CC BY
11
УДК 538.9
A. A. Ivanov, A. I. Ivanov, A. A. Kulagina
PHOTOLUMINESCENCE OF THE 15NV -CENTER CREATED BY IMPLANTATION
We study the properties of the spin states in single diamond 15NV--center at the ground state level anti-crossing. Our approach uses a complete set of commuting operators. We have shown that under certain conditions in 15NV--center it is possible to obtain a 100 % transfer of polarization from the electron spin to the spin of the 15N nucleus. We believe that these conditions can be satisfied for 15NV--centers obtained by implantation.
Мы изучаем свойства спиновых состояний в одиночном алмазном 15NV--центре на уровне пересечения основного состояния. Наш подход использует полный набор коммутирующих операторов. Мы показали, что при определенных условиях в 15NV--центре можно получить 100%-й перенос поляризации от спина электрона к спину ядра 15N. Мы полагаем, что эти условия могут быть выполнены для 15NV--центров, полученных имплантацией.
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Keywords: diamond, implantation, level anti-crossing. Ключевые слова: алмаз, имплантация, антипереход уровня.
The negatively charged nitrogen-vacancy (NV-) centers in diamond are used in a broad range of applications. They serve as qubits [1] or probes for various physical properties like magnetic field [2], electric field [3]. They can also be used to detect the properties of electronic and nuclear spins such as substitutional nitrogen (Pi) centers [4] or 13C atoms [5]. For all these applications it is important to know in detail the hyperfine structure of the NV--center.
The NV--center is an atom-like impurity in diamond crystal. The optical transitions of the NV- allow a high degree of spin polarization at room temperature via optical pumping. The electronic spin of the NV--centers is polarized into the ground-state magnetic sublevel ms = 0 under optical illumination and measured using optical detection techniques [6]. The NV--center has a ground state triplet l±1> and 10 > separated by zero-field splitting at D«2.87 GHz. In a magnetic field B along the N-V axis, the Zeeman splitting Ye B with electronic gyromagnetic ratio ye =28.025 GHz/T cancels the ground state zero-field splitting at a magnetic field at B ~ 1024 G, leading to a ground state level anti-crossing (LAC) between ms = l-1) and ms = 10} [7]. However, in the presence of magnetic field the NV- experiences a complex LAC, due to hyperfine interaction of the NV- electron spin with other spins.
The diamond lattice consist also other spins i. e., electronic and nuclear spins that cannot be initialized or read out optically. A key challenge is to
© Ivanov A.A., Ivanov A.I., Kulagina A.A., 2020
Вестник Балтийского федерального университета им. И. Канта. Сер.: Физико-математические и технические науки. 2020. № 1. С. 43 — 49.
transfer polarization controllably from bright NV spins to other spins. The authors of ref. [1] used the interaction of the electron spin NV--center with nearest 13C nuclear spin for the demonstration of quantum gate NOT and a conditional two-qubit gate. They used states of the form |ms)|mi), where ms = +1, mi = +1 /2. We will show that similar states can occur in a 15NV--center under appropriate conditions.
The basis of the presented approach of NV LAC investigation is a method based on a complete set of commuting operators (CSCO). To find an eigenvalues for a NV spin Hamiltonian, it is necessary to choose the spin basis functions. Usually, simple products of one-particle spin functions are used 44 as approximations for a many-particle functions. To obtain eigenvectors and eigenvalues of the spin Hamiltonian we introduce a method based on a complete set of commuting operators (CSCO). This method is well known in quantum mechanics for a long time, but has never been implemented in spectroscopy up until recently [8; 9]. The Hamiltonian in the presented approach is considered either a CSCO operator, or a function of CSCO. The properties of the spin states are uniquely determined by CSCO. Every eigenvector in this approach is determined by the unique value set of CSCO. Most of the resulting spin states are qualified as entangled spin states. The energy levels are found by solving a series of equations of less degree, than the ones found by diagonalizing the Hamiltonian using the numerical methods. It is also possible to obtain analytical expressions for some of the energy levels.
The NV--center in diamond consists of a nitrogen atom, which substitutes for a carbon atom, and a lattice vacancy. Its ground state is triplet state (S = 1) with an spin quantization axis provided by the NV--center axis of symmetry. We consider a single NV--center with a 15N nitrogen isotope having a nuclear spin I=1/2. The ground-state spin Hamiltonian of NV--center in the presence of magnetic field B reads as (as per [10], in frequency units):
H = D(SZ2 -S2/3) + A||SzIz + A±(SxIx + SyIy)+ yeSzBz + ynIzBz, (1)
where D«2870 MHz is the fine structure splitting, AM=3.03 MHz and A ± =3.65 MHz are the axial and non-axial magnetic hyperfine parameters, z-axis aligns with electronic spin quantization axis, y e = 28.025 GHz /T is the electron gyromagnetic ratio and yn = 4.316 kHz/mT is the 15N gyro-magnetic ratio.
To calculate the energy spectrum for the Hamiltonian we first determine the total spin operator:
J = S + I.
The operators J2,JZ,S2,I2 form a complete set of commuting operators (CSCO). The eigenvectors | J,Mz > of this CSCO takes the form
|3/2,3/2> = |1,1>|1/2,1/2>,
13/21/2) =
\
3|1,0)|1/21/2) + ^L|1,1)|1/2-1/2),
|3/2-1/2> = J 11,0)|1/2 -1/2) + ^ |1, -1>|1/2, 1/2),
|3/2-3/2) = |1,-1)11/2-1/2).
(2)
11/21/2) =
\
3|1,1)|1/2,-1/2)--1=|1,0)|1/2,1/2),
(3)
|1/2-1/2) = ^ 11,0)|1/2 -1/2) - J |1, -1)|1/21/2).
Note that the total spin J of the NV_-center is not preserved, as the Ham-iltonian (1) does not commute with the operator J2. At the same time, the Hamiltonian commutes with the projection of the total spin, the square of the electron spin and nuclear spin square:
[h,jz] = [ñ,§2] = [fí,í2] = o.
(4)
Set of operators H, Jz, S2, !2 is also a CSCO. This set has the unique system of eigenvectors: each eigenvector is characterized by a single set of commuting observables values. Consequently, the spin states of the NV--center are characterized by the energy E, the projection Mz of the total spin, the electron spin S, and the nuclear spin I: \E, MZ,S, /). Since for all of these states S = 1, I=1/2, then the equation for the eigenvalues and eigenvectors of the Hamiltonian (1) can be written as
H |E(0 E(0 |E(0 n pm,/ - m7 pm,/'
(5)
where the index (i) is introduced in order to distinguish the states with the even values of Mz and different values of energy E. Solving the equation (5) gives
E(±) _ 1/2 _
-D/3 - A, ,/2 + r e ±^D - A|| / 2 + r e - r n )2 + 2A,
/2,
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E(±) _ -1/2 _
-D/3 - A,, / 2 - r e ±J(D - A,, / 2 - r e + r n )2 +
2A,
/2,
E±3/2 = D/3 + A,,/2 ± r e ± fn/2,
(6)
|E3/2)= |U>|l/2,l/2>,
|E_3/2>= 11,-1)11/2,-1/2),
|E$) = Cjt|3/2, 1/2) + C±|1/2, 1/2), i-±i)/2) = Ci|V2, -1/2) + C2± 112, -1/2),
where re = yeBz, rn = ynBz. The states ^l^and |Ei~/2| are entangled 46 states.
The energy eM is a single-valued function of the total spin projection Mz. This means that if the projection Mz of the total spin in a certain state has a definite value, then the energy EM of this state also has a definite value. The opposite statement is also true: if the energy EM has a definite value, then the projection Mz also has a definite value. It is important to note that this function's uniqueness is violated, in particular, in a zero magnetic field. In this case the energy levels are pairwise equal and we have the complete level anti-crossing (CLAC):
E = E E(+) = E(+) E(-) = E(-) (7)
3/2 -3/2' 1/2 -1/2' M/2 _ -1/2- V'
When the equalities (7) are satisfied, all of these spin states do not have definite values for the total spin projection. The spin-lattice relaxation rates of all these states will increase, therefore, the ODMR spectrum of such an NV "-center will not have characteristic dips.
We further discuss the conditions under which one can observe the effective transfer of polarization from the electron spin to the nuclear spin in the
15NV "-center. Assuming A± =0 in (1), the Hamiltonian H of the 15NV "-center can be written in the following form:
H = D(SZ2 -S2/3)+A||SzIz + yeSzBz + ynIzBz. (8)
We note also that the set of the operators Sz, !z, S2, I2 is a CSCO. The Hamiltonian (8) commutes with all operators of this set. Consequently, the
operator H is a function of this operator set. The eigenvalues and eigenvectors of Hamiltonian (8) are:
E3/2 = D/3+A, ,/2 + re + rn/2, E-+/2 = D/3-A, ,/2-re + ^/2, E-3/2 = D/3 + A,, / 2 - fe - fn/2, E1/ =-2D / 3 + rn/2,
-3/2 ~ ^ / ^^ ^W / ^ * e M/2
E(1+/}2 = D/3 - A, ,/2 + re - rn /2, E--/2 =-2D/3 - rn /2, (9)
|E3/2)= |1,1>|1/2,1/2>, |Ei+1)/2) = |1,"1>|1/2,1/2>,
|E-3/2>= |1,-1>|1/2,-1/2>, |e^ = |1,0>|1/2,1/2>,
|1,1>|1/2,-1/2>, |Ei"1)/2)= |1,0>|1/2,-1/2>.
We see that in such a 15NV"-center there is a 100 % transfer of polarization from the electron spin to the spin of the 15N nucleus: all states are completely polarized and have the form |ms)|mj). States and under
optically pumped are effectively populated. The energies of these states do not depend on the parameter A,, . We note that in a zero magnetic field
these energies are equal e1/2 = E--/2 (LAC). Therefore, the observable Iz in
zero magnetic field is not a function of the observable E [11]. This means that
the quantum number mi in a zero magnetic field for the states and
is not has a definite value. We note that in a weak magnetic field (for
example, in the field of the Earth) these energies are close to each other:
E*/ « E-1)/2 (LAC). The energy splitting 5 = E*/ - E-1)/2at B = 0.5 G (Earth's
magnetic field at its surface) is only 200 Hz, therefore, the two peaks of the ODMR spectrum of such a 15NV"-center will overlap. This means that the quantum number mi in a weak magnetic field also is not a good quantum number. In contrast, the quantum number ms is a good quantum number.
For states and , the value of ms is 0 and, therefore, such a 15NV"-
center is a bright center. The quantum number mi for these states will be a good quantum number in a sufficiently strong magnetic field.
Typically, NV "-centers in diamond are mainly created by N+ ion implantation [12 — 14] or by nitrogen-doping during CVD growth [15]. Diamond substrates already contain NV"-centers. To distinguish between native and artificial NV "-centers, 15N isotopes are often used for the implantation. The different nuclear spins of I = 1 for 14N and I = 1/2 for 15N result in hyperfine triplet or doublet splittings in 14NV- and 15NV "-centers, respectively. In ref. [12] 14NV"-centers in diamond have been generated via 14N ions implantation and ODMR was used to measure the hyperfine splitting. The analysis of a number of NV "-color centers leads to the conclusion, that on average two nitrogen ions need to be implanted per 14NV"-center. 15NV"-centers were created in ref. [13] by implantation of the 15N+ ions. The analysis indicates that 1 in 40 implanted 15N+ ions give rise to an optically observable 15NV "-center. Many factors may influence the yields for 15NV "-centers. We consider an additional factor that can affect the yield of observable 15NV "-centers: LAC. The 15N ion beam used in the implantation process has strong anisotropy, therefore, it can be assumed that the isotropic contribution to the energy of the 15NV- center produced by implantation will be small.
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We investigated the properties of the spin states in single diamond 15NV--center at the LAC. Our approach uses a complete set of commuting operators. Each state is characterized by a single set of the values of CSCO. The uniqueness of this set of values is violated, in particular, in a zero magnetic field. In this case the energy levels are pairwise equal and can be considered a special case of a level anti-crossing. We have shown that under certain conditions in 15NV—center it is possible to obtain a 100 % transfer of polarization from the electron spin to the spin of the 15N nucleus. We believe that these conditions can be satisfied for 15NV--centers obtained by implantation.
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References
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The authors
Alexander A. Ivanov, Programmer, Kode LLC, Russia. Email: aivanov023@gmail.com
Prof. Ivanov I. Alexey, Immanuel Kant Baltic Federal University, Russia. E-mail: aivanov@kantiana.ru
Anastasia A. Kulagina, Assistant Professor, Immanuel Kant Baltic Federal Uni-versity,Russia.
E-mail: alebedkina@kantiana.ru
Об авторах
Александр Алексеевич Иванов — программист, ООО «Кодэ», Россия. E-mail: aivanov023@gmail.com
Алексей Иванович Иванов — д-р физ.-мат. наук, Балтийский федеральный университет им. И. Канта, Россия. E-mail: aivanov@kantiana.ru
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Анастасия Алексеевна Кулагина — университет им. И. Канта, Россия. E-mail: alebedkina@kantiana.ru
ст. преп., Балтийский федеральный
