Hiroki Ikeuchi

Hans De Raedt

Sylvain Bertaina

Seiji Miyashita
email: miyashita@phys.s.u-tokyo.ac.jp

Temperature-and Size-dependence of Line shape of ESR spectra of XXZ antiferromagnetic chain

Keywords: numbers: 05.30.-d, 75.10.jm, 76.30.-v ESR, XXZ model, line shape

The ESR (Electron Spin Resonance) spectrum of the XXZ spin chain with finite length shows a double-peak structure at high temperatures around the EPR (Electron Paramegnetic Resonance) resonance frequency. This fact has been pointed out by direct numerical methods (S. El Shawish, O. Cépas and S. Miyashita: Phys. Rev. B 81, 224421 (2010); H. Ikeuchi, H. De Raedt, S. Bertaina, and S. Miyashita: Phys. Rev. B 92, 214431 (2015)). On the other hand, at low temperatures. the spectrum has a single peak with a finite shift from the frequency of EPR as predicted by the analysis of field theoretical works (M. Oshikawa and I. Affleck: Phys. Rev. Lett. 82

I. INTRODUCTION

The ESR (Electron Spin Resonance) is one of the major tools to obtain information about the spin ordering. To understand the spectrum, parameter dependence of a concrete ESR spectrum for a specified system has to be clarified, including the temperaturedependence. To study these aspects theoretically, explicit forms of interactions of magnetic structure of the system such as spatial configuration of magnetic ions in the lattice must be taken into account. For example, modeling the ESR spectra of intrinsic defects in spin chains is an important problem for which data for finite but rather long chains are necessary [1].

The one dimensional S = 1 2 XXZ model is one of the most well-investigated systems in the field of ESR study. As for the resonance shift and the linewidth of the spectrum which include basic information of the system, a lot of theoretical research has been conducted [2,3]. Oshikawa and Affleck [4] developed an approach based on (1+1)-dimensional field theory, where they used the bosonization method and successfully derived the shift and the linewidth of the resonance peak at low temperatures in the thermodynamic limit [4]. This method has been also successfully used to investigate effects of the edge state [5].

There are other attempts utilizing the integrability of the XXZ model. Maeda et al. [6] derived the formula for the resonance shift which is exact up to the first order in anisotropy. By applying the Bethe ansatz technique to the formula, they obtained an analytic expression of the resonance shift over all the temperature region. Brockmann et al. [7] also obtained consistent results focusing on the moments of the spectral shape. In this way, a lot of information has been found in respect of the resonance shift and the linewidth.

However, when we need more explicit form of the spectrum which may have a complicated structure, e.g., satellite peaks, long tails, etc., we have to evaluate the Kubo formula for the system Hamiltonian of the interest directly. Such attempts have been also developed [8][9][10].

It has been pointed out that the line shape of the XXZ spin chain with even number of spins has a double-peak structure at high temperatures for the lattice with finite length L ≤ 16 [8], and the structure has been confirmed up to L = 26 [11]. But, its detailed dependence on temperature and size has not been known yet. Such information for finite sizes is important to study diluted systems which are ensemble of short chains [1], where temperature-dependence of the spectrum is also important.

In the present paper, first, we investigate how the high-temperature spectrum with the double-peak structure at the EPR (Electron Paramegnetic Resonance) position, i.e., hω EPR = γH (γ is the gyromagnetic ratio), changes to the low temperature spectrum with single peak at a shifted position. We present spectra for the intermediate-temperature region. We find a drastic change with the temperature. We also study the case of odd-number of spins, in which the high-temperature spectrum has a central peak at the EPR position with protuberances beside it. In this case we also found the spectrum changes to the lowtemperature shape which does not depend on the parity of the number of spin. The transition is well understood by the energy diagram as a function of the static magnetic field.

Next, we study origin of the structure of high-temperature spectrum, i.e., the double peak and small peaks beside it (protuberrances). We find that each peak can be attributed to a resonance between states with specified magnetizations (M and M ± 1). By making use of this fact, we introduce an extension of the above mentioned moment method [7] for each contribution from the transition specified by (M and M ± 1). With this analysis, we find that the deviation of the mean of the distribution for each contribution to the spectrum reduces as 1/N. This observation strongly suggests that the structure would shrink to the center in the thermodynamic limit. Indeed the numerical study up to N = 28 supports the dependence. But on the other hand, we also find that its variance converges to a certain finite value as N becomes large. Thus mathematically, there is some possibility of special shape of the spectrum in which the double peak may remain.

The outline of this paper is as follows. In Sec. II, we introduce the model and method. In Sec III, temperature-dependence of spectra is given. In Sec IV, we analyze structure of spectrum decomposing it into the contributions from transitions between specified sets of magnetizations. In Sec V, we study size-dependence of the structure of spectra by applying the moment method to individual contributions from the types of transitions. In particular, the estimation of the double peak's separation is an interesting problem, which is discussed in Sec V B. A summary and discussion of related problems are given in Sec. VI.

II. SYSTEM AND METHOD

We study a one-dimensional S = 1 2 XXZ model under a static magnetic field along the z-axis. We apply to the system an oscillating magnetic field parallel to the x-axis. The total Hamiltonian of the system is given by

H tot = J N-1 ∑ i=1 S S S i • S S S i+1 + ∆ N-1 ∑ i=1 S z i S z i+1 -H N ∑ i=1 S z i + λ 0 cosωt N ∑ i=1 S x i , ( 1
)
where ∆ represents the strength of the anisotropy. In this paper, we set J = 1K, ∆ = -0.08K (i.e., the XY-like anisotropy) and H = 5K (i.e., a sufficiently strong filed), and we impose the open boundary conditions. In the present study we do not include the dipole-dipole interaction, and thus the direction of the chain does not affect the results although the dipole-dipole interaction could cause an interesting angle dependence of the spectrum on the angle between the lattice direction and the fields like the Nagata-Tazuke dependence [3]. Thus we do not specify it in the present paper.

Here we remind important relations for the the ESR spectrum. According to the Kubo formula [12,13], the ESR spectrum, i.e., the absorption rate I x (ω) of the oscillating filed can be obtained with the dynamical susceptibility χ(ω) = χ ′ (ω) + iχ ′′ (ω) as follows:

I x (ω) = ωλ 2 0 2 χ ′′ (ω), (2)
χ ′′ (ω) = 1 -e -β ω 2 ∫ ∞ -∞ ⟨M x (0)M x (t)⟩ eq e -iωt dt, (3)
where M x (t) = e iH t M x e -iH t = e iH t ∑ i S x i e -iH t , H = J ∑ S S S i •S S S i+1 +∆ ∑ S z i S z i+1 -H ∑ S z i , and ⟨• • • ⟩ eq denotes the thermal average with respect to H at a temperature β -1 . By using the set of the eigenvalues and eigenvectors {E n , |n⟩} D n=1 of the Hamiltonian H (D is the dimension of the Hilbert space of the Hamiltonian), χ ′′ (ω) is readily given by

χ ′′ (ω) = ∑ m,n D m,n δ (ω -ω m,n ), (4)
where

D m,n ≡ π(e -β E n -e -β E m )|⟨m|M x |n⟩| 2 /Z, ω m,n ≡ E m -E n , Z = D ∑ n=1 e -β E n . ( 5
)
Thus, for small systems we obtain χ ′′ (ω) by direct numerical estimation of the fomula (4) as long as we can obtain all the eigenvalues and eigenvectors of the system [10,11]. However, the method is inevitably limited to small systems. Then timedomain methods have been introduced, where the spectrum is obtained by Fourier transform of the autocorrelation function of magnetization (AC method). There it is known that finite observation time causes artificial modification of the spectrum (say, the Gibbs oscillation). Then in our previous study [11], we proposed a new method (WK method) to make use of the Wiener-Khinchin relation with spectral density of magnetization fluctuation, in which the Gibbs oscillation is suppressed. However, in the same time, we found that that Gibbs oscillation is suppressed in a large system and the AC method works efficiently. Thus, in the present work, we obtained the spectrum with the AC method. In the AC method, we can study the double size of the case of the diagonalization theoretically. However, not the memory but the CPU time prevents us from treating large systems. In the present paper, we calculated up to N = 28 in Sec V. The methods are explained in references [9,11] in detail.

III. TEMPERATURE-DEPENDENCE OF SPECTRAL SHAPES

In our previous studies [8,11] we studied spectrum at high temperatures, where a double-peak structure was found around the EPR position, On the other hand, the field theoretical study [4] gives a single peak with a shift from the EPR position at low temperatures. Thus, it is interesting to study how the spectrum changes with the temperature. In this section, we investigate the temperature-dependence of the ESR spectrum.

Let us first discuss the case in which the number of spins is even. Spectrum for N = 20 is depicted in Fig. 1(Left) at a high temperature (β -1 = 100K), where we find a double-peak structure and two small protuberances next to it.The center of the structure is located at the EPR position (ω = 5). On the other hand, spectrum at a low temperature (β -1 = 1K) has a single peak at a shifted frequency (ω ≃ 5.08), which is depicted in Fig. 1(Right). Between them, the spectrum shows a significant change with the temperature as depicted in Fig. 1(Center).

In order to understand the change of spectrum, we study the structure of energy diagram as a function of static magnetic field. The number of states of a system with N spins is D = 2 N . For the large systems, D is too large to draw the diagram in a figure, and thus we draw the diagram for N = 6 in Fig. 2(Left) (green lines). There are 2 6 = 64 states. In the case of no anisotropy ∆ = 0, the eigenstates belong to some multiplets of spin S (S is a positive integer) which has 2(S + 1)-fold degenerate states (M = -S, -S + 1, • • • , S) at H = 0. The degenerate states develop with the field according to the Zeeman energy:

E(H, M) = E 0 -HM, M = -S, -S + 1, • • • , S
, where E 0 is the energy of the multiplet under no magnetic field.

Note that there exists only one state that has the magnetization M = S, which is 3 in the preset case, and the energy is located at the lowest in the present system as shown in Fig. 2

(Left).

When the anisotropy is introduced ∆ ̸ = 0, these 2(S + 1)-fold degenerate levels are split into a single state with M = 0 and S pairs of states which have opposite magnetization, i.e., {M, -M} = {1, -1}, {2, -2} and {3, -3} as shown by the red lines. The energy gap ∆E(M) between the doublets with M = 1, -1 and the single state with M = 0 at H = 0 is denoted by ∆E(1). Similarly, ∆E(2) and ∆E(3) are defined. Note that ∆E(0) = 0 by definition. Denoting the energy shift of the state with M = 0 due to anisotropy by δ E 0 (̸ = 0) , the energy of the state for {M, -M} ,

M = 1, 2, • • • S is given by E(H, M) = E 0 + δ E 0 + ∆E(|M|) -HM, M = -S, -S + 1, • • • , S : red lines in Fig. 2 (Right). (6)
In the left panel, detailed structure due to ∆E(M) is hardly seen, and thus the magnified structure is given in Fig. 2 (Right). At H = 5, the ground state is S = 3 and M = 3. Thus, the spectrum at low temperatures is mainly given by the resonance between this state and the excited state with S = 3 and M = 2. The resonant frequency is changed by ∆E(M = 3) from the EPR value γH. This resonance is the peak in Fig. 1(Right) with the shift ∆ω = ∆E(M = 3)/h. On the other hand, at high temperatures all the states are occupied with nearly the same probability, and the corresponding spectrum is that in Fig. 1(Left). Attribution of each resonance with ∆E(M) will be investigated in the next section. At intermediate temperatures, the population distributes with the Boltzmann weight, which gives the change of spectrum of Fig. 1. As the temperature decreases, the double-peak structure breaks down and the spectrum gradually gets shifted to the right side(Center).

Theoretically temperature-dependence of the line shape has not been studied in detail and so far this drastic change has not been recognized yet. But, this change is robust and it is expected to be observed in corresponding materials. Next we study the case of odd number of spins. In Fig. 3 the spectra for N = 21 are shown. The spectrum has a single sharp peak at the EPR position with protuberances besides it at the high temperature β -1 = 100K (Fig. 3 (Left)). The central peak is a characteristic of the odd number case. As the temperature is lowered, the spectrum changes to the high frequency side similarly to the case of even number (N = 20). (Fig. 3 (Center) and (Right)). As the corresponding energy diagram we give that for N = 5 in Fig. 4 (Left) and its magnified one in Fig. 4 (Right). The 6 states are split into 3 doublets, {M, -

M} = { 1 2 , -1 2 } , { 3

}

. The fact that the state with M = 0 does not exist is an important difference from the case of even number of spins. Because of this fact, the spectrum has a sharp peak at EPR position due to the transition between M = ±1/2.

IV. DECOMPOSITION OF THE SPECTRUM INTO CONTRIBUTIONS FROM TRANSITIONS SPECIFIED BY MAGNETIZATION

Without ∆, the spectrum has a single peak at the EPR position, and thus the structure of high temperature spectrum should be attributed to the energy structure lifted from the degeneracy. In what follows, we will reveal the origins of the characteristic shapes observed in the previous section by focusing on the energy diagrams of the systems given in Figs. 2 and4 In general, the resonance peaks in ESR spectrum are given by transition between states with M and M ′ = M ± 1. The most dominant contribution comes from the transitions between levels within the same multiplet in the case of ∆ = 0. The breakdown of SU(2) symmetry due to ∆ allows contributions from transitions between different multiplets, but the contributions from them are found very small. Thus, we will ignore those contributions in the analysis of data, although the contributions are included in spectra obtained by numerical method. The contributions from the transitions (M → M + 1) correspond to emission and it gives spectrum at negative ω. We do not study those contributions. Now we decompose the spectrum into contributions from transitions (M → M -1)of various values of M. Since the multiplet is separated into the pairs of {M, -M} and M = 0, the contributions of (M → M -1) and (-M + 1 → -M) has the same matrix elements. Thus we classify the contribution according to M. Here we adopt a system with N = 12 for which there are many states to form a continuous-like lineshape although we can still calculate the eigenstates by the exact diagonalization method.

In Fig. 5(Left), we show the contributions from M = 1, i.e., (M = 1 → M -1 = 0) (solid line) and (-M + 1 = 0 → -M = -1) (dotted line). We find the contribution of (M = 1 → M -1 = 0) gives the right peak of the double peak, and that of (-M + 1 = 0 → -M = -1) the left one. The separation of the peaks is given by the energy split ∆E(M). Here we concern the multiplet with maximum spin S = N/2 = 6, but the transitions of the pair with M = 1 exist in all the multiplets of S > 0. Thus number of transitions between M = 0 and ±1 is maximum, and the transition of M = 0 has the largest contribution. The energy gaps ∆E(M = 0) depend on the multiplet which the states belong to, and thus the resonance frequency distributes as we see in Figs. 5

(Left).

In the same way, the transitions of the pair with M = 2 give the second peak (protuberance) as depicted in Figs. -β E(H, m)) for m=M and -M + 1 are gradually getting difference, and thus the spectral shape becomes no more symmetric around ω EPR , and consequently the transitions between levels with large m = M > 0 become larger in the spectrum studied in the previous section. This explains the temperature-dependence of the spectrum.

In the low-temperature regime (Figs. 7), the state of M = S is mostly populated under a large filed, and the transition from it to M = S -1 gives the single peak with a shift.

= -S, -S + 1, • • • , -1/2, +1/2, • • • , S.
The transitions between M = 1/2 and -1/2 are most populated and they give the central peak. As the temperature increases, very similarly to the even number case, the peaks shift to the high frequency side, and finally become a single sharp peak with a shift.

V. SIZE-DEPENDENCE OF SPECTRAL SHAPES A. Numerical observation

Now, we examine the size-dependence of characteristic shapes of spectra found in the high-temperature region, such as the double-peak structure. We depicted spectra of various sizes N in Fig. 8, where we see rather systematic size-dependence. The systems with even numbers of spins show that the separation between double peaks ∆ω decreases as the system size becomes large. We may anticipate that in the thermodynamic limit, the double peak may get stuck together to become a single central peak. In fact, this problem has already posed in the reference [8], but its answer has not been concluded yet. Besides, the separation between two small protuberances also seems to decrease with the system size increased, and shrink to the center in the thermodynamic limit. The size-dependence of the separation ∆ω between the double peak is given in Fig. 9 (Left). The error bars in the figure denote the mesh size of ∆ω given by the observation time T as 2π/T . Here we find that in even systems the separation roughly decreases with the size as 1/N at least up to N = 22, and that the separations of N = 22, 24 and 26 are almost the same. But at N = 28 it again decreases. Thus, although we can have a rough picture about the finite size effect, we cannot conclude how the separation behaves in the large-size limit from the figure. In Fig. 9 (Center) we plot the size-dependence of separations of two small protuberances of both even and odd cases. This shows that the separation of protuberances decreases roughly proportionally to 1/N. In Fig. 9 (Right), we find the heights of the peaks of protuberances increase with the system size. There seems to be a tendency for separation of the double peak to decrease as N becomes large. But it might be saturated at some point. The green line denotes theoretical values calculated by Eq. ( 23). (Center) Size-dependences of the separation of two small protuberances. The red points denote odd-spin systems and the blue points even-spin systems. We can find that the small protuberances come close to the center ω = H as the system size N increases. The error bars denote the mesh size of ∆ω determined by the observation time T . The green lines denote theoretical values calculated by Eq. ( 23). (Right) Size-dependences of the height of two small protuberances. The height is related to the the intensity of the absorption. The protuberances become larger with N increasing.

B. Estimation of the separation

Now, we examine the size-dependence of characteristic shapes of spectra by making use of the moment method. The method of moments originated in van Vleck's paper [15] and has been used as a basic tool to investigate the spectral shapes [12,16]. In a recent study [7], the moments of the whole spectrum for the XXZ chain were discussed in detail. The width and position of a peak in spectrum have been studied by the the moment method. In the moment method, we can grasp characteristics of ESR spectra by focusing on n-th order moment m n of the spectrum.

Since we are considering properties at high temperatures, we use S xx (ω) instead of the susceptibility χ(ω):

S xx (ω) = ∫ ∞ -∞ ⟨M x M x (t)⟩ eq e -iωt dt. (7)
The moment m n is defined as [7]:

m n = ∫ ∞ -∞ dωω n ∫ ∞ -∞ dt⟨M x M x (t)⟩ eq e -iωt (8) = 2π⟨M x (ad H ) n M x ⟩ eq , (9)
where ad H • ≡ [H , •]. We could estimate these quantities by exact solutions or numerical methods. In particular, at the infinite temperature, we obtain the moments from combinatorics as we see later.

The intensity (i.e., the area of the spectrum), the mean position, and the linewidth are defined as intensity: m 0 , (10) mean position:

m 1 m 0 , (11) linewidth:
√ m 2 m 0 - ( m 1 m 0 ) 2 , ( 12
)
respectively. There is still room for argument about the definition of linewidth given in (12). Indeed, (12) may not exist for the spectrum with exact Lorentzian shape. But, as will be seen later, (12) always exists for the system we are considering and it is known that ( 12) could be regarded as a good approximation to the width of the Lorentzian distribution in some situations [12,14].

In this sense, we adopt (12) as the definition of the linewidth and other problems will be discussed in the last part of this section. So far the method has been used to study the whole shape of the spectrum. For our purpose, however, we need to improve this method because moments of the total spectrum are not very helpful in considering the detailed shape of spectrum. In other words, even if we know moments with small n of the whole spectrum, we could not conclude the size-dependence of the double-peak's split.

However, as we saw in the previous section, the spectrum can be decomposed to contributions from transitions specified by the magnetization M. Thus now we extend the method and investigate properties of each contribution to obtain information for the structure of the spectrum, e.g., the double peak. We will focus on the moments of partial spectrum made by only transitions (M → M -1), instead of the whole spectrum. For example, if we know the mean position of the partial spectrum from the transitions (0 → -1) (given by the solid line in Fig. 5(Left)), we can see how the double-peak structure behaves depending on the size N.

From now on, we set the external field H = 0K, because the external field brings about just a shift of the mean position of spectra and does not affect the shape (See Appendix A).

The whole spectrum at infinite temperature is given by

∫ ∞ -∞ ⟨M x M x (t)⟩ ∞ e -iωt = 1 2 N ∑ m,n |⟨m|M x |n⟩| 2 2πδ (ω -(E m -E n )), (13)
where

⟨• • • ⟩ ∞ = Tr [• • • ]/2 N .
The partial spectrum made by transitions (M → M -1) is obtained as follows:

∫ ∞ -∞ ⟨P (M z =M) M x P (M z =M-1) M x (t)P (M z =M) ⟩ ∞ e -iωt dt = 1 2 N ∑ m (M z m =M-1) ∑ n (M z n =M) |⟨m|M x |n⟩| 2 2πδ (ω -(E m -E n )), (14)
where P (M z =M) is a projection operator which projects states onto the subspace where M z = M. The spectrum from the specified transition is given by

S M xx ≡ ∫ ∞ -∞ ⟨P (M z =M) M x P (M z =M-1) M x (t)P (M z =M) ⟩ ∞ e -iωt dt, (15) (16)
and its n-th order moment

m M n = ∫ ∞ -∞ ω n S M xx (ω)dω (17) = 1 2 N π 2 ∑ σ σ σ (M z σ σ σ =M) ⟨σ σ σ |M + ad n H (M -)|σ σ σ ⟩, ( 18
)
where M ± ≡ M x ± iM y . As for the basis set {|σ σ σ ⟩}, we may use the up/down-spin representation, such as | ↑↓↓ • • • ↑↓↑⟩. Note that the scales are different between (Left) and (Right). According to Eq. ( 24), we consider 2 × ∆/ √ 2 as the linewidth, which is denoted by the black arrow.

VI. SUMMARY AND DISCUSSION

In the present paper, we studied the temperature-and size-dependence of the ESR spectrum for the XXZ chain. A drastic change from high temperature spectrum with a structure around the EPR position to the low temperature spectrum with a single peak at a shifted position from the EPR position. Temperature-dependence of the line shape has not been studied in detail and the drastic change has not been recognized yet in experiment. Although the structure may disappear in the thermodynamic limit as we discussed in the present paper, the structure definitely exists up to considerable length of lattice (say N < 30) and we hope the structure and the temperature-dependence will be observed in experiments.

Subsequently, we also investigated the size-dependence of the separation of the double peaks. The double-peak structure obtained by the AC method shows the tendency that the separation shrinks to zero as N → ∞. The size-dependence was analyzed by making use of an extended moment method. Since the XXZ model has a conserved quantity, i.e., the z-component of magnetization M, we can decompose the whole spectrum into the partial spectra specified by types of transitions. We applied the moment method to each contribution of the spectrum from a transition with specified magnetizations. As the results, we found that the mean positions of all spectrum approach to the center as ∆/N and become zero as N → ∞, which strongly indicates the double peak collapses at the center. The linewidth remains finite as ∆/ √ 2 even in the thermodynamics limit. Such observation prevents us from drowing the definite conclusion.

Finally, we put some comments towards the future works. In this paper, since we assumed the anisotropy to be small (∆/J = -0.08) and the transitions between different multiplets can be ignorable, the decomposition of the spectrum into contribution specified by magnetization was valid. But the analytical results derived with the moment method in the last section is valid for any J and ∆, and besides, according to our numerical simulation, the decomposition of the spectrum specified by magnetization somehow goes well for the wide range of XY-like anisotropy (0 < ∆/J < 1). On the other hand, for the Ising-like anisotropy (∆/J > 1), our method seems to be of limited use empirically.

The idea of the decomposition of the spectrum can be also used for other systems with magnetization conserved, e.g., systems with the Dzyaloshinskii-Moriya interaction whose direction is parallel to the static field. These applications will be studied elsewhere.

First, we consider the spectrum under a strong field H. Let

S > xx (ω -H, 0) = S > xx (H -ω, 0) = S < xx (ω -H, 0), (A9)
because of the symmetry of the shape of S > xx as seen in Sec. IV, and Eq. (A6). Then, by using Eq. (A3), we have S > xx (ω -H, 0) = 1 2 S xx (ω -H, 0). Therefore, it follows that S xx (ω, H) = 1 2 S xx (ω -H, 0), which is what we wanted to show. The difference between S xx (ω) and χ ′′ (ω) is just the presence of the factor (1e -β ω )/2 ∼ β ω/2. χ ′′ (ω) vanishes at infinite temperature because of this factor, but we are interested in the spectral shape, not in the exact value of the peak. So we can ignore this factor and consider only S xx (ω). Strictly speaking, the ω-dependence of the factor β ω/2 could deform the spectral shape, but this effect is also ignorable in the case where H is very large.

FIG. 1 .

 1 FIG. 1. (Color online) (Left) The ESR spectrum for N = 20 at β -1 = 100K. (Center) The spectra for N = 20 at β -1 = 100K, 50K, 30K, 10K, 7K, 5K and 3K. (Right) The spectra for N = 20 at β -1 = 1K and 0.1K in a large scale of χ(ω).

FIG. 2 .

 2 FIG. 2. (Color online) (Left) 64 energy levels for a 6-spin system. We accentuate the 7-fold degenerate levels by the red lines. (Right) The enlarged view of the Left figure.

FIG. 3 .

 3 FIG. 3. (Color online) (Left) The spectrum for N = 21 at β -1 = 100K. (Center) The spectra for N = 21 at β -1 = 100K, 50K, 30K, 10K, 7K, 5K and 3K. (Right) The spectra for N = 21 at β -1 = 1K and 0.1K.

FIG. 4 .

 4 FIG. 4. (Color online) (Left) 32 energy levels for a 5-spin system. We accentuate the 6-fold degenerate levels by the red lines. (Right) The enlarged view of the Left figure.

5 (

 5 Center), and transitions with M > 2 make the tail of the spectrum (Figs. 5(Right)). In the intermediate-temperature regime(Figs. 6), the Boltzmann factors exp(

FIG. 5 .

 5 FIG. 5. (Color online) Spectra for N = 12 classified by the magnetizations of transitions at β -1 = 100K. The Left figure is for the spectrum by transitions (1 → 0) (solid) and (0 → -1) (dotted). The Center figure is for (2 → 1) and (-1 → -2). In the Right figure, spectra for larger M's are given: Blue: (3 → 2), (-2 → -3), Magenta: (4 → 3), (-3 → -4), Cyan: (5 → 4), (-4 → -5), Black: (6 → 5), (-5 → -6).

FIG. 6 .FIG. 7 .

 67 FIG. 6. (Color online) Spectra for N = 12 classified by the magnetizations of transitions at β -1 = 10K. The Left figure is for the spectrum by transitions (1 → 0) (solid) and (0 → -1) (dotted). The Center figure is for (2 → 1) and (-1 → -2). In the Right figure, spectra for larger M's are given: Blue: (3 → 2), (-2 → -3), Magenta: (4 → 3), (-3 → -4), Cyan: (5 → 4), (-4 → -5), Black: (6 → 5), (-5 → -6).

FIG. 8 .

 8 FIG. 8. (Color online) Spectra of N = 20, 21, 22, 23, 24, 25, 26, 27 and 28 depicted on the unified scale. Note that the central peaks of odd systems are cut off in this figure.

FIG. 9 .

 9 FIG.9. (Color online) (Left) Size-dependence of the separation of the double peak. There seems to be a tendency for separation of the double peak to decrease as N becomes large. But it might be saturated at some point. The green line denotes theoretical values calculated by Eq. (23). (Center) Size-dependences of the separation of two small protuberances. The red points denote odd-spin systems and the blue points even-spin systems. We can find that the small protuberances come close to the center ω = H as the system size N increases. The error bars denote the mesh size of ∆ω determined by the observation time T . The green lines denote theoretical values calculated by Eq. (23). (Right) Size-dependences of the height of two small protuberances. The height is related to the the intensity of the absorption. The protuberances become larger with N increasing.

FIG. 10 .

 10 FIG.10. (Color online) (Left), (Right) Spectrum for N = 11, β -1 = ∞. Note that the scales are different between (Left) and (Right). According to Eq. (24), we consider 2 × ∆/ √ 2 as the linewidth, which is denoted by the black arrow.

  the simultaneous eigenvectors of H 0 + H ′ and H z be{ |E 0 n , M n ⟩ (H 0 + H ′ )|E 0 n , M n ⟩ = E 0 n |E 0 n , M n ⟩ H z |E 0 n , M n ⟩ = -HM n |E 0 n , M n ⟩, n = 1 • • • D. |⟨E 0 n , M n |M x |E 0 m , M m ⟩| 2 δ {ω -[(E m -E n ) -H(M m -M n )]} . (A2)Let us divide S xx (ω, H) into two parts in the following:|M + |E 0 m , M m ⟩| 2 δ {ω -[(E m -E n ) -H(M m -M n )]} , |M -|E 0 m , M m ⟩| 2 δ {ω -[(E m -E n ) -H(M m -M n )]} ,Noting that we are interested in the absorption, not the emission, we may focus on the region ω > 0. The peaks in this region need to satisfy the relationE m -E n > H(M m -M n ), then it follows that M m = M n -1, because H ≫ J > 0.Therefore, only S > xx (ω, H) contributes to the spectrum S xx (ω, H) in the region ω > 0: S

	} D n=1 :
					(A1)
	S xx (ω) is given by
	S xx (ω, H) =	2π Z ∑ m,n
					S xx (ω, H) = S > xx (ω, H) + S < xx (ω, H),	(A3)
	where
	S > xx (ω, H) ≡	2π Z ∑ m,n	1 4	|⟨E 0 n , M n (A4)
	S < xx (ω, H) ≡	2π Z ∑ m,n	1 4	|⟨E 0 n , M n (A5)
	and we find
					S > xx (-ω, H) = S < xx (ω, H).	(A6)

xx (ω, H) = S > xx (ω, H), ω > 0, (A7) S xx (ω, H) = S < xx (ω, H), ω < 0. (A8)

According to Eqs. (A7) and (A4), we have S xx (ω, H) = S > xx (ω -H, 0). On the other hand, it is shown that

ACKNOWLEDGEMENTS

The present work was supported by Grants-in-Aid for Scientific Research C (25400391) from MEXT of Japan, and the Elements Strategy Initiative Center for Magnetic Materials under the outsourcing project of MEXT. The numerical calculations were supported by the supercomputer center of ISSP of Tokyo University. We also acknowledge the JSPS Core-to-Core Program: Non-equilibrium dynamics of soft matter and information.

Because, in the infinite-temperature limit, the Boltzmann factor does not appear, and thus Eq. ( 18) can be calculated by counting the number of state (combinatorics). The zeroth, first, and second order moments are explicitly written as

From these quantities, we obtain the intensity (the area) of the spectrum, the mean position, and the linewidth of partial spectra from the transitions (M → M -1) in the following:

Note that these results are valid for any ∆ and J.

Let us analyze the numerical results again from the viewpoint of Eqs. ( 22) ∼ (24). The heights plotted in Fig. 9 (Right) correspond to the intensity of the spectrum and they relate to the quantity given by Eq. ( 22). Eq. ( 22) indicates that the area of the partial spectrum increases with N larger, which supports the size-dependence of the peak's height. Eq. ( 23) is consistent with the observation of 1/N in the numerical results shown in Fig. 9 (Left) and (Center). We added the green lines represented as 2(1 -2M)∆/N in Fig. 9. There are some differences between the numerical results and the green theoretical lines because Eq. ( 23) gives just the mean position, not the position of the maximum of the spectrum. Nevertheless, the mean of the peak due to the transitions between M = 0 and -1 tends to zero, which suggests that the peaks of the double peak shrink to the center. Thus the hypothesis is strongly supported that the double peak and small protuberances also get stuck as N → ∞.

As for the central peak in odd-spin systems, its mean position turns out to be always zero by substituting M = 1/2 into Eq. ( 23).

Based on these facts, we can draw the following conclusion about the separation of the double peaks. The mean position of spectrum for each transition exactly goes to zero in proportional to 1/N. This means the position of each peak coming from the transition (M → M -1), e.g., the peak shown by the solid line in Fig. 5, goes to zero. In particular, this seems to suggest that the double peak shrinks to the origin.

On the other hand, the interpretation of Eq. ( 24) includes a delicate problem. For instance, let us discuss the single peak in odd-spin systems. In Fig. 10 (Left), S xx (ω) of N = 11 is decomposed into different values of M as in the previous section. The linewidth of the spectrum due to the transition between M = ±1/2 (red line) in Fig. 10 looks very small, almost zero. Nevertheless, the linewidth calculated from Eq. ( 24) is shown by the arrow. The width estimated by Eq. ( 24) is much larger, which seems contradictory to the data. In order to resolve this problem we plot the spectrum in a different scale in which the small peaks can be seen (Fig. 10 (Right) ). We find small peaks far from the central peak, which cause the linewidth broader than we expected. In real experiments, the small peaks in Fig. 10 (Right) may be smeared out by noise and only the central peak can be observed. So in this sense, the linewidth calculated from Eq. (24) may not be appropriate in practical situation.

With this observation, small peaks away from the origin might give unexpected contributions, and thus we cannot deny the possibility that the tops of the double peak do not get stuck together in the thermodynamic limit. In Sec. V B, we investigated the spectral shapes by calculating S xx (ω) under no magnetic field, instead of the susceptibility χ ′′ (ω) under a finite static field H. This is valid for a sufficient strong field H ≫ J(> 0) and a sufficient high temperature β ∼ 0. In this appendix, we show this fact.


E Assaf

A Portavoce
email: alain.portavoce@im2np.fr

K Hoummada

M Bertoglio

S Bertaina

High Curie temperature Mn 5 Ge 3 thin films produced by nondiffusive reaction

Keywords: Germanium, Manganese, thin film, ferromagnetic, reaction

Polycrystalline Mn 5 Ge 3 thin films were produced on SiO 2 using magnetron sputtering and reactive diffusion (RD) or non-diffusive reaction (NDR). In situ X-ray diffraction and atomic force microscopy were used to determine the layer structures, and magnetic force microscopy, superconducting quantum interference device and ferromagnetic resonance were used to determine their magnetic properties. RD-mediated layers exhibit similar magnetic properties as MBE-grown monocrystalline Mn 5 Ge 3 thin films, while NDR-mediated layers show magnetic properties similar to monocrystalline C-doped Mn 5 Ge 3 C x thin films with 0.1  x  0.2. NDR appears as a CMOS-compatible efficient method to produce good magnetic quality high-curie temperature Mn 5 Ge 3 thin films.

The ferromagnetic compound Mn 5 Ge 3 is expected to allow spin current injection in Ge [1][2][3][4]. Consequently, the use of Mn 5 Ge 3 thin films as ferromagnetic contacts on the source and drain of Ge-based transistors is currently investigated in order to produce spintronic devices [4][5][6]. In the literature, Mn 5 Ge 3 films are generally obtained via reactive diffusion, reproducing the "Self-aligned silicide process" used in the Si Complementary metal oxide semi-conductor (CMOS) technology to produce ohmic contacts on transistor's source, drain and gate [5][6][7]. Usually, an Mn film is deposited on an Ge substrate by molecular beam epitaxy (MBE) [6,8] before to be in-situ thermally annealed under ultra-high vacuum, in order to form a Mn 5 Ge 3 layer at the Mn/Ge interface. The control of the annealing conditions (time, temperature) allows the Mn 5 Ge 3 layer to grow until the complete consumption of the initial Mn layer, without the formation of the next compound in the phase formation sequence Mn 11 Ge 8 . The obtained Mn 5 Ge 3 layer can be monocrystalline in epitaxy on Ge if the Mn film is deposited on a Ge(111) substrate [6,8]. This self-aligned germanide process has the benefit to be compatible with industrial CMOS processes [5]. However, the Curie temperature (T c ) of Mn 5 Ge 3 (~ 297 K) needs to be increased in order to obtain spin current injection at temperatures higher than room temperature (RT) [5,[9][10]. Usually, this is achieved using C [7,[10][11][12][13][14] or Fe doping [15], the dopant being co-evaporated with Mn on the Ge substrate, complicating the deposition process. Furthermore, ultra-high vacuum MBE growth is not currently used in CMOS process lines, due to its significant production cost. Metallic films used for ohmic contact fabrication in CMOS technology are deposited by magnetron sputtering. In addition, one of the central drawbacks of Mn 5 Ge 3 films grown on Ge is the weak stability of the film, which can transform into Mn 11 Ge 8 during a following annealing in following transistor fabrication process steps [5]. Though, electrical spin injection in n-type Ge through a 2 nm-thick Al 2 O 3 tunneling barrier was recently demonstrated, from a 40-nm thick Mn 5 Ge 3 C 0.8 polycrystalline layer grown by Mn, Ge, and C simultaneous magnetron sputtering at 400 °C [4].

In this work, the magnetic properties of Mn 5 Ge 3 films produced by means of two different methods compatible with CMOS technology were investigated by magnetic force microscopy (MFM), superconducting quantum interference device (SQUID) and Ferromagnetic resonance (FMR). The magnetron sputtering technique [4,9] is shown to produce Mn 5 Ge 3 films exhibiting similar magnetic properties as Mn 5 Ge 3 films produced by MBE. Furthermore, non-diffusive reaction (NDR) from an amorphous Mn 0.625 Ge 0.375 film leads to the formation of Mn 5 Ge 3 films exhibiting higher T c and better magnetic properties than Mn 5 Ge 3 films produced by reactive diffusion (RD) of Mn and Ge layers deposited in same conditions. Two types of samples were prepared using a commercial magnetron sputtering system exhibiting a base pressure of 10 -8 Torr and allowing simultaneous sputtering of three different targets. All the layers were deposited at RT in same conditions (Ar gas pressure…) using a 99.9999% pure Ar gas flow to sputter a 99.99% pure Ge target and a 99.9% pure Mn target.

The sputtered layers were deposited on a SiO 2 layer grown on an Si(001) substrate. The SiO 2 layer acts as a diffusion barrier, preventing any intermixing between the layers and the Si substrate. For RD experiments, a 31 ± 2 nm-thick polycrystalline Mn layer was deposited on top of a 202 ± 13 nm-thick amorphous Ge layer. A sputtered Ge substrate was preferred to a commercial Ge wafer, allowing the results obtained with this sample to be fully comparable with the results obtained with the second type of sample (sputtered Ge in the two cases). For NDR experiments, a 45 ± 4.5 nm-thick amorphous Mn 0.625 Ge 0.375 film was directly deposited on the SiO 2 layer. These two types of samples were in situ annealed under vacuum (10 -6 Torr) in an X-ray diffraction (XRD) setup following a heating ramp made of 5 °C per minute steps separated by 4 minutes-long XRD measurements at constant temperature (T), corresponding to an average heating ramp of 1 °C min 1 . The XRD measurements were performed between RT and 400 °C in the Bragg-Brentano geometry, using a Cu K  source ( K = 0.154 nm). The thicknesses of the films were measured before and after annealing using the X-ray reflectivity technique. The thickness (t) of the Mn 5 Ge 3 film in contact with Ge obtained by RD was found to be t ~ 48 ± 5 nm, while the thickness of the Mn 5 Ge 3 film in contact with SiO 2 obtained by NDR was found to be t ~ 47 ± 2.5 nm. The sample surface topography was studied using a NT-MDT SMENA atomic force microscope (AFM) in air in non-contact mode. The magnetization component perpendicular to the sample surfaces was studied at RT by MFM in air, while the magnetic properties of the films were studied using SQUID and FMR measurements performed at RT and at 200 K with a conventional Bruker EMX spectrometer operating at f = 9.62 GHz.

In situ XRD measurements (not presented here) showed the same phase formation sequence as in the case of RD between a polycrystalline Mn film and an amorphous Ge layer deposited by e-beam evaporation [5]. Only two phases are observed in the sequence, the first phase being Mn 5 Ge 3 and the second phase being Mn 11 Ge 8 . However, in the case of e-beam evaporation, Mn 5 Ge 3 was found to form at T ~ 210 °C and to transform into Mn 11 Ge 8 at T ~ 310 °C [5], while in the present case, Mn 5 Ge 3 was found to form at ~ 240 °C and to be stable up to temperatures as high as 400 °C. In the case of Mn 5 Ge 3 formation by NDR, in situ XRD measurements showed only the formation of Mn 5 Ge 3 at T ~ 225 °C. Fig. 1 presents the diffractograms obtained at RT after annealing the RD (a, red line) and NDR (b, black line) samples up to 400 °C. For the RD sample, three diffraction peaks corresponding to Ge (2 = 27.5°, 45.6°, and 53.9° corresponding respectively to the Ge planes (111), (220), and (311)) are easily detected (triangles in fig. 1). They correspond to the 200 nm-thick Ge layer that crystallized during annealing [5]. In addition, six diffraction peaks at 2 = 30.9°, 35.8°, 38.7°, 42.7°, 43.8°, and 56.7°, corresponding respectively to the atomic planes (111), (002), (210), (211), (112), and (311) of Mn 5 Ge 3 are also observed. No peak belonging to Mn 11 Ge 8 can be detected [5]. For the NDR sample, the same six diffraction peaks corresponding to Mn 5 Ge 3 are also observed, with an additional peak at 2 = 46.3°, corresponding to the Mn 5 Ge 3 (202) planes. The relative intensity of the Mn 5 Ge 3 peaks is different in the two samples. For example, the crystallographic direction exhibiting the maximum intensity in the RD sample is the (211), while it is the (002) direction in the NDR sample. Thus, the two Mn 5 Ge 3 layers do not possess the same texture. Fig. 2 presents AFM images of the sample surfaces. The grains are easily observed due to grain boundary (GB) grooving that occurred during annealing, resulting from equilibrium between the surface and interface energies at GBs [16][17][18][19]. For example, fig. 3a presents a one-dimensional profile (1D) measured on the NDR sample surface by AFM. GB grooves can exhibit a large width in the two samples (> 150 nm).

However, their depth is about 3 nm that is negligible compared to the Mn 5 Ge 3 layer thickness (~ 50 nm). The average grain width (L) was determined considering that the lateral size of a given grain corresponds to the distance between the bottoms of the grooves located on each side of the grain [16][17][18][19], as presented in fig. 3a. L was averaged over all the grains detected on AFM images of 25 × 25 µm 2 and 15 × 15 µm 2 for the RD and NDR samples, respectively [20][21]. L ~ 1.09 µm and the surface roughness (root mean squared) r ~ 1.5 nm for the Mn 5 Ge 3 layer produced by RD, while L ~ 0.4 µm and r ~ 1.2 nm for the Mn 5 Ge 3 layer produced by NDR. The Scherrer equation [22] applied to the XRD data of the RD-and NDRmediated layers gives for the grain size in the direction perpendicular to the surface d = 51.6 nm and d = 47.7 nm, respectively. Thus, the two layers present a similar surface roughness, and in both cases, the grains are columnar (t ~ d) and their lateral size is larger than the Mn 5 Ge 3 layer thickness (L  t). Though, Mn 5 Ge 3 grains are about two times larger in the layer obtained by RD. Fig. 4a and 4b present the MFM images (phase contrast) obtained on the sample surfaces at RT without external magnetic field. Magnetic domains are visible on the two samples, showing that the samples are ferromagnetic at RT. However, the magnetic domains are more easily detected on the NDR sample (fig. 3b), with a maximum contrast  ~ 1.7°, than on the RD sample, with a maximum contrast  ~ 1.4°. The domains do not follow the sample surface topologies. The average domain width is found to be ~ 0.65 ± 0.2 µm and ~ 0.39 ± 0.1 µm, and the average domain length is found to be ~ 1.5 ± 0.2 µm and ~ 0.9 ± 0.1 µm, on the RD and NDR samples, respectively. The angular () variation of the resonance field Hr is shown in fig. 4c. These measurements confirm that due to the film geometry, the samples possess a shape anisotropy with an easy axis in the direction parallel to the surface ( = 0°) and a hard axis in the direction perpendicular to the surface ( = 90°).

Despite that the samples were produced using the same deposition technique in same conditions, the Mn 5 Ge 3 layer produced by NDR exhibits a resonance field difference Hr between the hard (Hr  ) and easy axis (Hr // ) significantly higher than in the case of the Mn 5 Ge 3 layer produced by RD. In addition, the results obtained on the RD sample are similar to the results obtained on a monocrystalline Mn 5 Ge 3 layer in epitaxy on Ge(111) produced by RD of an Mn layer deposited by MBE on a Ge(111) substrate [14]. For the latter, Hr // and Hr  were found to be about ~ 2.7 and 5.0 kOe, respectively, giving Hr ~ 2.3 kOe at RT [14]. In our case, Hr // ~ 3.1, Hr  ~ 4.8, and Hr ~ 1.7 kOe for the polycrystalline Mn 5 Ge 3 layer obtained by RD. Hr depends on the ferromagnetic layer T c . For example, Hr ~ 2.3 kOe was shown to correspond to T c = 315  10 K for the epitaxial MBE-mediated Mn 5 Ge 3 layer [14]. This is in agreement with the expected T c ~ 297 K of the Mn 5 Ge 3 compound, as well as with our MFM measurements (fig. 4a) showing a faint signature of ferromagnetic domains at RT. Contrasting with Mn 5 Ge 3 epitaxied layers [14], the FMR measurements can be well simulated using the Chappert model (solid lines in fig. 4c) containing only two magnetic anisotropies [23]: the shape and magnetocrystalline anisotropies. This is probably due to the polycrystalline nature of the present Mn 5 Ge 3 films. Using the RT Mn 5 Ge 3 saturation magnetisation M = 360 emu cm 3 [24], the magnetocrystalline anisotropy constants (K 1 and K 2// ) of the polycrystalline RD-mediated Mn 5 Ge 3 layer were calculated considering a gyromagnetic ratio  /2 = 2.8 GHz kOe 1 [23]. We found K 1 = 0.57 × 10 6 erg cm 3 , and K 2// = 5.42 × 10 4 erg cm 3 . FMR measurements compared to SQUID measurements performed on C-doped epitaxial MBE-mediated Mn 5 Ge 3 C x layers showed that Hr ~ 3.7 kOe (Hr // ~ 2.3 and Hr  ~ 6.0 kOe) for x = 0.1 corresponds to T c = 345  10 K, and Hr ~ 7.0 kOe (Hr // ~ 1.6 and Hr  ~ 8.6 kOe) for x = 0.2 corresponds to T c = 450  30 K [14]. Hr // ~ 2.0, Hr  ~ 7.5, and Hr ~ 5.5 kOe in the polycrystalline Mn 5 Ge 3 layer obtained by NDR (fig. 4c), which suggests that the T c in the NDR-mediated Mn 5 Ge 3 layer is larger than in the RD-mediated layer, and is comprised between 345 and 450 K. SQUID measurements were performed on the samples in order to confirm the FMR measurements. As expected, the magnetization variations versus temperature presented in fig. 5 show that the Curie temperature is ~ 300 K and ~ 385 K, in the RD-and NDR-mediated Mn 5 Ge 3 films, respectively. FMR measurements were also performed at 200 K (not shown here). However, the signal from the RD-mediated layer being too broaden (red open symbols in the inset in fig. 4c), measurements could be performed only on the NDR-mediated Mn 5 Ge 3 layer (black solid symbols in the inset in fig. 4c). We measured Hr // ~ 1.3 kOe, Hr  ~ 9.1 kOe, and Hr ~ 7.8 kOe in this layer at 200 K. The significant FMR signal difference at 200 K between the two Mn 5 Ge 3 layers presented in the inset in fig. 4c shows clearly that the NDR-mediated layer exhibits better magnetic properties than the RD-mediated layer [25].

T c in Mn 5 Ge 3 is mainly related to Mn-Mn atoms interactions [26], and thus, depends on the distance between Mn atoms in the compound. This distance can be modified by stress and/or alloying with a third element. The Mn 5 Ge 3 bulk compound possess a hexagonal structure, with the standard (calculated) lattice parameters a = b = 0.7185 nm and c = 0.5053 nm [27]. These parameters were measured in the samples from the XRD data presented in fig. 1. For the RD mediated layer, the lattice parameters were found to be a = b = 0.7158 nm and c = 0.5014 nm. For the NDR mediated layer, the lattice parameters were found to be a = b = 0.7135 nm and c = 0.5016 nm. These data are very similar, the difference between the measurements being in the sub-angstrom range (10 2 angstrom), it is related to measurement accuracy. Consequently, the T c variations observed between the RD-and NDR-mediated samples cannot be explained by stress effect. In our case, there is no alloying effect, since Ge and Mn were the only elements sputtered on the substrates. However, sputtered layers are known to contain C and O atoms with concentrations that can reach several percent depending on the purity of the Ar gas and of the targets, as well as on the deposition rate and of the setup base pressure. The C doping effect allowing for T c increase in Mn 5 Ge 3 (C) was shown to be of electronic nature, interstitial C atoms in the Mn 5 Ge 3 compound promoting the appearance of the 90° ferromagnetic superexchange between Mn atoms [26]. Thus, the reason for the T c difference between the two Mn 5 Ge 3 layers could be related to a difference of C incorporation, related to the two reaction types RD and NDR. Usually, RD allows impurities such as C and O to be pushed-out to interfaces and to GBs during the layer growth, allowing the formation of thin films significantly less contaminated than the initial sputtered layers [28]. At the opposite, atomic diffusion being not necessary during NDR [29], C incorporation could be more effective during this process, leading to a T c increase in NDR-mediated Mn 5 Ge 3 layers.

In conclusion, Mn 5 Ge 3 layers were grown on SiO 2 using thin film magnetron sputtering followed by thermal annealing up to 400 °C. Two different reaction types were investigated: RD and NDR. Magnetron sputtering is shown to allow polycrystalline Mn 5 Ge 3 layers, exhibiting similar magnetic properties as MBE-mediated monocrystalline layers in epitaxy on Ge(111), to be produced. RD-mediated Mn 5 Ge 3 layers exhibit magnetic properties matching usual Mn 5 Ge 3 properties, while NDR-mediated Mn 5 Ge 3 layers show similar

FIGURE CAPTIONS FIG. 1 .FIG. 2 .FIG. 3 .FIG. 4 .FIG. 5 .

 CAPTIONS12345 FIGURE CAPTIONS

Figure 1 E

 1 Figure 1

Figure 2 E

 2 Figure 2

Figure 3 E

 3 Figure 3

Figure 4 E

 4 Figure 4

Figure 5 E

 5 Figure 5

magnetic properties as C-doped Mn 5 Ge 3 C x layers with 0.1  x  0.2 and T c ~ 385 K. The difference of magnetic properties between the layers produced by RD and NDR could be related to a difference of C incorporation in the Mn 5 Ge 3 layers, RD leading to a layer purification process decreasing C incorporation and, at the opposite, NDR supporting C incorporation. The NDR process appears as an interesting technique to produce low-cost high-Curie temperature Mn 5 Ge 3 layers. This work was supported by the French government through the program "Investissements d'Avenir A*MIDEX" (Project APODISE, no. ANR-11-IDEX-0001-02) managed by the National Agency for Research (ANR).