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Theoretical aspects of sputtering of magnetic materials near the Curie point
Vacuum 66 (2002) 123–132
Theoretical aspects of sputtering of magnetic materials near the Curie point Yu.N. Devyatkon, S.V. Rogozhkin Moscow Engineering and Physics Institute (the State University), Kashirskoe shosse 31, Moscow 115409, Russia Received 24 January 2002
Abstract A new approach for description of anomalies in sputtering of magnetic materials near the Curie temperature has been proposed. The energy of sublimation is shown to have no anomalies in this temperature range. The anomalies in sputtering of magnetic materials are connected with significant increase of evaporation of weakly-bounded surface atoms from the hot spots created by incident ions. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Magnetic; Sputtering; Curie temperature; Sublimation; Heisenberg model
1. Introduction Many experiments have shown significant difference in the sputtering yield Y of magnetic materials in ferro- and paramagnetic states (10– 20% under various experimental conditions) (see for example [1,2]). In addition, oscillations and sharp peaks (1.5–2 times) of the sputtering yield have been observed if magnetic is sputtered by noble-gas ions near the Curie temperature TC : Such anomalous behaviour for Ni takes place in the interval from 701 to 1001. The characteristic temperature dependences of the sputtering yield for Ni under Ne+ and Ar+ irradiation are given in Figs. 1 and 2 [2,3]. The Sigmund theory of linear cascades is commonly accepted for description of sputtering in the energy interval where yield anomalies are observed [4]. Its main assumptions are: the binary n
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character of collisions of moving atoms and existence of the energy threshold for target atom removal. It is also assumed that the threshold energy equals the sublimation energy es of the target material, being a few electronvolts in metals (for example es ¼ 3:8 eV for Ni). According to the Sigmund theory [4], the sputtering yield is defined as YZ ¼
SðE; M1 ; M2 ; Z1 ; Z2 Þ ; eas
ð1Þ
where aC1 at low and medium energies of bombarding particles; S is a function determined by kinematics of collisions; M1 ; M2 ; Z1 ; Z2 are the masses and charges of the target atom and incident ion, respectively. The general theory of magnetic phenomena, which includes the magnetic phase transition, is well developed. There are a number of works devoted to analyses of thermally activated processes near second-order phase transitions [5–7].
0042-207X/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 2 - 2 0 7 X ( 0 2 ) 0 0 1 7 5 - 6
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below. This is why, one cannot explain the 2–2.5 fold peak of sputtering near the Curie point within the framework of the Sigmund theory. This work deals with theoretical description of ion sputtering of magnetics near the Curie point.
2. Sublimation
Fig. 1. The temperature dependence of the (0 0 1) Ni facesputtering yield under 20 keV Ne+ ions incident in the (1 0 0) plane at 401 to the surface (current density 1.5 mA cm2) [1,2].
Fig. 2. The temperature dependence of the (0 0 1) Ni facesputtering yield under 20 keV Ar+ ions incident in the (1 0 0) plane, at 401 to the surface (current density 1.5 mA cm2) [1,2].
However, the above-mentioned phenomena are described using models; and sputtering anomalies observed experimentally have not yet been explained. This is connected with the low value of magnetic interactions in comparison with the energy of sublimation. Even in conditions of complete magnetization of the materials, the magnetic contribution to the energy is several times o1 eV. Furthermore, even this small contribution is achieved at temperatures several hundred K below the Curie point. As for magnetic fluctuations, they give an even smaller contribution as we will show
(A) Previously, the sputtering yield anomalies near the Curie point have been related to features of the sublimation energy es (see for example [2,8]). Therefore, it is reasonable to investigate the sublimation process of magnetic materials near TC to verify this possibility. Activated processes in the region of the magnetic phase transitions have been investigated for a long time (see for example [9]). However, there are only two experimental works [10,11] on the sublimation of a monatomic magnetic metal (it was Co) near TC : Fig. 3 shows the experimental temperature dependences of the deposition rate of evaporated Co [10] on a substrate. At a temperature below the Curie point, the magnetic contribution to the effective sublimation energy decreases, and this is clearly seen in Fig. 4 (circles). Assuming the Arrhenius dependence of the desorption rate on the temperature, the authors have shown that the magnetic contribution to the sublimation energy is well described by the mean-field approach to magnetic interaction. Only one fitting parameter A ¼ 2 is necessary to get a good agreement between the experiment and the calculations (the dashed line in Fig. 4). It follows from Fig. 4 that there is no evident anomaly of the sublimation energy near TC : The sublimation energy of Co in the paramagnetic state is 4.16 eV, and that in the totally ordered ferromagnetic state is estimated to be 4.36 eV. The main experimental results on evaporation of Co from [11] are shown in Figs. 5 and 6 (circles). Different curves in Fig. 5 correspond to different rates of either heating or cooling of the specimen. Using the assumption of the Langmuir dependence of the pressure on the temperature, the authors [11] obtained the magnetic contribution to the sublimation energy by subtraction of some
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Fig. 3. Logarithm of the co-deposition rate as a function of the inverse absolute temperature. Circles—the experiment [10]; solid line—linear approximation of the experimental data for T > Tc ; dashed line—extrapolation of the solid line to the region of ToTc :
‘‘equilibrium’’ dependence from the experimental one. A typical result is given in Fig. 6. The peculiar behaviour of Dln pðTÞ is obvious in the vicinity of the magnetic phase transition. A theoretical description of anomalous dependences of the sublimation rate behaviour was proposed in the work [12]. It was suggested that only those particles sublimate that have the spin inverted with respect to that of the domain. However, this suggestion was substantiated in no way and contradicts the results of [10]. At least two observations made in [11] cast doubt upon the interpretation of the experimental results. First, the sublimation curves in Fig. 5 depend on the rate and the sign of the temperature change. This indicates that there is no equilibrium in the experiment and its treatment is definitely incorrect in the Langmuir approach. Secondly, analysis of the curve in Fig. 6 at temperatures T >
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TC indicates that subtraction of the equilibrium dependence ln pðT > TC Þ was made in an incorrect way. It follows from the fact that the resulting curve depends on the temperature at T > TC : If one subtracts the dependence of the pressure in the paramagnetic interval pðT > TC Þ from the experimental data, the result should be the magnetic contribution to the sublimation energy. We reanalysed the experimental data of [11] and made an accurate subtraction. The result is represented in Fig. 7, where the dependence of Dln pðTÞ on T=TC is shown. What attracts additional attention, is the opposite sign of the effective magnetic contribution to the sublimation energy in Fig. 4 [10] and that in Fig. 7 obtained from the experiment [11]. The least-squares method of fitting to the Langmuir approach gives the paramagneticstate sublimation energy of about 4.5 eV for the data [11]. This value is close to that for the ferromagnetic state from [10]. The peaks near TC being about some hundredth of an electronvolt are too small to be recognised as reliable as they are within the experimental uncertainty. To sum up, accurate analyses of the two experiments on Co sublimation do not permit us to speak about any pronounced anomalies near the Curie point. No anomalies of the sublimation rate have been observed in the vicinity of the Curie point in [10] and these are negligible in [11]. Other results of [10,11] contradict each other. For example, the contribution of magnetic ordering to the effective sublimation energy in one experiment is opposite in sign to that in another. This is possible due to non-equilibrium conditions of the experiment [11]. (B) Let us use the general theory of magnetic phenomena to study the magnetic sublimation rate. The task of addressing the equilibrium between solid and gas phase of self-atoms is well known, see for example [13]. After minimising the free energy of the system with constant number of atoms, it is easy to calculate the equilibrium number of atoms in the gas phase Ng e m s m Ng ðTÞBexp ; ð2Þ T where mm is the contribution of magnetic interaction to the chemical potential of solid-state atom.
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Fig. 4. Magnetic contribution to the sublimation energy em for Co as a function of the reduced temperature T=T c. The dots are obtained from the experimental data [10] shown in Fig. 3. The dashed line for T=Tc r1 is the best fit of em ðTÞ ¼ ATc ðMðTÞ=Mð0ÞÞ2 to the data shown by dots and corresponds to A ¼ 2 (here MðTÞ is the magnetization at temperature T). The solid lines are the results of calculation according to (3) in the mean field approximation for S ¼ 1 (curve 1) and 2 (curve 2). The contribution of the fluctuations to the sublimation energy efluct (second term in (3) is displayed in the inset).
Hereafter, the temperature is given in eV in all the formulae. Obviously, additional attraction results in an increase of both the sublimation energy ðmm o0Þ and the slope of the experimental curves. According to the Heisenberg model, the contribution of the magnetic interaction in the longwave approximation is (see Appendix A) Z 1 a3 SðS þ 1Þ T d~ q mm ¼ J0 S 2 j2 þ 0 3 2 c 6ð2pÞ 2 njT þ cJ0 j þ cJ1~ q Jð~ qÞ : ð3Þ ln T Here j is the reduced magnetization, Jð~ q Þ is the Fourier transform of the exchange interaction (Jð~ q ÞEJ0 þ J1~ q 2 in the long-wave approximation), c is a constant determined by the total atomic magnetic moment S; a0 is the lattice constant. The first term in (3) is a mean-field contribution of magnetic ordering. The second term is a result of fluctuation of the magnetic subsystem. Integration over the first Brillouin zone is carried out in (3). The maximum contribution of magnetic ordering is ATC ; that is, it is determined by the Curie temperature. The parameter A was calcu-
lated according to the Heisenberg model (see Appendix A) A¼
9S 2 : 2SðS þ 1Þ
ð4Þ
This formula gives A ¼ 2:25 in the case of the total atomic magnetic moment S ¼ 1; and A ¼ 3 for S ¼ 2: The tabulated value of the atomic magnetic moment for Co given in [16] is 1.72. In the previous section, we mentioned that the fitting parameter A = 2 was used in [10] to describe the experimental dependence. The respective energies of magnetic ordering obtained using A (4) and TC are 0.27–0.36 eV for Co and 0.08 eV for Ni. The values of the energy of magnetic ordering, which are known from the literature, are: 0.2 eV for Co [10] and 0.08–0.09 eV for Ni (obtained from the exchange splitting D0 ¼ 0:33 0:37 eV [14,15]). One can see that agreement between the calculated numbers and the number known from literature is really good. The results of calculations of the effective energy jmm j; which takes into account magnetic interactions, are shown in Fig. 4 (the solid lines). The mean-field approach well describes the behaviour
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Fig. 6. The temperature dependence of Dln p ¼ ln pL ln peq [11]. Circles—the experimental data (the data I see Fig. 5); solid line—spline of the experimental data.
Fig. 5. Logarithm of the Co vopour pressure as a function of the inverse absolute temperature for three different heating rates. Dots—experimental data [11]; solid lines—results of calculation by (6). Dots I and line 1 are obtained by the cooling of the sample. Dots II, III and lines 2, 3 are obtained by heating of the sample. The respective rates of cooling and heating are 1 1 V1 : V2 : V3 ¼ 13 : 25 : 1:5 : The Solid line 0 is the equilibrium dependence V ¼ 0 (6).
of the experimental curve [10] without using any fitting parameter. Fluctuations contribute to (3) not only due to the energy change but also due to the entropy change at sublimation. These changes compensate each other, and the resulting contribution (for Co and Ni) does not exceed 0.01 eV, which is negligibly small (see the insert in Fig. 4). Thus, one should not expect any noticeable anomalies in the process of equilibrium sublimation of Co as well as of other monatomic magnetics near the Curie point. The difference between the curves corresponding to heating and cooling of the specimen
Fig. 7. The temperature dependence of magnetic contribution to the sublimation energy em ¼ Tðln pL ln pfit ). Here pL is the data from [11] (the data I see Fig. 5), pfit is a linear extrapolation to ToTc of a fitting line describing the data for ToTc :
indicates that the experiment [11] was performed in non-equilibrium conditions. In this case, one cannot restrict the consideration by the classical task of sublimation. The kinetics of non-equilibrium sublimation must be analysed. Taking into account the general mechanisms, the kinetic equation for the average gas concentration can be written as dngas ¼ aðTÞn0 Ongas bngas : dt
ð5Þ
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The first term in the right-hand side describes evaporation of atoms from the surface: aðTÞ ¼ a0 expðes =TÞ: The second term takes into account sorption of atoms by the walls of the experimental chamber. The third one is due to pumping of gas out of the experimental chamber. If the temperature changes linearly as T ¼ T0 þ vt (v is the rate of the temperature change), the gas concentration as a function of the temperature can be found in the first approximation with respect to v as aðTÞ es v 1 2 ngas Cn0 : ð6Þ Oþb T ðO þ bÞ The lines in Fig. 5 represent the results of the pressure calculations by using (6). The sublimation energy was obtained by applying the method of least squares within the chosen paramagnetic temperature interval. One can see that this simple model satisfactorily describes the experimental results at small rates of heating and cooling. The disagreement between the dots III and the line 3 in Fig. 5 is possibly due to high heating rate. This fact needs more thorough analyses of the sublimation kinetics. The results of theoretical calculations described above demonstrate that the anomalies of sublimation energy near the Curie point do not exceed 0.01 eV and cannot explain the magnetic sputtering anomalies [2] in the framework of the Sigmund theory.
3. Sputtering Ion sputtering is a non-equilibrium process. Ion bombardment produces knocking of atoms from the surface, cascades of atom–atom collisions, local heating, sublimation of atoms from the surface and implantation of impinging ions. These phenomena change the topography and properties of the surface layer. The characteristic energy of magnetization is of the order of the Curie temperature, which is about some tenths of eV. Magnetic fluctuations bring a much lesser contribution. On the other hand, the characteristic energy for sputtering is of the order of the sublimation energy es : This is why, within the Sigmund theory one cannot explain the experi-
mental sputtering anomalies [2]. To do this, the Sigmund theory should be modified. First of all, the Sigmund theory does not take into consideration the dependence of sputtering on temperature, though a remarkable increase of sputtering at high temperatures has been found experimentally [17]. Therefore, temperature sensitive processes must be considered. A possible reason for the temperature dependence may be connected with particle evaporation from hot spots. During cooling and spreading of the hot spot, which appears in the region of ion propagation, particles evaporate from the surface. The yield of these evaporated particles is YS Bexpðes =TLOC Þ where TLOC is the local surface temperature. This is determined by the energy released by the incident particle in the hot spot and the average surface temperature, which increases due to beam energy evolution. The evaporation yield is determined by particles, which escape the hot spot over the time period of its existence Z Z ob eb YS ¼ 2 exp ds dt; ð7Þ Tð~ r ; tÞ a0 where ob CoD is the frequency of vibrations of surface atoms (oD is the Debye frequency), eb is the binding energy. The temperature field Tð~ r ; tÞ in the hot spot was calculated using the equation of heat conductivity. It is important to underline that Eq. (7) is correct only in the time interval where the physical magnitude of the local temperature can be defined. The time interval B5 1013 s is considered as the initial one when the hot spot temperature equals the melting temperature Tm : Overlapping of hot spots and general warming-up of subsurface layers was not taken into account in those calculations. It should be remembered that these effects occur in experimental studies with high current densities (B1 mA/cm2). Since the temperature is measured far away from the region irradiated, sputtering anomalies are observed at temperatures below the Curie temperature TC (see Figs. 1 and 2). Furthermore, it is necessary to take into account that the binding energy of atoms sputtered from the surface eb does not equal the sublimation energy es : Generally speaking, the value eb is not a constant, as it is different in different
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atomic states. The change of topography results in appearance of weakly bounded atoms (self-interstitials and adsorbed atoms). Moreover, variation of the surface layer composition due to implantation of atoms from the beam (especially under irradiation by ions of heavy inert gases) significantly reduces the binding energy eb : If eb oes ; the sputtering and the evaporation yields become a function of eb : YZ ¼ YZ ðeb Þ and YS ¼ YS ðeb Þ: Computing of binding energies of atoms in weakly bounded states is not the task of this work. Our suggestion is that the main contribution to sputtering is both from linear cascades (calculated according to the Sigmund theory but with the binding energy eb ; instead of the sublimation energy) and from atomic evaporation. Our purpose at this stage of the work is to estimate the values of the characteristic parameters, which allow us to explain the observed anomalies of magnetic sputtering. To do this, the sputtering yield was calculated for different binding energies eb ; as well as for different concentrations of weakly bounded atoms. The energy eb in the expression for the sputtering yield fluctuates in the vicinity of the magneticphase transition. This circumstance was taken into account by averaging the sputtering yield over fluctuations of the binding energy: X Y¼ W ðeb ÞfYZ ðeb Þ þ YS ðeb Þg: ð8Þ eb
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the bulk of the material. Therefore, we used the expression for /De2b S calculated for the bulk of theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi material. The temperature dependence of q /De2b S calculated for Ni is shown in Fig. 8.
The peak of /De2b S near the Curie point leads to broadening of the Wðeb Þ distribution function. This is due to fluctuations of the magnetic subsystem. For sputtering of Ni by Ar+ ions with the energy of 20 keV, the results of the calculation of the sputtering yield at various binding energies eb (here eb is understood as the average value /eb S) are shown in Fig. 9. All the surface atoms were assumed to have the same binding energy in the course of the calculation. The absolute value of the experimental anomaly of sputtering can be explained at the binding energy eb C1:421:45 eV. This value is far less than the sublimation energy but it is not abnormally small. Moreover, this value is much higher than that of the formation of point defects on the surface [18,19]. The sputtering yields calculated by using three different models are given in Fig. 10. One can see that particle evaporation gives a really essential contribution at temperatures near the Curie point. If sputtering is only due to atoms weakly bound to the surface, a value of the binding energy less than that of calculated above (eb C1:4 eV) is required to describe the experimental sputtering anomalies. For example, if the concentration of weakly bounded atoms is 10%, the anomalies can
The distribution function W ðeb Þ can be found by microscopic consideration of material behaviour near the Curie point. In the Gauss approach, W ðeb Þ is determined both by the average binding energy /eb S of the atom in the state under consideration and by the correlator of fluctuations /De2b S: 1 ðeb /eb SÞ2 W ðeb Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp : ð9Þ 2/De2b S 2p/De2b S The values /eb Sand /De2b S were calculated according to the Heisenberg model (see Appendix A). Around TC ; fluctuations have a long wavelength and, therefore, surface atoms are in the field of fluctuations arising due to phase transitions in
Fig. 8. Temperature dependence of the mean square fluctuation q ffiffiffiffiffiffiffiffiffiffiffiffi of the energy s ¼ De2b near the Curie point.
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Fig. 9. The temperature dependence of the (0 0 1) Ni facesputtering under 20 keV Ar+ ions. Solid line—results of calculation by (8) for the binding energies of surface atoms: 1.5 eV (curve 1), 1.45 eV (curve 2), and 1.4 eV (curve 3); circles—results of the experiment [1,2].
energies. The calculations (Fig. 10) demonstrated that formation of 1–2 atoms with eb C0:8 eV in the hot spots region is good enough for explanation of anomalies of sputtering near TC : To sum up, non-equilibrium evaporation of surface atoms has been shown to give significant contribution to sputtering of surface atoms under ion irradiation. It is not reasonable to use the sublimation energy as the parameter that determines sputtering. Instead, one must use the average binding energy of surface atoms that depends upon the parameters of the ion beam. Let us mention that sputtering coefficients YZ and YS ; generally speaking, depend on different eb ; which are the characteristics of the two mechanisms. This does not change the results obtained because the main contribution to sputtering in the vicinity of TC is due to atom evaporation.
4. Conclusions
Fig. 10. Temperature dependence of the sputtering yield calculated for a binding energy of surface atoms of 1.45 eV. Curve 1—the mechanism of linear cascades (the Sigmond theory) without taking into account the magnetic fluctuation W (eb) (8); curve 2—mechanism of linear cascades with taking into account the magnetic fluctuations; curve 3—result of calculation by (8) taking into account the magnetic fluctuations, heat spots and two sputtering mechanisms: linear cascades and evaporation.
be explained using a binding energy eb C1:1 eV. The dependence of the fitting binding energy on the concentration of weakly bound atoms is shown in Fig. 11. Let us mention that the early stage of ion– surface interaction is accompanied not only by direct knock out of atoms, but also by formation and evaporation of atoms with low binding
Analyses of experimental data available from literature and theoretical calculations performed in this work have demonstrated that there are no pronounced anomalies of equilibrium sublimation near the Curie point. The contribution of magnetization into the sublimation energy is determined by the Curie temperature. The maximum contribution is achieved at T5TC and is of about several TC : Near the Curie temperature, this contribution is only of 0.01–0.02 eV. The contribution of fluctuations in the sublimation energy is even less. These calculations were performed within the framework of the Heisenberg model. The only parameters used were: the Curie temperature, the lattice constant and the value of the spin of the metal atom. It has been shown that sputtering near TC is determined by two mechanisms: linear collision cascades and non-equilibrium evaporation. Ion bombardment changes the structure and composition of the near-surface region. This is why, the atom-to-surface binding energy, but not the sublimation energy, has to be used in the theory of linear cascades [4]. The evaporation mechanism dominates in the temperature range under consideration due to simultaneous influence
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at several factors. Ion–solid interaction produced (along with linear cascades and surface modification) states of atoms with low binding energies as well as local hot spots. These factors take place at any temperature, but their influence is usually small except the region of the ferromagnetic phase transition. In this region, long lived (p109 s) fluctuations of the magnetic subsystem appear, which lead to fluctuations of the binding energy of surface atoms. Simultaneous acting of the factors mentioned makes the evaporation process to be the most important in sputtering. Magnetic fluctuations lead to evaporation of atoms, which have low binding energy, during expansion of the hot spot formed by an incident ion. Calculations demonstrated that the anomalies of sputtering of Ni near the Curie point, which were observed experimentally, can be explained if one has to use the binding energy of 0.8–1.5 eV in the hot spot region.
Acknowledgements The authors thank Professors V.E. Yurasova and A.A. Pisarev for useful discussion. This work was possible due to partial support of the Russian Foundation for Basic Researches (grant 01-0217934) and the Russian Ministry of Education (the program ‘Universities of Russia’, Grant 015.02.01.31).
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one can rewrite the Hamiltonian (A.1) as 1 X ~2 SÞ: Jð~ q ÞðSð~ q ÞSð~ q Þ N/S H¼ 2N ~q
ðA:3Þ
Here Jð~ q Þ is the Fourier transform of the exchange ~2 S is the mean square spin of potential, /S ~2 S ¼ ð1=NÞ P S ~2 : magnetic material: /S R To calculate the magnetic interaction contribution to the thermodynamic potentials, it is necessary to calculate the partition function: X H Z¼ exp : ðA:4Þ T ~ ~ fS ðR Þg
The summation in (A.4) is taken over various ~ in each point R ~: In the course of states of spin S calculation of the partition function (A.4), the method of collective variables developed by Galitskiy and Bogolubov [20] was used. The ~ðR ~Þg was replaced by integrasummation over fS tion over the collective variables: Z Y H ~ ~ðR ~ð~ ~Þ-S Z¼ dS ð~ q Þ exp IðS q ÞÞ ðA:5Þ T ~ q ~ðR ~ð~ ~Þ-S here IðS q ÞÞ is the Jacobian of change of variables: ~ðR ~ð~ ~Þ-S IðS q ÞÞ ¼
X Y
~ð~ d S q Þ:
~ðR q ~Þg ~ fS
X
!
~ðR ~Þ expði~ ~Þ : S qR
ðA:6Þ
~ R
Appendix A The Hamiltonian of magnetic interactions in the Heisenberg model is 1X ~ ~ðR ~ðR ~j ÞS ~i ÞS ~j Þ; H¼ JðR i R ðA:1Þ 2 i;j ~ðR ~Þ is an exchange potential, S ~Þ is the where JðR ~: The total spin operator of the atom in the point R summation in (A.1) is taken over the lattice ~: points R ~ to collective variables Changing from R X ~ð~ ~ðR ~Þexpði~ ~Þ; S qÞ ¼ S qR ðA:2Þ ~ R
Using the methods [21,22] for calculation of the Jacobian, one can obtain the partition function (A.5): Z Y 1 X T ~ Z ¼b dS ð~ q Þ exp JðqÞ þ 2NT ~q c ~ q 1 X ~ð~ ~ð~ ~2 S S q ÞS qÞ þ JðqÞ/S 2N ~q þ
X d ~ð~ ~ð~ ~ð~ ~ð~ q 2 ÞS q 3 ÞS q4Þ S q 1 ÞS 3 N ~q þ~q þ~q þ~q ¼0 1 2 3 4 !
þy :
ðA:7Þ
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The parameters b; c; d in (A.7) are determined by the magnetic model and the value of the total atomic spin S: So, in the case of single preferential direction, we have c ¼ SðS þ 1Þ=3: If there are three independent directions, c ¼ SðS þ 1Þ=9: The latter c value describes the experimental data [10] rather well, therefore it was used for further calculations. Expression (A.7) describes the magnetic-phase transition. This phase transition occurs if the ~ð~ ~ð~ coefficient of S q ÞS q Þ vanishes. First of all, this condition realizes for the wave vector ~ q ¼ 0: Below the phase transition point, the mode with the wave vector ~ q ¼ 0 corresponds to macroscopic magnetization and should be distinguished in (A.7). Let us note, that the expression for the Curie temperature can be obtained from the condition of phase transition: TC ¼ cjJð~ q ¼ 0Þj:
ðA:8Þ
In all the calculations made in this work, the following exchange potential Jð~ q Þ was used in the long-wave approximation: Jð~ q ÞEJ0 þ J1~ q2:
ðA:9Þ
The value of J0 ¼ Jð~ q ¼ 0Þ was determined by the Curie temperature (A.8). The nearest neighbour approach was used to calculate J1 ¼ 1=2J~q00¼0 : The expression for the partition function (A.7) ~ð~ contains series over S q Þ in the exponent index. These series can be reduced to the square approximation, which is the Gaussian one. This approximation agrees with the approach of the self-consistent field [21]. Taking the mode ~ q¼0 ~ð~ and integrating over S q a0Þ; one can obtain the expression for the partition function and calculate the corresponding chemical potential: Z 1 a30 SðS þ 1Þ T 2 2 mm ¼ J0 S j þ d~ q 3 2 c 6ð2pÞ 2 njT þ cJ0 j þ cJ1~ q Jð~ qÞ : ðA:10Þ ln T Here j is the magnetization of the P reduced ~ðR ~Þ=SN and n is the temperaspecimen j ¼ R~ S ture dependence coefficient n ¼ 1 at T > TC and n ¼ 2 at ToTC : The expression for the magnetic contribution to the energy of the atom can be
obtained by 1 a3 SðS þ 1Þ em ¼ J0 S 2 j2 þ 0 2 6ð2pÞ3 Z T d~ q Jð~ qÞ 1 : ðA:11Þ njT þ cJ0 j þ cJ1~ q2 The expression for the mean-square deviation /De2m S can also be obtained in the similar way.
References [1] Yurasova VE, Chernysh VS, Kuvakin MV, Shelyakin LB. Pis’ma Zh Exp Teor Fiz 1975;21:197. [2] Yurasova VE. Vacuum 1983;33:565. [3] Sheliakin LB, Martynenko TP, Bischoff A, Yurasova VE, Schaarschmidt G. Poverhnost’ (Russ J Surf Phys) 1983;6:65. [4] Sigmund P. In: Behrisch R, editor. Sputtering by particle bombardment. Berlin, Heidelberg, New York: Springer, 1981. [5] Suhl H. Phys Rev B 1975;11:2011. [6] Pregenzer AL, Suhl H. Phys Rev A 1985;31:1718. [7] Hentschel HGE, Procaccia I. J Chem Phys 1982;77:5234. [8] Ivanov VP, Samsonov GA, Tronin VN, Troyan VI. Pis’ma v JTF (Russ Lett J Tech Phys) 1991;17:56. [9] Hedvall JA, Hedin R, Person O. Z Phys Chem Abstr B 1934;27:196. [10] Sales BC, Turner JE, Maple MB. Phys Rev Lett 1980;44:586. [11] Handros VO, Bogolubov NA. Teplofiz Vys Temp (Russ J Therm Phys High Temp) 1983;21:600. [12] Borman VD, Pivovarov AN, Troyan VI. Poverhnost’ (Russ J Surf Phys) 1989;6:13. [13] Kubo R. Statistical mechanics. Amsterdam: North-Holland, 1965. [14] Hodges L, Ehrenreich H, Lang ND. Phys Rev 1966; 152:505. [15] Ponomarev BK, Tissen VG. Sov Phys JETP 1977;73:332. [16] Grigor’eva IS, Meilihova EZ. Physical values. Moscow: Energoatomizdat, 1991. [17] Roosendaal H. In: Behrisch R, editor. Sputtering by particle bombardment. Berlin, Heidelberg, New York: Springer, 1981. [18] Devyatko YN, Rogozhkin SV. Vacuum 2000;56:279. [19] Devyatko YN, Rogozhkin SV, Fadeev AV. Phys Rev B 2001;63:193. [20] Bogolubov NN. Problems of dynamical theory in statistical physics. Moscow, Leningrad: Gostehizdat, 1946. [21] Kudrin LP. Statistical physics of plasma. Moscow: Atomizdat, 1974. [22] Yukhnovsky IR. Second-order phase transitions. Kiev: Naukova Dumka, 1985.
Theoretical aspects of sputtering of magnetic materials near the Curie point Yu.N. Devyatkon, S.V. Rogozhkin Moscow Engineering and Physics Institute (the State University), Kashirskoe shosse 31, Moscow 115409, Russia Received 24 January 2002
Abstract A new approach for description of anomalies in sputtering of magnetic materials near the Curie temperature has been proposed. The energy of sublimation is shown to have no anomalies in this temperature range. The anomalies in sputtering of magnetic materials are connected with significant increase of evaporation of weakly-bounded surface atoms from the hot spots created by incident ions. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Magnetic; Sputtering; Curie temperature; Sublimation; Heisenberg model
1. Introduction Many experiments have shown significant difference in the sputtering yield Y of magnetic materials in ferro- and paramagnetic states (10– 20% under various experimental conditions) (see for example [1,2]). In addition, oscillations and sharp peaks (1.5–2 times) of the sputtering yield have been observed if magnetic is sputtered by noble-gas ions near the Curie temperature TC : Such anomalous behaviour for Ni takes place in the interval from 701 to 1001. The characteristic temperature dependences of the sputtering yield for Ni under Ne+ and Ar+ irradiation are given in Figs. 1 and 2 [2,3]. The Sigmund theory of linear cascades is commonly accepted for description of sputtering in the energy interval where yield anomalies are observed [4]. Its main assumptions are: the binary n
Corresponding author.
character of collisions of moving atoms and existence of the energy threshold for target atom removal. It is also assumed that the threshold energy equals the sublimation energy es of the target material, being a few electronvolts in metals (for example es ¼ 3:8 eV for Ni). According to the Sigmund theory [4], the sputtering yield is defined as YZ ¼
SðE; M1 ; M2 ; Z1 ; Z2 Þ ; eas
ð1Þ
where aC1 at low and medium energies of bombarding particles; S is a function determined by kinematics of collisions; M1 ; M2 ; Z1 ; Z2 are the masses and charges of the target atom and incident ion, respectively. The general theory of magnetic phenomena, which includes the magnetic phase transition, is well developed. There are a number of works devoted to analyses of thermally activated processes near second-order phase transitions [5–7].
0042-207X/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 2 - 2 0 7 X ( 0 2 ) 0 0 1 7 5 - 6
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below. This is why, one cannot explain the 2–2.5 fold peak of sputtering near the Curie point within the framework of the Sigmund theory. This work deals with theoretical description of ion sputtering of magnetics near the Curie point.
2. Sublimation
Fig. 1. The temperature dependence of the (0 0 1) Ni facesputtering yield under 20 keV Ne+ ions incident in the (1 0 0) plane at 401 to the surface (current density 1.5 mA cm2) [1,2].
Fig. 2. The temperature dependence of the (0 0 1) Ni facesputtering yield under 20 keV Ar+ ions incident in the (1 0 0) plane, at 401 to the surface (current density 1.5 mA cm2) [1,2].
However, the above-mentioned phenomena are described using models; and sputtering anomalies observed experimentally have not yet been explained. This is connected with the low value of magnetic interactions in comparison with the energy of sublimation. Even in conditions of complete magnetization of the materials, the magnetic contribution to the energy is several times o1 eV. Furthermore, even this small contribution is achieved at temperatures several hundred K below the Curie point. As for magnetic fluctuations, they give an even smaller contribution as we will show
(A) Previously, the sputtering yield anomalies near the Curie point have been related to features of the sublimation energy es (see for example [2,8]). Therefore, it is reasonable to investigate the sublimation process of magnetic materials near TC to verify this possibility. Activated processes in the region of the magnetic phase transitions have been investigated for a long time (see for example [9]). However, there are only two experimental works [10,11] on the sublimation of a monatomic magnetic metal (it was Co) near TC : Fig. 3 shows the experimental temperature dependences of the deposition rate of evaporated Co [10] on a substrate. At a temperature below the Curie point, the magnetic contribution to the effective sublimation energy decreases, and this is clearly seen in Fig. 4 (circles). Assuming the Arrhenius dependence of the desorption rate on the temperature, the authors have shown that the magnetic contribution to the sublimation energy is well described by the mean-field approach to magnetic interaction. Only one fitting parameter A ¼ 2 is necessary to get a good agreement between the experiment and the calculations (the dashed line in Fig. 4). It follows from Fig. 4 that there is no evident anomaly of the sublimation energy near TC : The sublimation energy of Co in the paramagnetic state is 4.16 eV, and that in the totally ordered ferromagnetic state is estimated to be 4.36 eV. The main experimental results on evaporation of Co from [11] are shown in Figs. 5 and 6 (circles). Different curves in Fig. 5 correspond to different rates of either heating or cooling of the specimen. Using the assumption of the Langmuir dependence of the pressure on the temperature, the authors [11] obtained the magnetic contribution to the sublimation energy by subtraction of some
Y.N. Devyatko, S.V. Rogozhkin / Vacuum 66 (2002) 123–132
Fig. 3. Logarithm of the co-deposition rate as a function of the inverse absolute temperature. Circles—the experiment [10]; solid line—linear approximation of the experimental data for T > Tc ; dashed line—extrapolation of the solid line to the region of ToTc :
‘‘equilibrium’’ dependence from the experimental one. A typical result is given in Fig. 6. The peculiar behaviour of Dln pðTÞ is obvious in the vicinity of the magnetic phase transition. A theoretical description of anomalous dependences of the sublimation rate behaviour was proposed in the work [12]. It was suggested that only those particles sublimate that have the spin inverted with respect to that of the domain. However, this suggestion was substantiated in no way and contradicts the results of [10]. At least two observations made in [11] cast doubt upon the interpretation of the experimental results. First, the sublimation curves in Fig. 5 depend on the rate and the sign of the temperature change. This indicates that there is no equilibrium in the experiment and its treatment is definitely incorrect in the Langmuir approach. Secondly, analysis of the curve in Fig. 6 at temperatures T >
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TC indicates that subtraction of the equilibrium dependence ln pðT > TC Þ was made in an incorrect way. It follows from the fact that the resulting curve depends on the temperature at T > TC : If one subtracts the dependence of the pressure in the paramagnetic interval pðT > TC Þ from the experimental data, the result should be the magnetic contribution to the sublimation energy. We reanalysed the experimental data of [11] and made an accurate subtraction. The result is represented in Fig. 7, where the dependence of Dln pðTÞ on T=TC is shown. What attracts additional attention, is the opposite sign of the effective magnetic contribution to the sublimation energy in Fig. 4 [10] and that in Fig. 7 obtained from the experiment [11]. The least-squares method of fitting to the Langmuir approach gives the paramagneticstate sublimation energy of about 4.5 eV for the data [11]. This value is close to that for the ferromagnetic state from [10]. The peaks near TC being about some hundredth of an electronvolt are too small to be recognised as reliable as they are within the experimental uncertainty. To sum up, accurate analyses of the two experiments on Co sublimation do not permit us to speak about any pronounced anomalies near the Curie point. No anomalies of the sublimation rate have been observed in the vicinity of the Curie point in [10] and these are negligible in [11]. Other results of [10,11] contradict each other. For example, the contribution of magnetic ordering to the effective sublimation energy in one experiment is opposite in sign to that in another. This is possible due to non-equilibrium conditions of the experiment [11]. (B) Let us use the general theory of magnetic phenomena to study the magnetic sublimation rate. The task of addressing the equilibrium between solid and gas phase of self-atoms is well known, see for example [13]. After minimising the free energy of the system with constant number of atoms, it is easy to calculate the equilibrium number of atoms in the gas phase Ng e m s m Ng ðTÞBexp ; ð2Þ T where mm is the contribution of magnetic interaction to the chemical potential of solid-state atom.
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Fig. 4. Magnetic contribution to the sublimation energy em for Co as a function of the reduced temperature T=T c. The dots are obtained from the experimental data [10] shown in Fig. 3. The dashed line for T=Tc r1 is the best fit of em ðTÞ ¼ ATc ðMðTÞ=Mð0ÞÞ2 to the data shown by dots and corresponds to A ¼ 2 (here MðTÞ is the magnetization at temperature T). The solid lines are the results of calculation according to (3) in the mean field approximation for S ¼ 1 (curve 1) and 2 (curve 2). The contribution of the fluctuations to the sublimation energy efluct (second term in (3) is displayed in the inset).
Hereafter, the temperature is given in eV in all the formulae. Obviously, additional attraction results in an increase of both the sublimation energy ðmm o0Þ and the slope of the experimental curves. According to the Heisenberg model, the contribution of the magnetic interaction in the longwave approximation is (see Appendix A) Z 1 a3 SðS þ 1Þ T d~ q mm ¼ J0 S 2 j2 þ 0 3 2 c 6ð2pÞ 2 njT þ cJ0 j þ cJ1~ q Jð~ qÞ : ð3Þ ln T Here j is the reduced magnetization, Jð~ q Þ is the Fourier transform of the exchange interaction (Jð~ q ÞEJ0 þ J1~ q 2 in the long-wave approximation), c is a constant determined by the total atomic magnetic moment S; a0 is the lattice constant. The first term in (3) is a mean-field contribution of magnetic ordering. The second term is a result of fluctuation of the magnetic subsystem. Integration over the first Brillouin zone is carried out in (3). The maximum contribution of magnetic ordering is ATC ; that is, it is determined by the Curie temperature. The parameter A was calcu-
lated according to the Heisenberg model (see Appendix A) A¼
9S 2 : 2SðS þ 1Þ
ð4Þ
This formula gives A ¼ 2:25 in the case of the total atomic magnetic moment S ¼ 1; and A ¼ 3 for S ¼ 2: The tabulated value of the atomic magnetic moment for Co given in [16] is 1.72. In the previous section, we mentioned that the fitting parameter A = 2 was used in [10] to describe the experimental dependence. The respective energies of magnetic ordering obtained using A (4) and TC are 0.27–0.36 eV for Co and 0.08 eV for Ni. The values of the energy of magnetic ordering, which are known from the literature, are: 0.2 eV for Co [10] and 0.08–0.09 eV for Ni (obtained from the exchange splitting D0 ¼ 0:33 0:37 eV [14,15]). One can see that agreement between the calculated numbers and the number known from literature is really good. The results of calculations of the effective energy jmm j; which takes into account magnetic interactions, are shown in Fig. 4 (the solid lines). The mean-field approach well describes the behaviour
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127
Fig. 6. The temperature dependence of Dln p ¼ ln pL ln peq [11]. Circles—the experimental data (the data I see Fig. 5); solid line—spline of the experimental data.
Fig. 5. Logarithm of the Co vopour pressure as a function of the inverse absolute temperature for three different heating rates. Dots—experimental data [11]; solid lines—results of calculation by (6). Dots I and line 1 are obtained by the cooling of the sample. Dots II, III and lines 2, 3 are obtained by heating of the sample. The respective rates of cooling and heating are 1 1 V1 : V2 : V3 ¼ 13 : 25 : 1:5 : The Solid line 0 is the equilibrium dependence V ¼ 0 (6).
of the experimental curve [10] without using any fitting parameter. Fluctuations contribute to (3) not only due to the energy change but also due to the entropy change at sublimation. These changes compensate each other, and the resulting contribution (for Co and Ni) does not exceed 0.01 eV, which is negligibly small (see the insert in Fig. 4). Thus, one should not expect any noticeable anomalies in the process of equilibrium sublimation of Co as well as of other monatomic magnetics near the Curie point. The difference between the curves corresponding to heating and cooling of the specimen
Fig. 7. The temperature dependence of magnetic contribution to the sublimation energy em ¼ Tðln pL ln pfit ). Here pL is the data from [11] (the data I see Fig. 5), pfit is a linear extrapolation to ToTc of a fitting line describing the data for ToTc :
indicates that the experiment [11] was performed in non-equilibrium conditions. In this case, one cannot restrict the consideration by the classical task of sublimation. The kinetics of non-equilibrium sublimation must be analysed. Taking into account the general mechanisms, the kinetic equation for the average gas concentration can be written as dngas ¼ aðTÞn0 Ongas bngas : dt
ð5Þ
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The first term in the right-hand side describes evaporation of atoms from the surface: aðTÞ ¼ a0 expðes =TÞ: The second term takes into account sorption of atoms by the walls of the experimental chamber. The third one is due to pumping of gas out of the experimental chamber. If the temperature changes linearly as T ¼ T0 þ vt (v is the rate of the temperature change), the gas concentration as a function of the temperature can be found in the first approximation with respect to v as aðTÞ es v 1 2 ngas Cn0 : ð6Þ Oþb T ðO þ bÞ The lines in Fig. 5 represent the results of the pressure calculations by using (6). The sublimation energy was obtained by applying the method of least squares within the chosen paramagnetic temperature interval. One can see that this simple model satisfactorily describes the experimental results at small rates of heating and cooling. The disagreement between the dots III and the line 3 in Fig. 5 is possibly due to high heating rate. This fact needs more thorough analyses of the sublimation kinetics. The results of theoretical calculations described above demonstrate that the anomalies of sublimation energy near the Curie point do not exceed 0.01 eV and cannot explain the magnetic sputtering anomalies [2] in the framework of the Sigmund theory.
3. Sputtering Ion sputtering is a non-equilibrium process. Ion bombardment produces knocking of atoms from the surface, cascades of atom–atom collisions, local heating, sublimation of atoms from the surface and implantation of impinging ions. These phenomena change the topography and properties of the surface layer. The characteristic energy of magnetization is of the order of the Curie temperature, which is about some tenths of eV. Magnetic fluctuations bring a much lesser contribution. On the other hand, the characteristic energy for sputtering is of the order of the sublimation energy es : This is why, within the Sigmund theory one cannot explain the experi-
mental sputtering anomalies [2]. To do this, the Sigmund theory should be modified. First of all, the Sigmund theory does not take into consideration the dependence of sputtering on temperature, though a remarkable increase of sputtering at high temperatures has been found experimentally [17]. Therefore, temperature sensitive processes must be considered. A possible reason for the temperature dependence may be connected with particle evaporation from hot spots. During cooling and spreading of the hot spot, which appears in the region of ion propagation, particles evaporate from the surface. The yield of these evaporated particles is YS Bexpðes =TLOC Þ where TLOC is the local surface temperature. This is determined by the energy released by the incident particle in the hot spot and the average surface temperature, which increases due to beam energy evolution. The evaporation yield is determined by particles, which escape the hot spot over the time period of its existence Z Z ob eb YS ¼ 2 exp ds dt; ð7Þ Tð~ r ; tÞ a0 where ob CoD is the frequency of vibrations of surface atoms (oD is the Debye frequency), eb is the binding energy. The temperature field Tð~ r ; tÞ in the hot spot was calculated using the equation of heat conductivity. It is important to underline that Eq. (7) is correct only in the time interval where the physical magnitude of the local temperature can be defined. The time interval B5 1013 s is considered as the initial one when the hot spot temperature equals the melting temperature Tm : Overlapping of hot spots and general warming-up of subsurface layers was not taken into account in those calculations. It should be remembered that these effects occur in experimental studies with high current densities (B1 mA/cm2). Since the temperature is measured far away from the region irradiated, sputtering anomalies are observed at temperatures below the Curie temperature TC (see Figs. 1 and 2). Furthermore, it is necessary to take into account that the binding energy of atoms sputtered from the surface eb does not equal the sublimation energy es : Generally speaking, the value eb is not a constant, as it is different in different
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atomic states. The change of topography results in appearance of weakly bounded atoms (self-interstitials and adsorbed atoms). Moreover, variation of the surface layer composition due to implantation of atoms from the beam (especially under irradiation by ions of heavy inert gases) significantly reduces the binding energy eb : If eb oes ; the sputtering and the evaporation yields become a function of eb : YZ ¼ YZ ðeb Þ and YS ¼ YS ðeb Þ: Computing of binding energies of atoms in weakly bounded states is not the task of this work. Our suggestion is that the main contribution to sputtering is both from linear cascades (calculated according to the Sigmund theory but with the binding energy eb ; instead of the sublimation energy) and from atomic evaporation. Our purpose at this stage of the work is to estimate the values of the characteristic parameters, which allow us to explain the observed anomalies of magnetic sputtering. To do this, the sputtering yield was calculated for different binding energies eb ; as well as for different concentrations of weakly bounded atoms. The energy eb in the expression for the sputtering yield fluctuates in the vicinity of the magneticphase transition. This circumstance was taken into account by averaging the sputtering yield over fluctuations of the binding energy: X Y¼ W ðeb ÞfYZ ðeb Þ þ YS ðeb Þg: ð8Þ eb
129
the bulk of the material. Therefore, we used the expression for /De2b S calculated for the bulk of theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi material. The temperature dependence of q /De2b S calculated for Ni is shown in Fig. 8.
The peak of /De2b S near the Curie point leads to broadening of the Wðeb Þ distribution function. This is due to fluctuations of the magnetic subsystem. For sputtering of Ni by Ar+ ions with the energy of 20 keV, the results of the calculation of the sputtering yield at various binding energies eb (here eb is understood as the average value /eb S) are shown in Fig. 9. All the surface atoms were assumed to have the same binding energy in the course of the calculation. The absolute value of the experimental anomaly of sputtering can be explained at the binding energy eb C1:421:45 eV. This value is far less than the sublimation energy but it is not abnormally small. Moreover, this value is much higher than that of the formation of point defects on the surface [18,19]. The sputtering yields calculated by using three different models are given in Fig. 10. One can see that particle evaporation gives a really essential contribution at temperatures near the Curie point. If sputtering is only due to atoms weakly bound to the surface, a value of the binding energy less than that of calculated above (eb C1:4 eV) is required to describe the experimental sputtering anomalies. For example, if the concentration of weakly bounded atoms is 10%, the anomalies can
The distribution function W ðeb Þ can be found by microscopic consideration of material behaviour near the Curie point. In the Gauss approach, W ðeb Þ is determined both by the average binding energy /eb S of the atom in the state under consideration and by the correlator of fluctuations /De2b S: 1 ðeb /eb SÞ2 W ðeb Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp : ð9Þ 2/De2b S 2p/De2b S The values /eb Sand /De2b S were calculated according to the Heisenberg model (see Appendix A). Around TC ; fluctuations have a long wavelength and, therefore, surface atoms are in the field of fluctuations arising due to phase transitions in
Fig. 8. Temperature dependence of the mean square fluctuation q ffiffiffiffiffiffiffiffiffiffiffiffi of the energy s ¼ De2b near the Curie point.
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Fig. 9. The temperature dependence of the (0 0 1) Ni facesputtering under 20 keV Ar+ ions. Solid line—results of calculation by (8) for the binding energies of surface atoms: 1.5 eV (curve 1), 1.45 eV (curve 2), and 1.4 eV (curve 3); circles—results of the experiment [1,2].
energies. The calculations (Fig. 10) demonstrated that formation of 1–2 atoms with eb C0:8 eV in the hot spots region is good enough for explanation of anomalies of sputtering near TC : To sum up, non-equilibrium evaporation of surface atoms has been shown to give significant contribution to sputtering of surface atoms under ion irradiation. It is not reasonable to use the sublimation energy as the parameter that determines sputtering. Instead, one must use the average binding energy of surface atoms that depends upon the parameters of the ion beam. Let us mention that sputtering coefficients YZ and YS ; generally speaking, depend on different eb ; which are the characteristics of the two mechanisms. This does not change the results obtained because the main contribution to sputtering in the vicinity of TC is due to atom evaporation.
4. Conclusions
Fig. 10. Temperature dependence of the sputtering yield calculated for a binding energy of surface atoms of 1.45 eV. Curve 1—the mechanism of linear cascades (the Sigmond theory) without taking into account the magnetic fluctuation W (eb) (8); curve 2—mechanism of linear cascades with taking into account the magnetic fluctuations; curve 3—result of calculation by (8) taking into account the magnetic fluctuations, heat spots and two sputtering mechanisms: linear cascades and evaporation.
be explained using a binding energy eb C1:1 eV. The dependence of the fitting binding energy on the concentration of weakly bound atoms is shown in Fig. 11. Let us mention that the early stage of ion– surface interaction is accompanied not only by direct knock out of atoms, but also by formation and evaporation of atoms with low binding
Analyses of experimental data available from literature and theoretical calculations performed in this work have demonstrated that there are no pronounced anomalies of equilibrium sublimation near the Curie point. The contribution of magnetization into the sublimation energy is determined by the Curie temperature. The maximum contribution is achieved at T5TC and is of about several TC : Near the Curie temperature, this contribution is only of 0.01–0.02 eV. The contribution of fluctuations in the sublimation energy is even less. These calculations were performed within the framework of the Heisenberg model. The only parameters used were: the Curie temperature, the lattice constant and the value of the spin of the metal atom. It has been shown that sputtering near TC is determined by two mechanisms: linear collision cascades and non-equilibrium evaporation. Ion bombardment changes the structure and composition of the near-surface region. This is why, the atom-to-surface binding energy, but not the sublimation energy, has to be used in the theory of linear cascades [4]. The evaporation mechanism dominates in the temperature range under consideration due to simultaneous influence
Y.N. Devyatko, S.V. Rogozhkin / Vacuum 66 (2002) 123–132
at several factors. Ion–solid interaction produced (along with linear cascades and surface modification) states of atoms with low binding energies as well as local hot spots. These factors take place at any temperature, but their influence is usually small except the region of the ferromagnetic phase transition. In this region, long lived (p109 s) fluctuations of the magnetic subsystem appear, which lead to fluctuations of the binding energy of surface atoms. Simultaneous acting of the factors mentioned makes the evaporation process to be the most important in sputtering. Magnetic fluctuations lead to evaporation of atoms, which have low binding energy, during expansion of the hot spot formed by an incident ion. Calculations demonstrated that the anomalies of sputtering of Ni near the Curie point, which were observed experimentally, can be explained if one has to use the binding energy of 0.8–1.5 eV in the hot spot region.
Acknowledgements The authors thank Professors V.E. Yurasova and A.A. Pisarev for useful discussion. This work was possible due to partial support of the Russian Foundation for Basic Researches (grant 01-0217934) and the Russian Ministry of Education (the program ‘Universities of Russia’, Grant 015.02.01.31).
131
one can rewrite the Hamiltonian (A.1) as 1 X ~2 SÞ: Jð~ q ÞðSð~ q ÞSð~ q Þ N/S H¼ 2N ~q
ðA:3Þ
Here Jð~ q Þ is the Fourier transform of the exchange ~2 S is the mean square spin of potential, /S ~2 S ¼ ð1=NÞ P S ~2 : magnetic material: /S R To calculate the magnetic interaction contribution to the thermodynamic potentials, it is necessary to calculate the partition function: X H Z¼ exp : ðA:4Þ T ~ ~ fS ðR Þg
The summation in (A.4) is taken over various ~ in each point R ~: In the course of states of spin S calculation of the partition function (A.4), the method of collective variables developed by Galitskiy and Bogolubov [20] was used. The ~ðR ~Þg was replaced by integrasummation over fS tion over the collective variables: Z Y H ~ ~ðR ~ð~ ~Þ-S Z¼ dS ð~ q Þ exp IðS q ÞÞ ðA:5Þ T ~ q ~ðR ~ð~ ~Þ-S here IðS q ÞÞ is the Jacobian of change of variables: ~ðR ~ð~ ~Þ-S IðS q ÞÞ ¼
X Y
~ð~ d S q Þ:
~ðR q ~Þg ~ fS
X
!
~ðR ~Þ expði~ ~Þ : S qR
ðA:6Þ
~ R
Appendix A The Hamiltonian of magnetic interactions in the Heisenberg model is 1X ~ ~ðR ~ðR ~j ÞS ~i ÞS ~j Þ; H¼ JðR i R ðA:1Þ 2 i;j ~ðR ~Þ is an exchange potential, S ~Þ is the where JðR ~: The total spin operator of the atom in the point R summation in (A.1) is taken over the lattice ~: points R ~ to collective variables Changing from R X ~ð~ ~ðR ~Þexpði~ ~Þ; S qÞ ¼ S qR ðA:2Þ ~ R
Using the methods [21,22] for calculation of the Jacobian, one can obtain the partition function (A.5): Z Y 1 X T ~ Z ¼b dS ð~ q Þ exp JðqÞ þ 2NT ~q c ~ q 1 X ~ð~ ~ð~ ~2 S S q ÞS qÞ þ JðqÞ/S 2N ~q þ
X d ~ð~ ~ð~ ~ð~ ~ð~ q 2 ÞS q 3 ÞS q4Þ S q 1 ÞS 3 N ~q þ~q þ~q þ~q ¼0 1 2 3 4 !
þy :
ðA:7Þ
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The parameters b; c; d in (A.7) are determined by the magnetic model and the value of the total atomic spin S: So, in the case of single preferential direction, we have c ¼ SðS þ 1Þ=3: If there are three independent directions, c ¼ SðS þ 1Þ=9: The latter c value describes the experimental data [10] rather well, therefore it was used for further calculations. Expression (A.7) describes the magnetic-phase transition. This phase transition occurs if the ~ð~ ~ð~ coefficient of S q ÞS q Þ vanishes. First of all, this condition realizes for the wave vector ~ q ¼ 0: Below the phase transition point, the mode with the wave vector ~ q ¼ 0 corresponds to macroscopic magnetization and should be distinguished in (A.7). Let us note, that the expression for the Curie temperature can be obtained from the condition of phase transition: TC ¼ cjJð~ q ¼ 0Þj:
ðA:8Þ
In all the calculations made in this work, the following exchange potential Jð~ q Þ was used in the long-wave approximation: Jð~ q ÞEJ0 þ J1~ q2:
ðA:9Þ
The value of J0 ¼ Jð~ q ¼ 0Þ was determined by the Curie temperature (A.8). The nearest neighbour approach was used to calculate J1 ¼ 1=2J~q00¼0 : The expression for the partition function (A.7) ~ð~ contains series over S q Þ in the exponent index. These series can be reduced to the square approximation, which is the Gaussian one. This approximation agrees with the approach of the self-consistent field [21]. Taking the mode ~ q¼0 ~ð~ and integrating over S q a0Þ; one can obtain the expression for the partition function and calculate the corresponding chemical potential: Z 1 a30 SðS þ 1Þ T 2 2 mm ¼ J0 S j þ d~ q 3 2 c 6ð2pÞ 2 njT þ cJ0 j þ cJ1~ q Jð~ qÞ : ðA:10Þ ln T Here j is the magnetization of the P reduced ~ðR ~Þ=SN and n is the temperaspecimen j ¼ R~ S ture dependence coefficient n ¼ 1 at T > TC and n ¼ 2 at ToTC : The expression for the magnetic contribution to the energy of the atom can be
obtained by 1 a3 SðS þ 1Þ em ¼ J0 S 2 j2 þ 0 2 6ð2pÞ3 Z T d~ q Jð~ qÞ 1 : ðA:11Þ njT þ cJ0 j þ cJ1~ q2 The expression for the mean-square deviation /De2m S can also be obtained in the similar way.
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