D Minenkov
email: minenkov.ds@gmail.com

M Rouleux
email: rouleux@univ-tln.fr

Quantum vorticity at positive temperature for spins systems with continuous symmetry

We propose a definition of vorticity at inverse temperature β for Gibbs states in quantum XY or Heisenberg spin systems on the lattice by testing exp[-βH] on a complete set of observables ("one-point functions"). Imposing a compression of Pauli matrices at the boundary, which stands for the classical environment, we perform some numerical simulations on finite lattices in case of XY model, which exhibit usual vortex patterns.

Introduction.

Consider the quantum XY or Heisenberg spin model for S = 1/2 on the 2-D lattice Z 2 , with nearest neighbor interactions. Marmin-Wagner, and Hohenberg theorems tell us that Gibbs states, for all inverse temperature β, are invariant under simultaneous rotation of spins (absence of continuous symmetry breaking in two dimensions). In the classical case, we know a bit more : although there is a unique Gibbs state, with rotational symmetry, which rules out the existence of first order transitions, a particular form for phase transition exists, characterized by a change of behavior in the correlation functions. For the 2-D rotator, it has been described by Berezinskii [3], and Kosterlitz-Thouless [11] in term of topological excitations, called vortices [9]. For Heisenberg model, we observe higher order topological defects, called instantons [2].

In this Report, we make a first attempt to answer the natural question : How can we observe vorticity in the quantum case?

Let us first consider a system in finite volume Λ ⊂ Z 2 . The Hamiltonians we are interested in are of the form H Λ (Φ) = -X⊂Λ Φ(X), where Φ is an "interaction" between sites in Λ. For nearest neighbor interaction, the contributing X are pairs i, j , and the Hamiltonian reads

H Λ = - 1 2 i,j ⊂Λ (σ x i ⊗ σ x j + σ y i ⊗ σ y j + uσ z i ⊗ σ z j ) ( 1
)
where u is a coupling constant, u > 0 in the ferromagnetic case (u = 1 when isotropic), u < 0 in the anti-ferromagnetic case (u = -1 when isotropic), and u = 0 is the XY model. We could also add an external magnetic field h i∈Λ σ z i to H Λ , and in case u = 0, allow for anisotropy between x and y components.

Though vortices can merge spontenaously in infinite volume, there are external fields that would certainly enhance vorticity. General external fields are defined within the notion of a "state" [START_REF] Simon | The Statistical Mechanics of Lattice Gas I[END_REF], Sect.II.3.

Throughout we denote by Tr the ordinary trace, and by tr the normalized trace, as tr(A) = 1 d Tr(A), where d is the dimension. Recall from [START_REF] Simon | The Statistical Mechanics of Lattice Gas I[END_REF], Sect.II.1 the partial trace: if A is a linear operator on K 1 ⊗K 2 , the partial trace tr K 1 or simply tr 1 is an operator L(K 1 ⊗K 2 ) → L(K 1 ) defined by the requirement :

tr K 1 B(tr 1 (A)) = tr K 1 ⊗K 2 ((B ⊗ 1)A), B ∈ L(K 1 ) (2)
A quantum state ρ assigns to each finite X ⊂ Z 2 an operator ρ X with Tr(ρ X ) = 1 and tr

H Y (ρ X∪Y ) = ρ X on H Y = ⊗ i∈Y C 2 i for all disjoint X, Y ⊂ Z 2 . (Instead of tr H Y (A)
, we use also the notation tr Y (A). ) Given a state and the finite interaction Φ(X), we define the Hamiltonian on all of

Z 2 H ρ Λ (Φ) = - X∩Λ =∅ Tr X\Λ (1 ⊗ ρ X\Λ )Φ(X) (3)
that couples Λ with the external field ρ through its nearest neighbors at the boundary. For A a quasi-local observable on Z 2 , we define the expectation value

A ρ β,Λ = Tr Λ exp -βH ρ Λ (Φ) A Tr Λ exp -βH ρ Λ (Φ) (4)
and A ρ β,Λ has a limit as |Λ| → ∞. Such a state has been constructed in [1] for the XY chain.

From a practical point of vue however, it is suitable to produce explicit "approximate states" that will favour the existence of vortices in finite volume; we proceed in the following way. Let Λ ⊂ Z 2 be the "small system", and ∂Λ ⊂ Z 2 its "environment", both finite. On ∂Λ, we "compress" the spin operators, so that the measure of observable "direction of spin" is deterministic for j ∈ ∂Λ, and quantum for j ∈ Λ. The resulting Hamiltonian H Λ∪∂Λ (Φ) accounts for the interaction with the approximate "external field" on ∂Λ, as in . Now, in finite volume Λ ∪ ∂Λ, the only (normalized) Gibbs state is given by

A → ω β (A) = tr(exp[-βH Λ∪∂Λ ]A) tr(exp[-βH Λ∪∂Λ ] (5)
and called the "canonical Gibbs state". We shall actually define vorticity at inverse temperature β by decomposing the linear form ω β on a canonical (orthonormal) basis of observables.

Vorticity matrices

Gibbs state (1) for spin S = 1/2 systems, as a linear form on the C * -algebra of observables

O = ⊗ j∈Λ∪∂Λ o j , o j = M 2×2 (C)
("quasi-local observables" if we were to consider the thermodynamical limit, ) can be decomposed in a canonical basis. The simplest way is to restrict to "one-point functions", i. a) The XY model

We first consider the XY model, where we can restrict to real 2×2 matrices. The compression of Pauli matrices on ∂Λ is given by the orthogonal projector

Π(θ) = cos 2 θ sin θ cos θ sin θ cos θ sin 2 θ (6)
where θ parametrizes a point on the unit circle, accounting for the prescribed direction of "vorticity" on ∂Λ. Let again O R ⊂ O be a real sub-algebra O, of real dimension 4N .

Example 1: O R is the "canonical" algebra, generated by real matrices (D i ) i∈Λ∪∂Λ , whose all non-diagonal 2 × 2 blocks vanish, and all diagonal 2 × 2 blocks vanish, except this supported on site i that takes values in {δ 1 , δ 2 , δ 3 , δ 4 }, where

δ 1 = 1 0 0 0 , δ 2 = 0 1 0 0 , δ 3 = 0 0 1 0 , δ 4 = 0 0 0 1
So the family of block-diagonal 2N × 2N matrices with 2 × 2 entry δ j , 1 ≤ j ≤ 4 at the i:th place, 1

≤ i ≤ N D i j i∈Λ∪∂Λ = 0 ⊕ • • • ⊕ δ j ⊕ • • • ⊕ 0 (7)
gives an orthonormal basis (ONB) of 1-point functions O R .

Example 2: O R is the (real) algebra generated by Pauli matrices ( D i ) i∈Λ∪∂Λ with diagonal block supported on site i that takes values in {Id, iσ x , iσ y , iσ z }.

We shall restrict to the canonical algebra, whose generators enjoy the nice property of being real matrices. Let also o R ⊂ o be the algebra of 2 × 2 matrices with real coefficients, endowed with the scalar product (A|B) = Tr(B * A), which is isometric with R 4 . By extension, the basis δ = {δ 1 , δ 2 , δ 3 , δ 4 } of o R will be called an "elementary basis" of O R , since N copies of δ, attached to each site i, give a basis (D i j ) i∈Λ∪∂Λ,1≤j≤4 of O R . We say the same thing of any other ONB

b = {b 1 , b 2 , b 3 , b 4 } of o R ,
and of the corresponding basis (B i j ) i∈Λ,1≤j≤4 of O R , where B i j is defined as in (7), with b j instead of δ j . Actually, the order of the elements of b matters, so we prefer to think of b as an "array", namely

b = b 1 b 2 b 3 b 4 (8)
where b k is of the form

b k = b 1k b 2k b 3k b 4k ∈ M 2×2 (R)
which we identify with the vector b k = t b 1k , b 2k , b 3k , b 4k . We will not use the algebraic structure of o R . With the notations above (partial traces), we see easily that :

tr 1 (b) = trb 1 trb 2 trb 3 trb 4 (9)
which justifies the interpretation of b as a matrix (operator). So tr 1 are the components, in some matrix representation, of the usual trace ("tracial state") on o R . For simplicity we set T (b) = tr 1 (b) and call it the "matrix of traces". An important rôle will be played with symmetric basis. has a degenerate eigenvalue, that is, is a multiple of identity.

Most of the basis are not symmetric, but occasionally we can make them symmetric, by permuting or multiplying by -1 some elements. We can characterize δ-symmetric basis: namely, if b is δ-symmetric, then modulo such transformations, there exists discrete or one-parameter families P s ∈ O(2; R) such that b = b s = t P s δP s (10) (where the product is understood as if b j 's were numbers).

So far we have constructed "one point functions", i.e. a basis of O R . In the sequel we content with Hamiltonians of type (1) which are of second order in the interactions; if we were to include the linear term i∈Λ σ z i we would write it as i,j

1 i ⊗ σ z j . Embed O R into O R ⊗ O R by the usual coproduct ∆, and set x = ∆(x) = 1 2 (1 ⊗ x + x ⊗ 1) ∈ o R ⊗ o R , for x ∈ o R . So we have "lifted" b = ∆(b) as a family of O R ⊗ O R by ( B i j ) i∈Λ,1≤j≤4 , with B i j = ∆(B i j )
. With the notations of ( 8) and ( 9) we have

B i = B i 1 B i 2 B i 3 B i 4 ∈ M 4N ×4N (R), tr 1 ( B i ) = tr B i 1 tr B i 2 tr B i 3 tr B i 4 ∈ M 2×2 (R) (11)
In the same way, we form e -βH B i j , so we can map to each site i ∈ Λ ∪ ∂Λ a 2 × 2 matrix :

tr 1 (e -βH B i ) = tr(e -βH ) ω β ( B i 1 ) ω β ( B i 2 ) ω β ( B i 3 ) ω β ( B i 4 )
Definition 1.2: We call vorticity matrix at site i, relative to the basis b, at inverse temperature β, the matrix :

Ω i β (b) = tr 1 (e -βH B i ) tr(e -βH )
The traceless matrix

Ω i β (b) = Ω i β (b) -tr Ω i β (b) Id ( 12
)
is called the reduced vorticity matrix at site i.

Example: Λ = {1, 2} is a lattice with 2 sites, ∂Λ = ∅, one has Ω 1 β (δ) = Ω 2 β (δ) = 0. This is observed also numerically for all Λ, with ∂Λ = ∅, although vortices could merge sponteanously in infinite volume. If b is a symmetric basis of o R , then Ω i β (b) and Ω i β (b) are Hermitean since H is self-adjoint (real symmetric if moreover H has real coefficients), and

Ω i β (b) 2 = det Ω i β (b)Id ( 13
)
Thus Ω i β (b) enjoys the nice property, to be diagonalizable with real (opposite) eigenvalues for all sites i, and all inverse temperature β. Viewing these as a field of matrices over the lattice, we can figure out the "vorticity" of the system, by simply looking at their principal directions. We will see that it also gives a measure of vorticity, i.e. numbers (integers) that should be independent of the choice of "elementary" basis b. Next we define vortices as the set of sites where the reduced vorticity matrix is singular. By construction, all sites are vortices when β = 0. Now we turn to consistency of Definitions 1.2 and 1.3 relatively to the choice of b within δ-symmetric basis. That b is a δ-symmetric basis is a natural requirement for computing the degree of Ω i (b), see Sect.2. With b written as in (8), and P ∈ O(2; R), we set with the notations of (10) a = t P bP . The same holds after taking the co-product ∆ of each term, i.e. a = t P bP . This defines conjugacy classes, which pass to the partial traces (9), i.e. T (a) = t P T (b)P , and

T ( a) = t P T ( b)P . Moreover, if X ∈ L(R 2 ), we have (1 ⊗ X)b = Xb 1 Xb 2 Xb 3 Xb 4 = (1 ⊗ X)P a t P (14)
After lifting a and b to O R ⊗ O R , (11) becomes

A i = t P B i 1 B i 2 B i 3 B i 4 P ∈ M 4N ×4N (R), tr 1 ( A i ) = t P tr B i 1 tr B i 2 tr B i 3 tr B i 4 P ∈ M 2×2 (R) (15)
and ( 14) also extends when taking X ∈ L(R 4N ) and replacing a by A i , b by B i . Let now X = e -βH , we obtain that conjugacy classes again pass to vorticity matrices, i.e.

Ω i β (a) = t P Ω i β (b)P, Ω i β (a) = t P Ω i β (b)P (16)
and ( 10) eventually gives :

Proposition 1.4: Definitions 1.2 and 1.3 are consistent, i.e. vorticity matrices relative to all δ-symmetric ONB b are related by (16) for some P s ∈ O(2; R), and in particular ξ is a vortex relative to δ iff this is a vortex relatively to any δ-symmetric b.

Moreover we have the numerical evidence that, among all δ-symmetric basis b, the canonical basis δ is most "faithful", in the sense that Ω i β (δ) have on the boundary lattice ∂Λ the same principal directions as the directions along which Pauli matrices are compressed (associated with the eigenprojector Π j ).

b) Heisenberg model

The algebra goes essentially along the same lines, except for the fact that the basis b cannot be real. Again, this relies on the observation that the spin representation of SU( 2 where (θ, ϕ) parametrizes a point on Bloch sphere, accounting for the prescribed direction of "vorticity" on ∂Λ.

For both XY and Heisenberg model, we observe that (reduced) vorticity matrices belong to su(2) = {M ∈ M 2×2 (C) : M * = M, trM = 0} which is the tangent Lie algebra of SU(2).

Topological degree and holonomy on SU(2)

The natural idea is to consider vorticity matrices as a map Λ ∪ ∂Λ → su (2) and "integrate" it, so to get some topologial invariant, such as the local degree. This assumes a thorough knowledge of discrete analysis on the lattice, with values in su (2). For advanced results on Differential Calculus on lattices in the scalar case, see [START_REF] Smirnov | Discrete complex analysis and probability[END_REF]. The non-commutative discrete case has still to be set up, so we pass here to an idealistic continuous limit, where vorticity matrices would be defined as a smooth field on R 2 (away from vortices), valued in the Lie algebra su(2). Our purpose is to integrate such fields vanishing at some points, and define a "non-commutative degree".

We consider Heisenberg model (Hermitean vorticity matrices), the XY model (real symmetric vorticity matrices) will be treated as a particular case. So let M :

D ⊂ R 2 → su(2), x → M (x) be a C 1 map, M (x) 2 = λ(x)Id, λ(x) ≥ 0, and consider ρ ∈ Λ 1 (R 2 ; su(2)) the 1-form ρ(x) = 1 2 (M -1 (x)dM (x) -dM (x)M -1 (x)) (20)
(anti-symmetrized Maurer-Cartan form). We have

M -1 (x)dM (x) + dM (x)M -1 (x)) = dλ(x) λ(x) . Let M = a b c -a , we compute dρ = -λ -2 (adb ∧ dc + bdc ∧ da + cda ∧ db)M (21)
(so dρ = 0 if M is symmetric). On the other hand, computing the structure coefficients for the Lie algebra su(2), we find dρ

+ [ρ, ρ] = 0 (22)
Recall that if G is a Lie group, and G its Lie algebra, ω the canonical Maurer-Cartan form on G, invariant by left translations, we define Darboux differential of the map f ∈ C 1 (D; G) by π f = f * ω. The fundamental existence theorem ("Poincaré lemma"), with a differential form ρ ∈ Λ 1 (D; G) verifying (22), assigns (locally) a map f ∈ C 1 (D; G), whose Darboux differential is precisely equal to ρ. Moreover this map is unique when prescribing its value at a point x 0 ∈ D.

Applying this result to (20), gives local primitives of ρ, whenever λ(x) = 0, called a "logarithm" of M , which belong to SU(2).

Remark: For these computations we can also use the isomorphism θ

: (R 3 , ∧) → (su(2), i 2 [•, •]
), where ∧ is the usual wedge-product on R 3 .

For the XY model on R 2 ("idealized" lattice Z 2 ) we can define the square of the "local degree" at an isolated singularity (vortex) ξ ∈ R 2 , by

s 2 ξ = det 1 2π γ ρ(x) (23)
where γ encircles ξ, and similarly, when M elliptic at infinity |λ(x)| ≥ C > 0, |x| ≥ r 0 the square of the "total degree"

s 2 ∞ = det 1 2π |x|=r ρ(x), r > r 0 (24)
Since the fundamental group of the universal covering of SU( 2) is Z, we can conclude that s ξ , s ∞ ∈ Z.

Example: For the symmetric matrices

M a (x) = a cos nθ sin nθ sin nθ -a cos nθ we have deg ∞ (M ) = n. The 1-form ρ associated wih M 0 is simply 0 n -n 0 dθ.
This makes sense also for Heisenberg model on Riemann's sphere S 2 , provided λ(x) > 0 everywhere, but we should speak of "instantons" rather than of "vortices", see [2], [7].

Example: For Hermitean matrices

M a (x) =
a cos θ e idϕ sin θ e -idϕ sin θ -a cos θ

where x = cos θ e iϕ sin θ e -iϕ sin θcos θ ≈ (θ, ϕ) ∈ S 2 , we have deg ∞ (M ) = d. The condition M (0, ϕ) = -M (π, ϕ) reproduces the condition of [2] that all (classical) spins point upwards at infinity (θ = 0) while they point downwards at 0 (θ = 0), so that the equilibrium state at inverse temperature β is a spin wave, or "instanton", of degree d (the number of coverings of the sphere).

Other topological defects, such as "lines of vortices" occur in Heisenberg model on R 3 ("idealized" lattice Z 3 ).

Among possible extensions, we mention: (1) the orbital compass model, which has reflection posivity, but no rotation invariance [4]; (2) Hubbard model with continuous symmetry (hopping and spin interaction) [10], with an application to the dynamics of Cooper pairs in a supraconductor bulk, or the dynamics of electron/hole pairs in SNS junctions. Vorticity can also defined for maps R (2n+1)n → sp(2n; R) (Hamiltonian matrices) and as well in general gauge sigma models [6].

Numerical tests for the XY model

Recall we have completed the lattice Λ with an environment ∂Λ ⊂ Z 2 where Pauli matrices are compressed in directions (θ j ) j∈∂Λ , i.e. we change σ by Π θ σΠ θ , where Π θ as in (6). Thus

σ x i (θ i ) = (sin 2θ i )Π θ i , σ y i (θ i ) = 0
Hamiltonian (1) with nearest neighbor interaction has too large a kernel, to be suitable for numerical simulations, even when modified by an external field. As in QFT we could try to remove the "artificial" part of ker H by reducing the Hilbert space H = C 4N to a "physical space", but H is not positive in the form sense. So a first attempt to lift the degeneracy of the spectrum of the Hamiltonian, and enhance the effects of the external field on vorticity, is to change (1) to the anisotropic XY model. So for n, k > 0, we consider . We consider rectangular lattices of minimal sizes to exclude important volume effects, with sufficiently large ∂Λ to constrain the "quantum system" within Λ. We choose θ j = dω j + φ where ω j is the polar angle representing the vector j ∈ ∂Λ.

H (n,k) (σ|∂Λ) = -1 2(n+k) i,j ;i,j∈Λ (n σ x i ⊗ σ x j + k σ y i ⊗ σ y j ) (26) -1 2(n+k) i,j ;(i,j)∈Λ×∂Λ n (σ x i ⊗ σ x j (θ j ) + σ x j (θ j ) ⊗ σ x i ) -1 2(n+k) i,j ;i,j∈∂Λ n σ x i (θ i ) ⊗ σ x j (θ j ) so H (n,k) (
We study Gibbs state at inverse temperature β, with significant results provided β ranges in some interval, for which however, there is no evidence (even in an approximate sense) of a second order phase transition. Computing 1 2π γ ρ as a discrete integral along a contour γ ∈ Λ, not too far from ∂Λ (in practice, 2 or 3 layers), it turns out that the computed degree is close to this we would obtain in (25). The main flaw affecting the computations is due to the fact that eigenvalues of Ω i β (b) are decaying exponentially when approaching the center of Λ. This we partially compensate by considering the anisotropic model. To make our vorticity patterns more demonstrative we consider the case of high anisotropy with k = 10. In Table 1 below we give the results for degree, computed along a cycle γ ⊂ Λ consisting of the rectangle of the first or the second neighbors to the boundary for different values of the anisotropy parameter n = 1, k = 2; 10. Inverse temperature is β = 1. Let us finally discuss the antiferromagnetic model. It is known that on Z 2 , the unitary transformation U consisting in flipping the spins at sites i with i odd (i.e. indices i = (i 1 , i 2 ) such that |i| = |i 1 | + |i 2 | is odd) intertwines the ferro with the antiferromagnetic models. More precisely, -H = U * HU . The reason is that Z 2 e and Z 2 o (the even and odd lattices) are swapped into each other by symmetries on the lines x = n + 1/2 or y = m + 1/2 (called the "chessboard symmetry"). There follows that tr exp[βH]A = tr exp[-βH]U AU * , and if A = D i (the canonical basis), we can check U AU * = A so the matrices of vorticity (for the Hamiltonian with free boundary conditions) are the same. This equivalence holds also in the case of the torus, but not on Λ ⊂ Z 2 with an odd number of sites. Of course, when ∂Λ = ∅, H and -H are not so simply related; nevertheless, we may observe (numerically) that the relation Ω i β (δ) = Ω i -β (δ) holds with a very good accuracy.

  e. the set O ⊂ O of 2N × 2N , block-diagonal 2 × 2 matrices (Pauli matrices are spin representations of SU(2) of dimension 2S + 1 = 2), supported on individual sites of Λ ∪ ∂Λ, N = |Λ ∪ ∂Λ|.

Definition 1 . 1 :

 11 We call the ONB b symmetric iff the corresponding matrix b in (9) is Hermitian, i.e. b 1 = b * 1 , b 4 = b * 4 , and b 3 = b * 2 . We call it δ-symmetric if moreover b is real, and T (b)

Definition 1 . 3 :

 13 We say that ξ ∈ Λ is a vortex at inverse temperature β relative to the δsymmetric ONB b iff Ω ξ β (b) has a degenerate eigenvalue, i.e. Ω ξ β (b) = 0. We call regular the other points.

= δ 1 δ 2 δ 3 δ 4 , we consider the 8 × 8 array e = e 1 e 2 e 3 e 4 θ 2 e -iϕ sin θ 2 cos θ 2 e -iϕ sin θ 2 cos θ 2 sin 2 θ 2 (

 4842222 ) is 2-D, so the "one-point functions" can be simply parametrized by 2 × 2 matrices. Instead of 4 × 4 arrayb 3 = (e ′′ 2 ) *(18)this choice being non-unique. Moreover the e j 's have the right dimension for quadratic interaction, so we don't need to take co-product as in the case of XY model. Compression of Pauli matrices on ∂Λ can be obtained by the orthogonal projector Π(θ, ϕ) = cos 2

  σ|∂Λ) is self-adjoint and real. For k = 1, we call H (1,k) (σ|∂Λ) the anisotropic XY model. Only when ∂Λ = ∅, H (1,k) is unitarily equivalent to H (k,1)

Table 1 .

 1 Table of calculated degree for different values of anisotropy factor k for the first and the second neighbors to the boundary. Here the case of |Λ ∪ ∂Λ| = 23 × 33 with 2 boundary layers is considered, and β = 1. st neighbours 1 st neighbours 2 nd neighbours 2 nd neighbours

	Given	k = 2	k = 10	k = 2	k = 10
	degree 1 1	1.05	1.09	0.89	1.05
	2	1.98	2.03	1.70	1.78
	3	2.76	2.75	2.01	2.50


A Savoyant

H Alnoor

O Pilone

O Nur

M Willander

Core-defect reduction in ZnO nanorods by cobalt incorporation

Keywords: numbers: 61.46.Hk, 61.72.uj, 75.75.Cd, 76.30.Fc

Zinc oxide (ZnO) nanorods grown by the low-temperature (90 • C) aqueous chemical method with different cobalt concentration within the synthesis solution (from 0 % to 15 %), are studied by electron paramagnetic resonance (EPR), just above the liquid helium temperature. The anisotropic spectra of substitutional Co 2+ reveal a high crystalline quality and orientation of the NRs, as well as the probable presence of a secondary disordered phase of ZnO:Co. The analysis of the EPR spectra indicates that the disappearance of the paramagnetic native core-defect (CD) at g ∼ 1.96 is correlated with the apparition of the Co 2+ ions lines, suggesting a gradual neutralization of the former by the latter. We show that only a little amount of cobalt in the synthesis solution (about 0.2 %) is necessary to suppress almost all these paramagnetic CDs. This gives insight in the experimentally observed improvement of the crystal quality of diluted ZnO:Co nanorods, as well as into the control of paramagnetic defects in ZnO nanostructures.

I. INTRODUCTION

Zinc oxide (ZnO) micro and nanostructures are highly interesting materials since they offer a wide variety of morphologies 1 (rods, dots, rings, etc.), tunable semiconducting and optical properties when doped 2,3 , are health friendly and of low fabrication cost 4 . In particular, ZnO nanorods (NRs) are the focus of great attention because their one-dimensional character allows researchers to obtain coherent orientation of them, while keeping high surface area/volume ratio and interesting piezoelectric properties. Currently, ZnO NRs are involved in many functional electronic devices such as light emitting diode 5 , fast UV detectors and pressure sensors 6 . In these applications, native (intrinsic) or intentionally introduced (extrinsic) defects play a major role. For instance, in all luminescence phenomena, intrinsic and extrinsic defects in ZnO are involved as steps in the excitation and/or recombination paths, hence influencing the absorbed or emitted light properties (photon energy, relaxation time, etc.). On the other hand, electrical conduction in semiconductors is entirely driven by the ability of these defects to provide delocalized electrons (holes) in the conduction (valence) bands. Consequently, the knowledge and control of intrinsic and extrinsic defects, as well as the relation between both, are key issues for the electronic and optical optimization of ZnO nanostructures.

The case of paramagnetic defects is of particular interest because, when subjected to an external applied magnetic field, a majority of them (depending on temperature) have up (or down) spin projection, which then restricts the possible recombination paths via these polarized defects to those allowed by the Pauli exclusion principle.

Regarding these paramagnetic defects, electron paramagnetic resonance (EPR) is a precious tool to characterize them and to study their occurrence as a function of the various synthesis parameters (synthesis temperature, stirring time, dopant, etc.). Simultaneously, EPR spectra can give information about the spin environment, the crystal quality at the atomic scale, and the macroscopic ordering. However, applying EPR spectroscopy on paramagnetic nanostructures is difficult, mainly because of the low amount of resonating centers and the large spin relaxation. Moreover, as most of the studies deal with nonoriented nanoparticles, information on magnetic anisotropy is often lost [7][8][9] . Nevertheless, in a previous study 11 based on EPR measurement of well-oriented ZnO:Co NRs, we have successfully used the Co 2+ signal to prove the incorporation of these ions in well-oriented crystalline NRs phase, and have suggested that the drastic reduction of the intrinsic core defects (CD) at g ∼ 1.96 of ZnO nanostructures observed in Co-doped samples was related to this cobalt incorporation. The fine structure and magnetic anisotropy of the g ∼ 1.96 signal has been in addition determined.

In this work, by a systematic examination of the EPR spectra of ZnO:Co NRs grown with varying the cobalt concentration in the synthesis solution, from 0 % to 15 %, we prove that the apparition of the substitutional Co 2+ signal is correlated with the disappearing of the paramagnetic CD signal at g ∼ 1.96. The analysis of the Co 2+ signal brings information about the ordered phase of NRs and suggests the presence of an additional disordered phase of the same ZnO:Co system. In addition to give some insight into the observed augmented crystal quality of low Co-doped ZnO NRs 12 , this work opens the path to accurately control the number of these paramagnetic CDs with relatively small amount of dopant incorporation.

II. METHODS

The samples under study are made of ZnO:Co NRs of length and diameter 3.5 µm and 200 nm, respectively, grown on a ZnO nanoparticles (20-50 nm of diameter) seed layer, deposited on a 3mm×3mm×0.43mm sapphire substrate. These samples, with different Co concentration within the synthesis solution, were grown by the lowtemperature (90 • C) chemical aqueous method described in Ref. 11. Scanning Electron Microscopy (SEM) images of 0.2, 2, 10 and 15 % samples are shown in Fig. 1. The EPR measurements were performed on a conventional Bruker ELEXSYS continuous wave spectrometer operating at X-band (ν = 9.38 GHz) using a standard TE 102 mode cavity. The orientation of the samples was insured by manual goniometer with a precision of 0.5 • , and allow to set the angle between the applied magnetic field B and the caxis of the wurtzite ZnO nanocrystals, parallel to the NRs and perpendicular to the substrate surface. Due to the low Co and CD absolute number, signals have been accumulated to increase the signal/noise ratio. All measurements have been performed at a temperature of 6 K. As no reference line was added to the spectra, the exact magnetic field is not known with precision lower than 2-4 G, and the absolute g-factor values may have to be slightly corrected by an amount of 0.002. The scope of this work is to compare the EPR line's intensity of the substitutional Co 2+ ions to that of the CD when the Co concentration within the synthesis solution is varied, in order to compare the respective number of these extrinsic and intrinsic defects. The EPR intensity of a given line can be expressed as I = K.W.f (T ).N , where K is a constant depending on experimental condition, W is the transition probability, f (T ) is a temperature-dependent function, and N is the number of resonating spins (see Ref. 11). By recording all spectra at the same temperature (6 K), scaled to the same experimental conditions (10 dB attenuation, 5 G modualation), and assuming that transition probabilities are of the same order (W co ∼ W cd ), the ratio I co /I cd will give the ratio of extrinsic/intrinsic defects numbers. Even if these transition probabilities are substantially different, as they are regarded as constant quantities, the evolution of the intensities ratio will reflect the evolution of the defects numbers ratio. In order to determine the EPR intensities, two main methods are usually employed: (1) the double integration procedure or (2) the fitting of each spectrum by an appropriate simulation. However, the first method demands high signal/noise ratio with negligible baseline, while the latter requires precise and certain models for each lines. Both of these methods are inappropriate in our case, firstly because the recorded signals are very noisy due to the low resonant centers quantity in NRs and, secondly, because some of the lines are a superposition of several lines (see below), which makes the simulation difficult by increasing the number of adjustable parameters. In addition to these constraints, the chosen method must be identically used for each line. We then adopt a third solution, which is based on roughly estimate a line's intensity by multiplying its width by its peak-to-peak height. This method is only rigorously valid for a single 1/2 spin line, symmetric and without any substructure. Nevertheless, in the following, all intensities will be calculated by this third method, and it will be assumed to give reasonable estimation for the relative intensities and their evolution.

III. RESULTS AND DISCUSSION

The typical EPR spectra of the samples under study are shown in Fig. 2. The substitutional Co 2+ ions give highly anisotropic signal, characterized by an eight-lines structure at B ∼ 300 mT for θ = 0 • (Fig. 2-c) and a broad asymmetric single line at B ∼ 150 mT for θ = 90 • (Fig. 2-b), as previously observed in similar samples 11 . When the Co concentration within the synthesis solution is decreased, the Co 2+ signal decreases (Fig. 2-a) while the CD signal begins to appear at B ∼ 346 mT (g ∼ 1.96). This latter reaches a maximum intensity for pure ZnO NRs. In some of the samples (10 % and 0.5 %), point defects in the crystalline sapphire substrate give line very closed to the CD signal for the θ = 0 • orientation (see Fig. 2). Fortunately, this sapphire defect (SD) line cannot be mistaken with the CD one, because of its great and known anisotropy: rotating the sample by 90 • with respect to the magnetic field shifts the SD line to B ∼ 175 mT field, while the CD line practically does not move (Fig. 2). For this reason, the θ = 90 • orientation is preferable in order to study the CD line's intensity without signal interference. In the following, we begin by analyzing the Corelated signal and discuss some of its characteristics and, after that, we focus on the CD signal. The Co 2+ spectra for both orientations, and for each sample, are all shown in Fig. 3. Their intensities, line width and position (g-factor) are reported in Tab. I. Regarding these Co 2+ spectra, two interesting facts have to be pointed out. Firstly, and contrary to what can be expected, the I ∥ co /I ⊥ co ratio is not constant in each sample, and, secondly, the Co 2+ line at 150 mT (for θ = 90 • ) displays a pronounced asymmetry (positive and negative parts are not equal). We will see that these two points are likely to reflect the presence of a secondary, disordered, phase of ZnO:Co. Regarding the first point, although the number of substitutional Co 2+ varies from one sample to another, if these ions are sufficiently diluted in the same ordered ZnO matrix, the I ∥ co /I ⊥ co must be constant. As it is not the case in our samples, it is interesting to evaluate this ratio for a perfectly crystalline ZnO:Co system. We do that by estimating the I ∥ co /I ⊥ co ratio with the same method in single crystalline ZnO:Co microwire grown by optical furnace method (see Fig. 2-d), and find a ratio of I ∥ co /I ⊥ co = 1.3. This value has to be seen as a lower limit because the concerned microwire has been proved to be inhomogeneous in Co doping 13 , so that several Co 2+ pairs contribute to the 150 mT line at θ = 90 • leading to an overestimation of I ⊥ co . Compared to this value, the estimated ratio in all our samples is systematically lower: 0, 0.85, 0.96, and 0.55 for the 0.5 %, 2 %, 10 %, and 15 %, respectively. This variation can be explained if, in addition to the well-orientated ZnO:Co NRs giving rise to I ∥ and I ⊥ , there exists another phase of the same ZnO:Co system which only contributes to I ⊥ by augmenting its intensity and breaking the line symmetry. The variation of the I ∥ co /I ⊥ co ratio would then reflects the variation in the relative proportion of the two phases. The other mentioned point (asymmetry) tends to support and to specify this hypothesis. Indeed, while the eight-lines structure recorded for the θ = 0 • orientation clearly arise from substitutional Co 2+ ions in orientated crystalline ZnO NRs, this is not the case for the broad line at B ∼ 150 mT. Simulation and experiment on bulk ZnO:Co (Fig. 2-d) show that the asymmetry and the relative intensity of this latter line are not compatible with the simple rotation of the orientated ZnO:Co NRs, which would give a symmetric line of weaker intensity. Keeping in mind that the EPR lines are derivative of the absorption lines, the observed asymmetry is the consequence of a wide distribution in some of the Co 2+ parameters, which is most likely to be an angle distribution, indeed very closed to that of a powder spectra. Such a powder spectrum would precisely gives an isotropic and asymmetric line at 150 mT and, actually, a trace of it can be seen in the spectrum of Fig. 2-c (10 %, 0 • ), unfortunately not sufficiently resolved. We are then led to conclude that, in addition to the well-orientated ZnO:Co NRs phase, there exists a little disordered phase of the same system which contributes to the 150 mT Co 2+ line. The better evidence for that is the 0.2 % sample spectra, in which only the broad line at 150 mT is visible (see Fig. 3). The nature of this disordered ZnO:Co phase has not been elucidated, but is likely to be a set of ZnO:Co nanoparticles or badly oriented NRs distributed within the sample. To accurately quantify the relative amount of these two phases would require precise simulations of betterresolved spectra, recorded with additional intermediate orientations. Such simulations would be difficult to perform because of the anisotropy of the line width (not taken into account in common powder simulations) and of the required knowledge of the perpendicular hyperfine constant A ⊥ which, moreover, may be different in the ordered and disordered phases. As a consequence, we state that while the Co intensity at θ = 0 • is proportional to the number of Co 2+ ions in the well-orientated phase, the Co intensity at θ = 90 • is proportional to the whole number of Co 2+ , within the ordered and disordered phases. Then, for the aim of comparing the total number of incorporated Co 2+ ions to the number of CDs, we will use the cobalt intensity recorded at θ = 90 • .

Table I: Evolution of EPR lines parameters with increasing concentration. All line widths are in mT and the intensities are in arbitrary unit scaled so that it is 100 for the CD line of pure ZnO NRs. The examination of the intensities evolution indicates that the total number of incorporated Co 2+ ions (∝ I ⊥ co ) continuously increases, while their number in the ordered phase (∝ I ∥ co ) reaches a maximum for 10 % of Co in the synthesis solution. This is consistent with the previous study on ZnO:Co NRs grown with the similar low-temperature method, which has stated that the maximum number of incorporated Co 2+ arise at about 1 % of cobalt in the synthesis solution, and that above this value the crystal quality is degraded 12 . This may be understood by noting that ZnO:Co is not an equilibrated thermodynamic system, so that low-temperature growth method can only incorporate a small amount of Co in the ZnO matrix. Indeed, the 10 % sample has also the closest I ∥ co /I ⊥ co ratio (0.96) to the single crystal one (1.3), thus appearing as a better ordered and crystalline sample. This is consistent with the SEM images of the different samples (Fig. 1), which show that the 10 % sample has the highest Co concentration while preserving the NRs integrity. Regarding the evolution of the other Co 2+ parameters, we note that the parallel hyperfine constant (A ∥ ), defined by the spacing between two hyperfine lines, is the same in all samples where it can be detected (16×10 -4 cm -1 ), and is almost exactly that of the bulk case, thus indicating a high crystallinity of the ordered phase. The perpendicular hyperfine constant (A ⊥ ) is not resolved in the 150 mT broad line because it is lower than the perpendicular line width. For the g-factors, we recall that the position of the 150 mT broad line gives an effective value,

Co 2+ CD x g ⊥ co w ⊥ co I ⊥ co g ∥ co w ∥ co I ∥ co g ⊥ cd w ⊥ cd I ⊥ cd 0 - - 0 - - 0
g ⊥ ef f = 2g ⊥ co (see Ref. 11
). The g ⊥ co and g ∥ co are closed to their bulk values (2.279, 2.243), but due to the asymmetry of the 150 mT line, there is a notable uncertainty in the g ⊥ co value and conclusions about a variation of the anisotropy are difficult to draw. The perpendicular line width (w ⊥ co ) is practically constant and does not bring much information since it results from a lines superposition and unresolved hyperfine coupling (A ⊥ ).

On the contrary, the augmentation of the parallel line width (w ∥ co ) indicates a continuous increase in the Co 2+ concentration within the ordered phase, due to dipolar broadening. By comparing the 10 and 15 % sample, we see that the latter has a lower number of orientated Co 2+ , which are however more concentrated: this can be interpreted as a reduction of the ordered phase volume in the 15 % sample, consistent with its lowest I ∥ co /I ⊥ co ratio (0.55) and SEM imaging (Fig. 1).

We now turn to the study of the intrinsic defects signal, which gives lines around g ∼ 1.96. Previously, several defect signals have been detected close to this value, both in bulk and nano-sized ZnO materials [14][15][16][17] . Their identification still remains controversial but two main assignments are usually made: (1) a single-ionized zinc interstitial (Zn + i ) at g ⊥ = 1.9595 and g ∥ = 1.9605 17,18 , and (2) the so-called D * center at g ⊥ = 1.9605 and g ∥ = 1.9565 19,20 . Besides, a g ∼ 1.96 signal has been proved to decrease when reducing the ZnO nanoparticles size, thus being attributed to a shallow donor CD signal [21][22][23] . However, the nanoparticles not being oriented, the information on anisotropy is lost so that the signal may arise from defects mentioned in (1), (2), or both. On the contrary, when dealing with oriented nanos-tructures, it is valuable to characterize the defects by the anisotropy (if any) of their g-factor components. Defining ∆g = g ⊥ -g ∥ , the two abovementioned defects have ∆g = -0.0010 (easy-axis) and ∆g = +0.0040 (easy-plane), respectively. The difference in the anisotropy type of these defects points to their different nature. Regarding our samples, we first note that in all of them, excepting the g ∼ 1.96 CD signal, no other defect or pollution-related lines are detected, pointing to the high crystal quality and purity of the NRs under investigation. The CD signal detected on the pure ZnO NRs (Fig. 4-top) is made of at least two closed lines, which collapse into a single asymmetric line for θ = 90 • . Here, one of these observed lines appears to be almost isotropic, while the other one displays a slight easy-axis anisotropy signed by the displacement of the positive peak. This shift allows us to estimate ∆g for the anisotropic line to be -0.0014, thus likely to be related to a Zn + i defect (∆g = -0.0010) within the ordered phase of NRs. Regarding the isotropic line, although an isotropic defect cannot be excluded, it is likely to be related to the previously mentioned disordered phase. If so, the line may arise from a powder of the anisotropic defect, with slightly modified parameters: this could be the case for disordered ZnO:Co large nanoparticles or partially grown NRs. We note that in a previous study on similar samples 11 , we have observed a three-line structure at g ∼ 1.96, denoting the presence of another kind of resonant center. This additional line displays an easy-axis anisotropy closed to that of the (probable) Zn + i signal, so that it must arise from the ordered phase of NRs as well. Concerning the intensities estimation, we keep the same simple method as before, which is better applied for the θ = 90 • CD spectra, when a single asymmetric line is present and the SD signal is shifted toward low field (Fig. 4). We then determine the g-factor (g ⊥ cd ), line width (w ⊥ cd ) and height, thus obtain the I ⊥ cd intensity (Tab. I), proportional to the whole number of g ∼ 1.96 CDs.

When a small amount of Co is introduced within the synthesis solution, the total CD signal is drastically reduced by a factor ∼20 in the 0.2 % sample, and a factor ∼50 in the 0.5 % sample (see Tab. I). In these Co-doped samples, the two-lines structure of the θ = 0 • orientation is not anymore visible and there only remains a broader line, whose asymmetry indicates an underlying structure (Fig. 4-bottom). This broad line displays an easy-axis anisotropy ∆g = -0.0012, so that it is reasonably assumed to arise from the same anisotropic (probable Zn + i ) defect present in the pure sample (∆g = -0.0014). The isotropic line of the pure ZnO NRs discussed above is likely to be responsible for the observed right shoulder of the reduced line. We note that the two lines (anisotropic and isotropic) of the pure ZnO NRs samples are similarly reduced in their intensity, and augmented in their line width. The line width augmentation is certainly due to the presence of polarized Co 2+ ions which cause a dipolar broadening, but also an increased spin-spin relaxation due to the Co 2+ open d-shell.

Regarding the evolution of the CD signal for θ = 90 • , figure 5 clearly shows that the CD intensity (number) decreases with the increase of Co in the synthesis solution. However, the comparison of absolute intensity in different samples does not lead to significant information since the total number of the NRs may vary from a sample to another, mainly due to differences in the substrate coverage. Consequently, in order to indubitably correlate this decrease with the increase of total number of incorporated Co 2+ , it is preferable to plot the CD's intensity (I ⊥ cd ) with respect to the Co intensity at θ = 90 • (I ⊥ co ), rather than with respect to the Co concentration within the synthesis solution. This has been done in the inset of Fig. 5, which clearly shows that the incorporation of Co 2+ ions (in ordered and disordered phases) is accompanied by the progressive disappearance of the the CD signal. This anti-correlation does moreover follow qualitatively the Co content within the synthesis solution so that, for example, the 10 % sample has more incorporated Co and less CD than the 2 % sample.

To analyze this, three main explanations are pos- sible: (1) the presence of cobalt in the synthesis solution may, in a way or another, simply inhibit the formation of this defect during the growth, (2) the paramagnetic incorporated Co 2+ ions and CD may form a non-magnetic complex, which is then not detectable by EPR, or (3) another kind of nonmagnetic incorporated cobalt may destroy the CD, for example by filling a vacancy. It has also been suggested that the disappearing of the g ∼ 1.96 signal when transition metal ions are incorporated in ZnO is only due to a large broadening due to spinspin relaxation with the 3d electrons 23 . However, it is difficult to imagine a Co 2+ ion interacting so with a CD without being altered in its magnetic parameters, so that, as only Co 2+ ions with bulk parameters are detected, this suggestion may be finally equivalent to the case (2). Indeed, the physics of ZnO intrinsic and extrinsic defects is so rich that many kinds of defects complexes are likely to occur. Consequently, at this stage, we cannot decide an exact answer for this question, in particular because the exact nature of the CD is not known with certainty. Further experimental and theoretical investigation have to be done. For instance, ab initio theoretical study of ZnO supercells containing various kind of paramagnetic defects (compatible with the slight easy-axis magnetic anisotropy observed in this work) and substitutional Co 2+ ions in various relative positions would be able to give insight into these questions.

IV. CONCLUSION

We have detected the EPR signal of substitutional Co 2+ ions in the ZnO matrix of NRs. The analysis of the anisotropic spectra indicates a high crystallinity of the NRs, which are moreover mostly well-oriented perpendicularly to the substrate surface. The line-shape analysis of the Co 2+ signal reveals the probable presence of a secondary disordered phase of ZnO:Co. Besides, the resolved anisotropy of the CD signal at g ∼ 1.96 in pure NRs strongly suggests that this defect signal arises, at least partially, from the Zn + i shallow donor. By a systematic examination of the EPR spectra recorded just above liquid helium temperature on samples grown with varying cobalt concentration within the synthesis solution, we prove that the apparition of the cobalt signal is anti-correlated with that of the CD signal, thus suggesting a gradual neutralization of the CD by Co 2+ . On one hand, these results can explain the augmentation of the ZnO:Co NRs crystal quality observed in low Co-doped samples and, on the other hand, it will allow researchers to finely tune the number of the paramagnetic defects in ZnO nanostructures, as well as to give insight into the identification of the g ∼ 1.96 signal.

Figure 1 :

 1 Figure 1: (color online) Top-view of the 0.2, 2, 10 and 15 % samples recorded by SEM. All images have 5×6 µm dimensions.

Figure 2 :

 2 Figure 2: (colour online) EPR spectra of 10 and 2 % Co-doped samples, showing the anisotropy of the Co 2+ signal, recorded at T = 6 K. Sample holder (SH), sapphire defect (SD) and core defect (CD) lines are indicated.

Figure 3 :

 3 Figure 3: (color online) Evolution of the Co 2+ signal with the cobalt concentration within the synthesis solution, for both orientation, recorded at T = 6 K. Spectra for θ = 0 • are divided by 3.

Figure 4 :

 4 Figure 4: (colour online) Anisotropy of the EPR CD signal in pure (0 % Co) and Co-doped (0.5 %) samples, with respect to the magnetic field orientation, recorded at T = 6 K. Sapphire defect (SD) signal present in the 0.5 % sample is shown for the θ = 0 • orientation.

Figure 5 :

 5 Figure 5: (color online) Evolution of the CD signal with the cobalt concentration within the synthesis solution. Line of the 0 % is divided by 100. The 15 % sample spectra is not shown for clarity (similar to the 10 % one). Inset shows I ⊥ cd vs I ⊥ co .

V. ACKNOWLEDGEMENT

This work was supported by the NATO project Science for Peace (SfP) 984735, Novel nanostructures. A. Savoyant and O. Pilone acknowledge Sylvain Bertaina for useful discussion about EPR measurement.