→ 𝑫 (eV) ZPL (eV) ∆S (meV) ∆AS (meV)

𝑁𝑉− 2.21 1.74 1.96 258 217

𝑂𝑉𝐵0 1.75 1.47 1.60 150 130

HSE calculations for NV and OVB:

Shengbai Zhang, RPIDynamic Jahn-Teller Effect

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Origin of the optical similarities

Nearly identical

“atomic”-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 𝑎1 state than

diamond does.

𝑎1 𝑒

BN

Dia

mond

B

N

O

C

OVB offers a small variation to NV → good for studying NV physics.

Shengbai Zhang, RPIDynamic Jahn-Teller Effect

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Anything else that supports the GC-2 assignment?

Define ∆𝑹𝑖 = 𝑹𝑖 𝑄𝑒 − 𝑹𝑖 𝑄𝑔 the displacement vector, 𝒑𝛼𝑖 the

polarization vector of the 𝛼th phonon mode on the ith atom, and

𝜂𝛼 = 𝑖1

𝑚𝑖𝒑𝛼𝑖 ∙ ∆𝑹𝑖 the projection of the displacement vector

Calculated local vibration modes (LVM) and projections: The first three

LVMs, weighted

by 𝜂 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

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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 “equivalent” in the 2D counting

Primary point defects to consider: 𝑉𝐵(-3), 𝑉𝑁(-3), 𝑂𝑁(+1), 𝐶𝐵(+1), as

well as 𝑁𝐵(0) and 𝐵𝑁(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 𝑂𝑉𝐵0 in cBN for its great similarity to 𝑁𝑉−

and suggested it as the experimental GC-2 center.