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Electronic structure and magnetic properties of CoTi1−xAlx alloys
PHYSICA ELSEVIER
Physica B 205 (1995) 397 402
Electronic structure and magnetic properties of CoTil_ xAlx alloys A. Jezierski a'*, G.
Borstel b
alnstitute of Molecular Physics, Polish Academy of Sciences, 60-179 Poznan, Poland UDepartment of Physics, University of Osnabriick, D 49069 Osnabriick, Germany
Received 16 June 1994; revised 27 August 1994
Abstract The band structure of ordered CoTi, CoAl and CozTiA1 alloys is calculated by the spin-polarized TB LMTO method. We do not observe a magnetic moment on the Co atom in CoTi and CoAl compounds with the CsC1 type structure. The Co2TiA1 alloy with the L21 type structure is ferromagnetic with a magnetic moment on the Co atom mco= 0.27/~B and a small opposite induced magnetic moment on the Ti atom mt~ = - 0.05/~B. The electronic densities of states of the disorder CoTil -xAlx alloys are computed by the TB LMTO CPA method. We find that the magnetic moment has the maximum for x = 0.5 and it decreases to zero for x = 0 and x = 1.0. From the values of the density of states at the Fermi level we calculate the electronic specific heat coefficient ?. The general trend of 7 is close to the experimental data.
1. Introduction The various types of Heusler Co2XY alloys (X = Mn and Y = nonmagnetic element) have been investigated in the last years I-1-6]. The magnetic properties of Co2MnY alloys is due to Mn sublattice. The electronic and magnetic properties of such systems were studied by Ishida et al. [ 7 - 9 ] and Kubler et al. [10]. The other group of interesting Heusler type alloys are Co2TiY systems. Ishida et al. [7] have performed the band calculations for Co2TiY (Y = Sn, A1, Ga) alloys. They observed a small magnetic moment on Co atom. Recently, DiMasi et al. [6] studied the dependence of the Curie temperature of Co2TiA1 ordered alloy. The intermetallic compounds CoTi and CoAl crystallise in the CsC1 (B2) type of structures. Co atoms in B2 structures have eight nearest-neighbours A1 or Ti atoms located at the corners of
*Corresponding author.
a simple cubic sublattice. Measurements [2, 4] indicate that the part of atoms Co occupy the A1 or Ti sites (the Co antistructure atoms). The magnetic and electronic properties depend strongly on the homogeneity of samples and sample preparation. The magnetic measurements of CoTi and CoAl compounds indicate that these systems are paramagnetic without magnetic moment on the Co atom. Webster and Ziebeck [1] have shown by neutron diffraction experiments that the Co2TiA1 alloy with the L21 type structure is ferromagnetically ordered with a Curie temperature Tc = 135 K and a magnetic moment on the Co atom mco= 0.35/~B. These results suggested that the magnetic moment on Co atom depends strongly on the concentration of A1 and Ti atoms as well as on the local environment. Endo et al. [2] studied the magnetic properties of pseudobinary CoTil xAlx ordered and disordered alloys for 0 < x < 1.0. The main problem was the nature of the magnetic moment on Co atom. They presented two suggestions:
0921-4526/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 0 9 0 4 - X
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A. Jezierski, G. Borstel / Physica B 205 (1995) 397 402
the magnetic moment on Co atom depends on the local atomic arrangement of the nearest-neighbours Ti or A1 atoms around Co, and - the magnetic moment is connected to the change of the concentration in the alloy. The measurements of Endo et al. [2] suggested that Ti and A1 atoms occupied the F C C sublattic at any composition. They observed the m a x i m u m of the magnetic moment in Co2TiAI ordered L21 type structure for x = 0.4. The Arrott plots of the magnetization had shown that for x = 0.2 and x = 0.6 the systems were inhomogeneous ferromagnets. For the interpretation of the dependence of the local magnetic moment on the Co atom as the function of nearest neighbours of A1 and Co, in CoTil-xAlx alloys, Endo et al. [2] applied the Jaccarino and Walker model [11]. F r o m the numerical analysis they found that the Co atom had the magnetic moment, when it was surrounded by 4,5 or 6Ti(A1) atoms in the nearest-neighbour distance. The lattice parameter in CoTil xAlx alloys changes linearly with the composition. Only a small deviation is observed about x = 0.5. The Curie temperature had the maximum at x = 0.4 and then its value decreased and reached zero for x = 0.2 and x = 0.6 (Fig. 4 in Ref. [2]). The dependence of the magnetic moment on concentration was similar as that of the Curie temperature. The magnetic moment had its maximum value for x = 0.4 and decreased to zero for x = 0.2 and x = 0.6. The specific heat coefficient 7 was examined by Waterman and Franse [43. The value of the specific heat coefficient 7 was decreases from x = 0 (7 = 9 . 4 m J tool - 1 K -2) to x = 0 . 5 (7 -- 6.1mJ m o l - l K 2). Then its value reaches the maximum for x = 0.6 (7 = 9.4mJ m o 1 - 1 K - 2 ) . F r o m 0.6 < x < 1.0 the values of 7 decrease. The strong dependence of the 7 coefficient on the concentration indicates significant changes of the density of states at the Fermi level. The very interesting dependence of the magnetic moment and specific heat coefficient on the concentration motivated the present theoretical study the electronic and magnetic properties of pseudobinary CoTi~_~AI~ alloys.
2. Electronic structure of the ordered alloys The electronic structure of the ordered CoTi, CoAl and Co2TiAl alloys were calculated by the spin-polarized TB L M T O method [12]. The exchange correlation potential was assumed in the form proposed by von Barth and Hedin 1-13]. The band structure calculations were performed for the experimental lattice parameters [2]. We assumed the following initial configuration: Co-3d 7, 4s2;A1-3s2, 3pl; Ti-5d 1°, 6s 2, 6p 1. The values of the Wigner-Seitz (W-S) radii were estimated from the experimental dependence of the lattice parameters [2]. In the atomic sphere approximation (ASA) the W S cell is replaced by the sphere with the same volume. The average W S radius is determined by the condition Sav ----a(3/4rtn) 1/3, where a denotes the experimental lattice parameter and n is the number of atom in cell. The values of the W S radii for Co, Ti and A1 were estimated from the relation Y,(S,]Sav) 3 = 1, where the summation is over all type of atoms in cell. Using the experimental values of the lattice parameter for CoTi, CoAl and Co2TiAl ordered compounds [2] we obtained the following values of the W - S radii: Sco = 2.367 a.u., SAI ~ 2.90 a.u., Sxi = 3.097 a.u. For the CoTi and CoAl ordered L2~ type alloys we do not observe any magnetic moment on the Co atom. The spin-polarized band calculation performed for the CozTiA1 ordered L2~ type structure have shown a magnetic moment on the Co atom mco= 0.27/~a and a small negative induced moment on the Ti atom m~ri= - 0 . 0 5 # a . The total magnetic moment in Co2TiAl was small, mx = 0.49#B/f.u. From the self-consistent band calculation we estimated the values of the Stoner parameter for the Cobalt atom (in Co2TiA1) /Co = 0 . 0 7 R y and hence the Stoner condition IcoN(EF) = 1.8. The density of states at the Fermi level in CoAl ordered alloy is lower than in CoETiAI (N(EF) = 9.07 states/Ry spin cell) and when we assume the same value of the Stoner parameter lco we get /Co N(EF) = 0.63. In the case of CoTi alloy the density of states at the Fermi level is higher than for CoETiA1 system, but the measurements [1] and our self-consistent spin-polarized L M T O calculation have shown that CoTi ordered alloy is paramagnetic. It seems that the magnetic moment on the Co atom can be connected to the local environment as suggested Endo et al. [2].
399
A. Jezierski, G. Borstel / Physica B 205 (1995) 397 402 CoTi
CoTi 250
200
! 150
-
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-03 ===.~---- ~ -04
I
~
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6
-05
-04
-03
-02
-01
0
01
-06 -07 L
ENERGY [R'~]
G
X
W
K
G
W
K
G
CoAl
CoAl 250
oi; /
200
~ o
150
0
ii
100
\ -06
-0.5
-0.4
-03
-02
-O I
[ -l 01
....... L
G
X
ENERGY [Ry]
Fig. 1. The plot of the total density of states for ordered CoTi (a) and CoAl (b) alloys. The vertical line (EF) denotes the Fermi level.
Fig. 2. The band energy for the ordered CoTi (a) and CoAl (b) alloys. The Fermi level is marked by horizontal line. 8O
60[ In Fig. 1 (a) we present the total density of states for the CoTi ordered alloy. The Fermi energy is located on the left side of a broad peak. In Fig. 2(a) we show the corresponding energy band in irreducible part of the FCC Brillouin zone. The total density of states (DOS) and the band structure for CoAl have been seen in Fig. l(b) and 2(b), respectively. The Fermi level is located near a valley and the value of the density of states at the Fermi level is small compared to CoTi. The density of states for the ferromagnetic CozTiAI is plotted in Fig. 3. We observe a small gap near E = - 0.5 Ry. This situation is typical for the many Heusler alloys [8-10].
2O
0
-20
40
-60
-80 -07
-06
-05
-04
-03
-02
-01
0
ENERGY [ Ry ]
Fig. 3. ThetotaldensityofstatesfortheferromagneticCo2TiA1 ordered alloy. The Fermi level is marked by vertical line.
01
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A. JeziersM, G. Borstel / Physica B 205 (1995) 397 402
3. The electronic density of states and the magnetic moment in the disordered CoTil xAix alloys .0
-02
~
-03
~
-05
The electronic densities of states (DOS) for the disordered CoTil_xAlx alloys were calculated by the TB L M T O CPA method [14]. The values of the potential parameters were chosen as follows: for the disordered CoTi, CoAl and Co2TiA1 alloys we take the values of potential parameters as obtained from the self-consistent TB L M T O band calculations for the same ordered systems. In the region of concentration 0 < x < 0.5 and 0.5 < x < 1.0 we assume a linear interpolation for the values of the potential parameters. This assumption seems reasonable because the lattice parameter change linearly with the concentration. In Fig. 5 we plot the total (spin integrated) density of states for the different concentrations. We observe for x = 0.1 a broad peak located between - 0.3 Ry < E < - 0.2 Ry and a second sharp peak between - 0.1 Ry < E < 0.0 Ry. The Fermi level (the arrow) is located in peak near the E = - 0 . 1 Ry. With the change of the concentration the value of the Fermi level removes towards E = 0 and we observe the significant change of the density of states at the Fermi level N(EF). In Fig. 6 we present the dependence of the experimental and calculated values of the electronic
...........
/
-0.7 -08 L
G
X
W
K
G
Fig. 4. The energy band for spin up (solid curve) and spin down (dotted curve) for CozTiAl. The horizontal line denotes the Fermi level.
The Fermi level is located on the right side of a valley. The corresponding energy bands are presented in Fig. 4 for spin up (solid curve) and for spin down electrons (dotted curve). The values of the occupation numbers for the ordered systems are listed in Table 1. The band structure of the ordered Co2TiA1 alloy was close to that obtained by Ishida et al. [7-8] by the A P W method.
Table 1 Occupation numbers for s, p and d electrons for majority (1") and minority (,L) spin electrons Co2TiAI I
CoAl
CoTi
s
p
d
s
p
d
0.237 0.248 0.011
0.255 0.255 0.000
3.594 3.857 0.263
0.221 0.221 0.0
0.252 0.252 0.0
3.760 3.760 0.0
ntT Moment
0.359 0.354 - 0.004
0.518 0.517 - 0.001
1.473 1.427 - 0.046
nl,L nil" Moment
0.545 0.544 - 0.001
0.892 0.896 0.004
0.297 0.289 - 0.008
p
d
0.272 0.272 0.0
0.226 0.226 0.0
3.682 3.682 0.0
0.379 0.379 0.0
0.528 0.528 0.0
1.414 1.414 0.0
Co
nl~, nl• Moment
Ti
nt;
AI 0.538 0.538 0.0
0.885 0.885 0.0
0.340 0.340 0.0
A. Jezierski, G. Borstel / Physica B 205 (1995) 397-402
N~
,..,, E
n u')
10 9 8 7~ 6 5 4 3 2 1
t-
0
401
theory (CPA)- disorder
0.2
0.4 0.6 concentration x
0.8
Fig. 6. Thedependenceoftheelectronicspecificheatcoefficient 7 on the concentration in COTil_xAix disordered alloys. The upper curve (exp) presents the experimental data [2]. The solid line shows the theoretical values obtained for the disordered alloys. The symbols (O) denote the theoretical value of 7 coefficient obtained for ordered CoTi, Co2TiAI and CoAl alloys.
L
o LJ
o (23 ._J
< iI
o
I--
x =0.8
x=0.9 I
-0.6
1
-0.4
1
-0.2
ENERGY
1
0
[Ry]
Fig. 5. The total density of states for disordered CoTit_xAlx alloys. The arrow denotes the position of the Fermi level.
specific heat coefficient y on the concentration in CoTil-xAlx disordered alloys. The points denote the values obtained for the ordered systems. The theoretical values of Y decrease from x = 0 to x = 0.5. For CozTiA1 we observe the minimum of 7 and then the values of 7 increase and reach the maximum for x = 0.7. Our theoretical calculations were performed for the completely disordered alloys. The values of Y coefficient obtained for the ordered system are smaller than the experimental ones [4] for x = 0 (CoTi) and x = 0.5 (Co2TiAI), however, for x = 1 (CoAl) we got the similar value. The similar tendency was observed by Kubler et al.
[10] for other Heusler alloys. From the relation between the electronic specific heat coefficient 7 and the N(Ev) ( y = 1/3rtk2N(Ev)(1 +2), k is the Boltzmann constant), we estimated the electronphonon enhauncement factor 2 in CoTi (2 = 0.33), CoAl (2 = 0.03) and Co2TiA1 (2 = 0.40). In Fig. 7 we presented the dependence of the local magnetic moment on the Co atom in CoTit xAlx disordered alloys. We observe the maximum value of the magnetic moment for x = 0.5. In the region of concentration 0 < x < 0.5 and 0.5 < x < 1.0 the values of the magnetic moment decrease up to zero for x = 0 and x = 1.0. In Fig. 7 we have shown also the experimental values (O) and the values calculated for the ordered alloys (I). From our theoretical results we can conclude: (a) For x = 0.5 the system is partially ordered. The value of the local magnetic moment on the Co atom mco obtained for the disordered alloy is greater than for the ordered system. The experimental value is located between these two values. (b) Near x = 0 and x = 1.0 the alloy is not completely disorder. The experimental data indicate that the magnetic moment vanishes for x = 0.2 and x = 0.6. Such behaviour of the local magnetic moment on the Co atom can be connected with the specific local configuration of Ti and A1 atoms (clustering effect).
402
A. Jezierski, G. Borstel / Physica B 205 (1995) 397-402
0.6
connected to the number of the nearest neighbours of Ti and A1 atoms.
Acknowledgements
0.4
/I0
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I I
\~
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E 0.2
iI I
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0
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Q
I
I
I
0.5
IO
I
t
One of authors (A. J.) would like to acknowledge the hospitality of the members of the Physics Department, Osnabrueck University, where a major part of his work was carried. The self-consistent band calculations were performed by using the TB LMTO code of Dr. O. Jepsen and coworkers from Max-Planck Institute in Stuttgart. One of us (A.J.) thanks Dr. Josef Kudrnovsk~ for the useful discussion and for the possibility of using his TB LMTO-CPA code.
i
1.0
References
X Fig. 7. The dependence of the local magnetic moment on the Co atom in the disordered CoTil-xAlx alloys. The points (O) denote the experimental data 1,2] and the symbol ( i ) shows the theoretical value obtained for the ordered Co2TiAI alloy. The broken curve and points (O) present the theoretical values of mco for the disordered alloys.
4. Summary We have studied the electronic and magnetic properties of the ordered and disordered Co-Ti-A1 alloys. For the ordered CoTi, CoAl and Co2TiA1 compounds the spin-polarized self-consistent TB LMTO results are in a good agreement with the experimental data [i 4]. The general shape of the density of states is close to the previous theoretical band calculations [7, 8, 10]. The calculations of the magnetic and electronic properties of the disordered CoTil xAlx alloys have shown that these systems are not quite disordered. Probably we may expect some cluster effect as in CuNi alloys. Our conclusion is close to the Endo et al. [2] observation that the magnetic moment on the Co atom is
[1] P.J. Webster and K.R.A. Ziebeck, J. Phys. Chem. Solids 34 (1973) 1647. [2] K. Endo, I. An and A. Shinogi, J. Phys. F 7 (1977) L99. I-3] A.K. Grover, S.K. Dhar, E.V. Sampathkumaran, L.C. Gupta and S.K. Malik, Solid State Commun. 30 (1979) 141. [4] E.H. Waterman and JJ.M. Franse, J. Phys. F 10 (1980) 947. I-5] J.V. Yakhmi, I.K. Gopalakrishnan and A.K. Grover, Phys. Stat. Sol. A 85 (1984) K89. [6] E. DiMasi, M.C. Aronson and B.R. Coles, Phys. Rev. B 47 (1993) 14301. [7] S. Ishida, S. Akazawa, Y. Kubo and J. Ishida, J. Phys. F 12 (1982) 1111. I-8] S. Ishida, S. Asano and J. Ishida, J. Phys. Soc. Japan 53 (1984) 2718. [9] S. lshida, S. Sugimura, S. Fujii and S. Asano, J. Phys.: Condens. Matter 3 (1991) 5793. [10] J. Kiibler, A.R. Williams and C.B. Sommers, Phys. Rev. B 28 (1983) 1745. [11] V. Jaccarino and LR. Walker, Phys. Rev. Lett. 15 (1965) 258. 1-12] O.K. Andersen, O. Jepsen and M. fob, in: Electronic Structure and Its Applications, ed. M. Yuassoff (Springer, Berlin, 1987) p. 2. [13] U. von Barth and L. Hedin, J. Phys. C 5 (1972) 1629. [14] J. Kudrnovsk~, and V. Drchal, Phys. Rev. B 41 (1990) 7515. [15] P.J. Webster and K.R.A. Ziebeck, in: Landolt Bornstein New Series 1, Group III, Vol. 19, Magnetic Properties of Metals, Subvolume C.
Physica B 205 (1995) 397 402
Electronic structure and magnetic properties of CoTil_ xAlx alloys A. Jezierski a'*, G.
Borstel b
alnstitute of Molecular Physics, Polish Academy of Sciences, 60-179 Poznan, Poland UDepartment of Physics, University of Osnabriick, D 49069 Osnabriick, Germany
Received 16 June 1994; revised 27 August 1994
Abstract The band structure of ordered CoTi, CoAl and CozTiA1 alloys is calculated by the spin-polarized TB LMTO method. We do not observe a magnetic moment on the Co atom in CoTi and CoAl compounds with the CsC1 type structure. The Co2TiA1 alloy with the L21 type structure is ferromagnetic with a magnetic moment on the Co atom mco= 0.27/~B and a small opposite induced magnetic moment on the Ti atom mt~ = - 0.05/~B. The electronic densities of states of the disorder CoTil -xAlx alloys are computed by the TB LMTO CPA method. We find that the magnetic moment has the maximum for x = 0.5 and it decreases to zero for x = 0 and x = 1.0. From the values of the density of states at the Fermi level we calculate the electronic specific heat coefficient ?. The general trend of 7 is close to the experimental data.
1. Introduction The various types of Heusler Co2XY alloys (X = Mn and Y = nonmagnetic element) have been investigated in the last years I-1-6]. The magnetic properties of Co2MnY alloys is due to Mn sublattice. The electronic and magnetic properties of such systems were studied by Ishida et al. [ 7 - 9 ] and Kubler et al. [10]. The other group of interesting Heusler type alloys are Co2TiY systems. Ishida et al. [7] have performed the band calculations for Co2TiY (Y = Sn, A1, Ga) alloys. They observed a small magnetic moment on Co atom. Recently, DiMasi et al. [6] studied the dependence of the Curie temperature of Co2TiA1 ordered alloy. The intermetallic compounds CoTi and CoAl crystallise in the CsC1 (B2) type of structures. Co atoms in B2 structures have eight nearest-neighbours A1 or Ti atoms located at the corners of
*Corresponding author.
a simple cubic sublattice. Measurements [2, 4] indicate that the part of atoms Co occupy the A1 or Ti sites (the Co antistructure atoms). The magnetic and electronic properties depend strongly on the homogeneity of samples and sample preparation. The magnetic measurements of CoTi and CoAl compounds indicate that these systems are paramagnetic without magnetic moment on the Co atom. Webster and Ziebeck [1] have shown by neutron diffraction experiments that the Co2TiA1 alloy with the L21 type structure is ferromagnetically ordered with a Curie temperature Tc = 135 K and a magnetic moment on the Co atom mco= 0.35/~B. These results suggested that the magnetic moment on Co atom depends strongly on the concentration of A1 and Ti atoms as well as on the local environment. Endo et al. [2] studied the magnetic properties of pseudobinary CoTil xAlx ordered and disordered alloys for 0 < x < 1.0. The main problem was the nature of the magnetic moment on Co atom. They presented two suggestions:
0921-4526/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 0 9 0 4 - X
398
A. Jezierski, G. Borstel / Physica B 205 (1995) 397 402
the magnetic moment on Co atom depends on the local atomic arrangement of the nearest-neighbours Ti or A1 atoms around Co, and - the magnetic moment is connected to the change of the concentration in the alloy. The measurements of Endo et al. [2] suggested that Ti and A1 atoms occupied the F C C sublattic at any composition. They observed the m a x i m u m of the magnetic moment in Co2TiAI ordered L21 type structure for x = 0.4. The Arrott plots of the magnetization had shown that for x = 0.2 and x = 0.6 the systems were inhomogeneous ferromagnets. For the interpretation of the dependence of the local magnetic moment on the Co atom as the function of nearest neighbours of A1 and Co, in CoTil-xAlx alloys, Endo et al. [2] applied the Jaccarino and Walker model [11]. F r o m the numerical analysis they found that the Co atom had the magnetic moment, when it was surrounded by 4,5 or 6Ti(A1) atoms in the nearest-neighbour distance. The lattice parameter in CoTil xAlx alloys changes linearly with the composition. Only a small deviation is observed about x = 0.5. The Curie temperature had the maximum at x = 0.4 and then its value decreased and reached zero for x = 0.2 and x = 0.6 (Fig. 4 in Ref. [2]). The dependence of the magnetic moment on concentration was similar as that of the Curie temperature. The magnetic moment had its maximum value for x = 0.4 and decreased to zero for x = 0.2 and x = 0.6. The specific heat coefficient 7 was examined by Waterman and Franse [43. The value of the specific heat coefficient 7 was decreases from x = 0 (7 = 9 . 4 m J tool - 1 K -2) to x = 0 . 5 (7 -- 6.1mJ m o l - l K 2). Then its value reaches the maximum for x = 0.6 (7 = 9.4mJ m o 1 - 1 K - 2 ) . F r o m 0.6 < x < 1.0 the values of 7 decrease. The strong dependence of the 7 coefficient on the concentration indicates significant changes of the density of states at the Fermi level. The very interesting dependence of the magnetic moment and specific heat coefficient on the concentration motivated the present theoretical study the electronic and magnetic properties of pseudobinary CoTi~_~AI~ alloys.
2. Electronic structure of the ordered alloys The electronic structure of the ordered CoTi, CoAl and Co2TiAl alloys were calculated by the spin-polarized TB L M T O method [12]. The exchange correlation potential was assumed in the form proposed by von Barth and Hedin 1-13]. The band structure calculations were performed for the experimental lattice parameters [2]. We assumed the following initial configuration: Co-3d 7, 4s2;A1-3s2, 3pl; Ti-5d 1°, 6s 2, 6p 1. The values of the Wigner-Seitz (W-S) radii were estimated from the experimental dependence of the lattice parameters [2]. In the atomic sphere approximation (ASA) the W S cell is replaced by the sphere with the same volume. The average W S radius is determined by the condition Sav ----a(3/4rtn) 1/3, where a denotes the experimental lattice parameter and n is the number of atom in cell. The values of the W S radii for Co, Ti and A1 were estimated from the relation Y,(S,]Sav) 3 = 1, where the summation is over all type of atoms in cell. Using the experimental values of the lattice parameter for CoTi, CoAl and Co2TiAl ordered compounds [2] we obtained the following values of the W - S radii: Sco = 2.367 a.u., SAI ~ 2.90 a.u., Sxi = 3.097 a.u. For the CoTi and CoAl ordered L2~ type alloys we do not observe any magnetic moment on the Co atom. The spin-polarized band calculation performed for the CozTiA1 ordered L2~ type structure have shown a magnetic moment on the Co atom mco= 0.27/~a and a small negative induced moment on the Ti atom m~ri= - 0 . 0 5 # a . The total magnetic moment in Co2TiAl was small, mx = 0.49#B/f.u. From the self-consistent band calculation we estimated the values of the Stoner parameter for the Cobalt atom (in Co2TiA1) /Co = 0 . 0 7 R y and hence the Stoner condition IcoN(EF) = 1.8. The density of states at the Fermi level in CoAl ordered alloy is lower than in CoETiAI (N(EF) = 9.07 states/Ry spin cell) and when we assume the same value of the Stoner parameter lco we get /Co N(EF) = 0.63. In the case of CoTi alloy the density of states at the Fermi level is higher than for CoETiA1 system, but the measurements [1] and our self-consistent spin-polarized L M T O calculation have shown that CoTi ordered alloy is paramagnetic. It seems that the magnetic moment on the Co atom can be connected to the local environment as suggested Endo et al. [2].
399
A. Jezierski, G. Borstel / Physica B 205 (1995) 397 402 CoTi
CoTi 250
200
! 150
-
,oo[
-03 ===.~---- ~ -04
I
~
-05
50
!
oL -0
6
-05
-04
-03
-02
-01
0
01
-06 -07 L
ENERGY [R'~]
G
X
W
K
G
W
K
G
CoAl
CoAl 250
oi; /
200
~ o
150
0
ii
100
\ -06
-0.5
-0.4
-03
-02
-O I
[ -l 01
....... L
G
X
ENERGY [Ry]
Fig. 1. The plot of the total density of states for ordered CoTi (a) and CoAl (b) alloys. The vertical line (EF) denotes the Fermi level.
Fig. 2. The band energy for the ordered CoTi (a) and CoAl (b) alloys. The Fermi level is marked by horizontal line. 8O
60[ In Fig. 1 (a) we present the total density of states for the CoTi ordered alloy. The Fermi energy is located on the left side of a broad peak. In Fig. 2(a) we show the corresponding energy band in irreducible part of the FCC Brillouin zone. The total density of states (DOS) and the band structure for CoAl have been seen in Fig. l(b) and 2(b), respectively. The Fermi level is located near a valley and the value of the density of states at the Fermi level is small compared to CoTi. The density of states for the ferromagnetic CozTiAI is plotted in Fig. 3. We observe a small gap near E = - 0.5 Ry. This situation is typical for the many Heusler alloys [8-10].
2O
0
-20
40
-60
-80 -07
-06
-05
-04
-03
-02
-01
0
ENERGY [ Ry ]
Fig. 3. ThetotaldensityofstatesfortheferromagneticCo2TiA1 ordered alloy. The Fermi level is marked by vertical line.
01
400
A. JeziersM, G. Borstel / Physica B 205 (1995) 397 402
3. The electronic density of states and the magnetic moment in the disordered CoTil xAix alloys .0
-02
~
-03
~
-05
The electronic densities of states (DOS) for the disordered CoTil_xAlx alloys were calculated by the TB L M T O CPA method [14]. The values of the potential parameters were chosen as follows: for the disordered CoTi, CoAl and Co2TiA1 alloys we take the values of potential parameters as obtained from the self-consistent TB L M T O band calculations for the same ordered systems. In the region of concentration 0 < x < 0.5 and 0.5 < x < 1.0 we assume a linear interpolation for the values of the potential parameters. This assumption seems reasonable because the lattice parameter change linearly with the concentration. In Fig. 5 we plot the total (spin integrated) density of states for the different concentrations. We observe for x = 0.1 a broad peak located between - 0.3 Ry < E < - 0.2 Ry and a second sharp peak between - 0.1 Ry < E < 0.0 Ry. The Fermi level (the arrow) is located in peak near the E = - 0 . 1 Ry. With the change of the concentration the value of the Fermi level removes towards E = 0 and we observe the significant change of the density of states at the Fermi level N(EF). In Fig. 6 we present the dependence of the experimental and calculated values of the electronic
...........
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G
X
W
K
G
Fig. 4. The energy band for spin up (solid curve) and spin down (dotted curve) for CozTiAl. The horizontal line denotes the Fermi level.
The Fermi level is located on the right side of a valley. The corresponding energy bands are presented in Fig. 4 for spin up (solid curve) and for spin down electrons (dotted curve). The values of the occupation numbers for the ordered systems are listed in Table 1. The band structure of the ordered Co2TiA1 alloy was close to that obtained by Ishida et al. [7-8] by the A P W method.
Table 1 Occupation numbers for s, p and d electrons for majority (1") and minority (,L) spin electrons Co2TiAI I
CoAl
CoTi
s
p
d
s
p
d
0.237 0.248 0.011
0.255 0.255 0.000
3.594 3.857 0.263
0.221 0.221 0.0
0.252 0.252 0.0
3.760 3.760 0.0
ntT Moment
0.359 0.354 - 0.004
0.518 0.517 - 0.001
1.473 1.427 - 0.046
nl,L nil" Moment
0.545 0.544 - 0.001
0.892 0.896 0.004
0.297 0.289 - 0.008
p
d
0.272 0.272 0.0
0.226 0.226 0.0
3.682 3.682 0.0
0.379 0.379 0.0
0.528 0.528 0.0
1.414 1.414 0.0
Co
nl~, nl• Moment
Ti
nt;
AI 0.538 0.538 0.0
0.885 0.885 0.0
0.340 0.340 0.0
A. Jezierski, G. Borstel / Physica B 205 (1995) 397-402
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401
theory (CPA)- disorder
0.2
0.4 0.6 concentration x
0.8
Fig. 6. Thedependenceoftheelectronicspecificheatcoefficient 7 on the concentration in COTil_xAix disordered alloys. The upper curve (exp) presents the experimental data [2]. The solid line shows the theoretical values obtained for the disordered alloys. The symbols (O) denote the theoretical value of 7 coefficient obtained for ordered CoTi, Co2TiAI and CoAl alloys.
L
o LJ
o (23 ._J
< iI
o
I--
x =0.8
x=0.9 I
-0.6
1
-0.4
1
-0.2
ENERGY
1
0
[Ry]
Fig. 5. The total density of states for disordered CoTit_xAlx alloys. The arrow denotes the position of the Fermi level.
specific heat coefficient y on the concentration in CoTil-xAlx disordered alloys. The points denote the values obtained for the ordered systems. The theoretical values of Y decrease from x = 0 to x = 0.5. For CozTiA1 we observe the minimum of 7 and then the values of 7 increase and reach the maximum for x = 0.7. Our theoretical calculations were performed for the completely disordered alloys. The values of Y coefficient obtained for the ordered system are smaller than the experimental ones [4] for x = 0 (CoTi) and x = 0.5 (Co2TiAI), however, for x = 1 (CoAl) we got the similar value. The similar tendency was observed by Kubler et al.
[10] for other Heusler alloys. From the relation between the electronic specific heat coefficient 7 and the N(Ev) ( y = 1/3rtk2N(Ev)(1 +2), k is the Boltzmann constant), we estimated the electronphonon enhauncement factor 2 in CoTi (2 = 0.33), CoAl (2 = 0.03) and Co2TiA1 (2 = 0.40). In Fig. 7 we presented the dependence of the local magnetic moment on the Co atom in CoTit xAlx disordered alloys. We observe the maximum value of the magnetic moment for x = 0.5. In the region of concentration 0 < x < 0.5 and 0.5 < x < 1.0 the values of the magnetic moment decrease up to zero for x = 0 and x = 1.0. In Fig. 7 we have shown also the experimental values (O) and the values calculated for the ordered alloys (I). From our theoretical results we can conclude: (a) For x = 0.5 the system is partially ordered. The value of the local magnetic moment on the Co atom mco obtained for the disordered alloy is greater than for the ordered system. The experimental value is located between these two values. (b) Near x = 0 and x = 1.0 the alloy is not completely disorder. The experimental data indicate that the magnetic moment vanishes for x = 0.2 and x = 0.6. Such behaviour of the local magnetic moment on the Co atom can be connected with the specific local configuration of Ti and A1 atoms (clustering effect).
402
A. Jezierski, G. Borstel / Physica B 205 (1995) 397-402
0.6
connected to the number of the nearest neighbours of Ti and A1 atoms.
Acknowledgements
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One of authors (A. J.) would like to acknowledge the hospitality of the members of the Physics Department, Osnabrueck University, where a major part of his work was carried. The self-consistent band calculations were performed by using the TB LMTO code of Dr. O. Jepsen and coworkers from Max-Planck Institute in Stuttgart. One of us (A.J.) thanks Dr. Josef Kudrnovsk~ for the useful discussion and for the possibility of using his TB LMTO-CPA code.
i
1.0
References
X Fig. 7. The dependence of the local magnetic moment on the Co atom in the disordered CoTil-xAlx alloys. The points (O) denote the experimental data 1,2] and the symbol ( i ) shows the theoretical value obtained for the ordered Co2TiAI alloy. The broken curve and points (O) present the theoretical values of mco for the disordered alloys.
4. Summary We have studied the electronic and magnetic properties of the ordered and disordered Co-Ti-A1 alloys. For the ordered CoTi, CoAl and Co2TiA1 compounds the spin-polarized self-consistent TB LMTO results are in a good agreement with the experimental data [i 4]. The general shape of the density of states is close to the previous theoretical band calculations [7, 8, 10]. The calculations of the magnetic and electronic properties of the disordered CoTil xAlx alloys have shown that these systems are not quite disordered. Probably we may expect some cluster effect as in CuNi alloys. Our conclusion is close to the Endo et al. [2] observation that the magnetic moment on the Co atom is
[1] P.J. Webster and K.R.A. Ziebeck, J. Phys. Chem. Solids 34 (1973) 1647. [2] K. Endo, I. An and A. Shinogi, J. Phys. F 7 (1977) L99. I-3] A.K. Grover, S.K. Dhar, E.V. Sampathkumaran, L.C. Gupta and S.K. Malik, Solid State Commun. 30 (1979) 141. [4] E.H. Waterman and JJ.M. Franse, J. Phys. F 10 (1980) 947. I-5] J.V. Yakhmi, I.K. Gopalakrishnan and A.K. Grover, Phys. Stat. Sol. A 85 (1984) K89. [6] E. DiMasi, M.C. Aronson and B.R. Coles, Phys. Rev. B 47 (1993) 14301. [7] S. Ishida, S. Akazawa, Y. Kubo and J. Ishida, J. Phys. F 12 (1982) 1111. I-8] S. Ishida, S. Asano and J. Ishida, J. Phys. Soc. Japan 53 (1984) 2718. [9] S. lshida, S. Sugimura, S. Fujii and S. Asano, J. Phys.: Condens. Matter 3 (1991) 5793. [10] J. Kiibler, A.R. Williams and C.B. Sommers, Phys. Rev. B 28 (1983) 1745. [11] V. Jaccarino and LR. Walker, Phys. Rev. Lett. 15 (1965) 258. 1-12] O.K. Andersen, O. Jepsen and M. fob, in: Electronic Structure and Its Applications, ed. M. Yuassoff (Springer, Berlin, 1987) p. 2. [13] U. von Barth and L. Hedin, J. Phys. C 5 (1972) 1629. [14] J. Kudrnovsk~, and V. Drchal, Phys. Rev. B 41 (1990) 7515. [15] P.J. Webster and K.R.A. Ziebeck, in: Landolt Bornstein New Series 1, Group III, Vol. 19, Magnetic Properties of Metals, Subvolume C.