Self-consistent spin-density-functional - Science Direct

Journal of Magnetism and Magnetic Materials 20 (1980) 277-284 0 North-Holland Publishing Company

SPIN-DENSITY FUNCTIONAL

CALCULATIONS

FOR CHROMIUM

J. KijBLER Ruhr-Universitiit Bochum, Abteilung fiir Physik, 4630-Bochum, Fed. Rep. Germany Received 29 November 1979; in revised form 6 February 1980

Spin-density functional theory is used to calculate the electronic structure of chromium whose ground state is - in separate sets of calculations - assumed to be paramagnetic (P-Q) (not spin polarized), ferromagnetic (F-Cr) and antiferromagnetic (AF-Cr) (ignoring the spin-density wave). In the self-consistent calculations the magnetic moment of F-Cr is found to converge to zero. The magnetic moment of Af-Cr is found to be O.59&$. The total energy favors the antiferromagnetic ground state by 0.06 eV. The equilibrium lattice constant, cohesiveenergy, bulk modulus, magnetic moment, its pressure derivative, nesting wave vector and its pressure derivative are given and compared with experimental data.

1. Introduction

allow easy treatment with band structure theory. This is the approach taken in this paper. It differs from the classical paper by Asano and Yamashlta [ 141 merely in that SDF theory is used self-consistently. Modern computational techniques then allow groundstate properties like the total energy to be determined quite reliably. In contrast to the calculations for S-Mn [4], the present results can be compared with a considerable body of experimental data. One has to keep in mind; however, that the SDW is neglected. Input to the present calculation is the atomic number of Cr and the crystal structure. The lattice constant, obtained from the minimum of the total energy, is ‘one of the results which can be compared with experiment. In separate sets of calculations the groundstate is (a) assumed to be non-magnetic (called paramagnetic, P-Cr) with a bee unit cell,.(b) ferromagnetic (F-Cr) also with a bee unit cell, and (c) antiferromagnetic (AF-Cr) with a chemical bee unit cell but a magnetic CsCl-unit cell. Any initially ferromagnetic spin polarizations are found to converge quickly to zero so that calculations (a) and (b) result in identical electronic structures: aligning the magnetic moments of Cr metal ferromagnetically results in zero moments. The antiferromagnetic calculations (c) are in the spirit of Slater [ 181 and use a Fagnetic superlattice, the CsCl-structure. They give a self-consistently determined IW&netiC InOment 0f0.59pB per atOm wltha total energy that is lower by 0.06 eV per atom than non-magnetic Cr (P-Cr). The band structure allows

Self-consistent spin-density-functional (SDF) theory has recently been extremely successful in calculations of the magnetic moments of the ferromagnetic metals Fe, Co and Ni [l-3]. An exploratory application to 6-Mn [4] showed that SDF theory can be used to study antiferromagnetic metals self-consistently, too, but the results of ref. [4] could not be compared with experimental data since &Mn does not exist at low temperatures. Neutron diffraction experiments [S-8] with chromium reveal that below a Nbel temperature of 3 11.5 K Cr has a magnetic structure in which the magnetic moments, cc,of nearest-neighbor atoms are oriented antiparallel in the [OOl] direction but their magnitude varies sinusoidally with a long period incommensurate with the periodicity of the crystal lattice. The variation of ~1can be written as ~(=~(~cosQ*r,

(1)

where cc0is the maximum value of p and the magnitude of the wave vector Q is slightly less than 2n/a (a is the lattice constant, Q = 0.952(2n/~)(l,O,O)). It is now widely accepted that the magnetic structure of Cr results from an instability of the itinerant electrons with respect to the formation of a spin density wave (SDW) [9-171. If one ignores the slight departure from 2n/a of the wave vector, then the ground-state of Cr is antiferromagnetic and is simple enough to 277

278

J. Kiibler /Spin-density functional calculationsfor Cr

an interpretation of these results that is in line with that of Asano and Yamashita [ 141.

2. Computational details All calculations reported in this paper are performed using the method of Williams et al. [ 191, which is a descendant of Andersen’s LMTO-technique [20]. This former method app!oximates the metal by a dense packing of spherical atoms compressed into finite volumes appropriate to the solid. All contributions to the total energy are obtained from independent, spin-polarized compressed-atom calculations. Interatomic interactions enter the calculations through boundary conditions which give the atomic valence states finite widths and through the distribution of spin-polarized valence charge among s, p, d, etc. states. These environmental constraints are obtained from energy-band calculations which use expansions of the Bloch functions in terms of linear combinations of energy-independent’augmented spherical wavks (ASW). Details of the method and estimates of its accuracy can be found in ref. [ 191, for spin- polarized applications in ref. [21]. The combined ASW compressed-atom calculations are iterated to selfconsistency, the energies being stable to better than 0.02 eV and the magnetic moment to three significant figures. The unpolarized calculations are performed using the local density approximation of Hedin and Lundqvist [22] to the density functional theory of Kohn and Sham [23]. For both the ferromagnetic and antiferromagnetic calculations the. local Spindensity approximation of von Barth and Hedin [24] is used, but the constants in their functional form are changed according to Janak [25] so that the exchange-correlation energy reduces to that of Hedin and Lundqvist in the unpolarized knit. For the calculations of ferromagnetic Cr the converged potential, V, of paramagnetic Cr is used to generate two potentials V+ and V.. , by applying a magnetic field. Spin-up and spin-down electrons are then essentially treated independently in the potentials V+ and V_., respectively [24]. The magnetic field is removed and in the following iterations the system relaxes to equilibrium, which in the present calculations is identical to that of paramagnetic Cr. For the calculations of antiferromagnetic Cr the

Fig. 1. C&l structure as magneticunit cell for Cr.

converged potential, V, of the paramagnetic metal is again used to generate two potentials V+ and V_ by initially applying a magnetic field. Spin-up electrons states are then calculated in a field made up of V+ on the cube corners and V_ in the body center of a C&l-unit,cell, see fig. 1, with the same lattice constant as bee-Cr. For spin-down electrons V+ and V_ are interchanged. The spin-polarized wave functions so obtained give spin-up and spin-down charge densities and hence potentials V+ and V_ so that the process can be iterated to self-consistency resulting in a new equilibrium.

3. Results and discussion The self-consistent band structure of paramagnetic Cr with the experimental lattice constant of a = 2.884 A [26] is shown in fig. 2. The agreement with other self-consistent [ 171 and non-self-consistent band calculations [ 14,16,27] is good. The corresponding total density of .st?tes as well as the partial p- and d-densities of states ,are shown in fig. 3. The partial s-density of states (not .shown) is dominating from the band bottom up to about 0.45 Ry. To prepare for the, antiferromagnetic band structure, the energy states of fig. 2 are shown again in fig. 4a but now forded in the simple cubic Brillouin zone (fig. 4b) appropriate to the C&l-unit cell. Along the axis l?M one now finds the states of the axes l?N and HN of the bee structure, along MR the degeneracy is doubled compared with NP of fig. 2, along RF one finds the states of PI’ and PH, and along I’X those of I’H folded at the midpoint of this axis. Visible in the panels l-&land RT of fig. 4a is the intersection of

J. Kiibler /Spin-density

functional calculations for Cr

279

EF

H

~romogn

0.6

O=~.EELA bee

I

r

/

I4

r

R to)

i

c

NOi

A

(bl

Fig. 4. a) Energy bands of non-magnetic CI folded into scBrillouin zone. b) bee (solid lines) and SC(dashed lines) Brillouin zones.

A

i

I x

Fig. 2. Energy bands of non-magnetic Cr at experimental lattice constant.

electron-states with the Fermi energy EF that is one point of the electron Fermi surface centered at P (see e.g. ref. [ 141, fig. 1) as well as the intersection of hole-states (dotted) with EF that is one point of the hole surface centered at H. These two Fermi surfaces have “nesting” character [ 10,141 and give rise to the instability of the paramagnetic ground-state with respect to the formation of a spin-density wave (SDW). The crossing point of these two bands along I’M and RI? is seen to occur slightly above the Fermi energy,

TOTAL

Cr poramagnetic 0=Z.B84A

00s

\

U.4

0.6

^

01

ElRy

Fig. 3. Total density of states (DOS) and partial d- and p-DOS of non-magnetic Cr at experimental lattice constant.

forming small triangles with EF. The base of the triangle in the panel I’M gives the wave-vector mismatch which causes the SDW-period to be incommensurate with the periodicity of the crystal lattice. If the conduction electron concentration is changed slightly by allowing Cr with V, Mn or Fe, this wave mismatch can be changed and hence the period of the SDW. This has been verified by a number of experiments [28-311. The self-consistent band structure of antiferromagnetic Cr with the experimental lattice constant is shown in fig. 5. It is emphasized that these bands are manifestly spin-degenerate just like those of the paramagnetic metal. The evolution of the antiferromagnetic band structure from the folded band structure of fig. 4a is readily apparent: the degeneracies at the zone boundary are lifted, see M, R and X in fig. 4a and fig. 5; degeneracies inside the zone are lifted through hybridization. Most notable is the energy gap at EF along I’M and RI’, i.e. the nesting Fermi surfaces have disappeared [ 141. Partial densities of states (DOS) for antiferromagnetic Cr are shown in fig. 6: for Cr in the cube comer (Cr+ - fig. 1) it shows the spin-up DOS and that of spin-up-d electrons in the lower panel. In the upper panel it shows the spin-down DOS and that of the spin-down-d electrons. For Cr in the body center of the cube (Cr_ - fig. 1) the lower panel applies to

280

J. Kiibler /Spin-dens@ functioncll calculationsfor Cr

Cr antiferrom.

o=z.wA CSCI

MTR

A

iA

i

Fig. 5. Energy bands of antiferromagnetic Cr at experimental lattice constant.

spin-down electrons and the upper to spinup electrons. Clearly, therefore, fig. 6 equally well shows the site decomposed DOS of one spin-direction only, for instance, the DOS of spin-up electrons on Cr+ in the lower panel and the DOS of spin-up electrons of Cr_ in the upper panel.

3-l

--0.2

0.L

0.6

0.6

ElRy

Fig. 6. Densityof states (DOS) of spin-upand spin-down electrons, partial d-t-DOS and partial d-h-DOS of antiferromagnetic Cr at experimental lattice constant (see also text).

1.0

The reason for this is the spin degeneracy (emphasized in connection with fig. 5) which is due to the fact that spin-up electron states and spin-down electron states are transformed into one another by inversion half way between Cr+ and Cr_ . Since the partial density of states, N&T), is defined by [ 191 No”

=q

6 (E - E@))&)

(2)

where 8‘) are the eigenenergies and the &) are the angular momentum, I, and site, u, decompositions of the single electron norms associated with each eigenstate (&.& =$), the visible panel asymmetry of fig. 6 - when read as the site decomposition of the DOS - reflects the site asymmetry of &I, i.e. it reflects the. site asymmetry of the wave function of one spin direction. By repeating the entire sets of calculations at three different lattice constants, the total energy is minimized and the equilibrium atomic volume determined. The first column of table 1 gives the calculated lattice constants in the nonmagnetic (P-Cr) and in the antiferromagnetic state (AF-Cr). While the magnetic and nonmagnetic states have almost identical lattice constants, they are somewhat smaller than the experimental one. A small correction due to the zero-point vibration has been added [3]. It was found earlier [3] that the calculated equilibrium lattice constants for the ferromagnetic 3d transition metals were smaller than the experimental ones. The present results are seen to suffer from the same short coming of the local spin-density functional approximation. The cohesive energy of the metal is obtained by subtracting the total energy of atomic Cr [3] from the total energy of the solid. The results are shown in the second column of table 1, where an experimental value is also listed. The systematically lower calculated values are again familiar from the ferromagnetic calculations of ref. [3]. More significant than a comparison with the experimental value is the difference of AE = -0.06 eV between the cohesive energy of P-Cr and AP-Cr: the antiferromagnetic state has lower total energy and is thus realized by nature unless a state of even lower energy is available. One can make an attempt to analyze the calculations in order to extract the leading terms in the energy difference AE that stabilize the antiferromagnetic

J. Kiibler /Spin-density functional cakulations for Cr

281

Table 1 Ground-state properties of pammagnetic (P-Cr) and antiferromagnetic (AF-Cr) chromium: a0 - lattice constant, &oh - cohesive energy, B - bulk modulus, ho - magnetic moment, and (~/MO)dpo/dP]o - pressure derivative of magnetic moment a0 (A)

P-Cr AF-Cr

B (Mbar)

Ecob (ev)

(l/r01

PO O”B)

~O/d&Wad

talc.

exp.

talc.

exp.

CdC.

exp.

talc.

exp.

CdC.

2.854 2.855

2.884 a)

-4.55 -4.61

-4.09 b,

2.55 2.13

1.62 c,

0.59

0.59 d)

-0.50x10-2

exp.

-2x10-2

a) Tebble and Craik [ 261. b, Gschneider [ 321. ‘) Bolef and de Klerk [ 331. d, Bacon [ 81. e, Umebayashi et al. [ 341.

state. The total energy per atom in the magnetic state is obtained from E&y= g

--

:

(2lt

s” 47rr'p VH dr 0

tc =

I)EftifNl(EY’EdE

C

j

=

0

4nr2puv& dr

(3)

s

P*)*, s4~r2puex,@t,

The third and fourth terms, &?H and AE”xc, are the so-called double-counting terms that make a rigorous angular momentum decomposition of the total energy difference impossible. A qualitative analysis, however, can still be made. The sum of the first four terms of eq. (5) results from the kinetic energy of a system of non-interacting particles and the electrostatic energy of electron-electron, electron-nucleus and nucleusnucleusinteractions. At the theoretical lattice constant, this is obtained as

0

where u denotes the spin index, I$ is the core energy of angular momentum I and Nl(E)” is the partial density of states (see, e.g., fig. 6) which is defined by eq. (2) (but the site index v has been dropped). The energy origin for Ej’, Ep and Nl(E)O is the zero of the local Hartree potential, p [ 191. Furthermore, p is the charge density, v& is the exchange-correlation potential, exCbt, p*) is the exchange-correlation energy density, and S is the radius of the atomic sphere. An expression analogous to eq. (3) for the total energy per atom, Erot, of the non-magnetic state can be written down. Then AE=Eg-EEtP,t

(4)

is obtained as a sum of five terms corresponding to those of eq. (3):

=

0.06 eV .

It is the exchange-correlation contribution in eq. (5) which gives Al&, = -0.12 eV and thus formally stabilizes the antiferromagnetic state. The five separate contributions in eq. (5) are AE = (0.62 - 0.03 - 0.69 t 0.16 - 0.12) eV which simply shows that the double-counting terms are important. Unfortunately, the energy gain of -0.03 eV from the valence electrons is accidental since it vanishes entirely at larger lattice constants where the magnetic moment is larger. Still, there are two terms in the valence contribution to AE, J# s

+mxc*

E%

(5) __

: s NdE)Ee,

Nl(E)“E dE

e,

282

J. Kiibler /Spin-density fknctionol calculationsfor Cr

obtained from a pressure change of up to 6 kbar. ‘Ihe calculated value is based on pressure changes that are roughly a factor of 20 larger. Thusj the discrepancy may be due to an exponential change of cc0with the volume, compare eq. (2) of ref. [34] with ref. [35]. But this is not at all certain. A more likely reason for the discrepancy is the neglected SDW. In table 2 some relevant band structure data are listed. They apply at the theoretical lattice constants. In the first group of data the electron count ni is given, added for the s, p and f electrons (the latter being a small correction). The difference between the t and 4 line of AF-Cr is the magnetic moment of 0.59pu. Partial and total densities of states, N,, at the Fermi energy are given in the next group. The total density of states are compared with experimental values as suggested originahy by Asano and Yamashita [14]. For non-magnetic Cr (P-Cr) the experimental number is the specific heat value of Cheng et al. [37] for Cr with 5 at% V which is paramagnetic. For antiferromagnetic Cr (AF-Cr) the specific heat value for pure Cr of Hulm and Blaugher [36] is given. The agreement is good. A somewhat smaller value of Nfir’= 0.59 eV_’ atom-’ for AF-Cr has been measured by Heiniger et al. [38], whereas they estimate NteoXp = 1.22 eV_’ atom -’ for P-Cr. The lower value for the density of states of antiferromagnetic Cr is due to the loss of Fermi surface caused by the formation of the energy gap whose value and relative pressure derivative is also included in table 2. An indirect

which, although they nearly cancel each other, are large separately. These result from the d-electrons, LL??t valence, d = -0.42 eV , ‘#a,ence,d = 0.39 ev , the contribution from s, p and f electrons being insignificant. The absolute value of the difference of these two terms is found to be proportional to the magnetic moment as the calculations at other lattice constants show; so is the absolute value of the exchange-correlation energy, A&,. Therefore, the statement is reasonable, that the exchange-correlation energy gain is a response to the above large changes in the d-band energies. The latter, finally, are of the order of the energy-gap at EF (see below), which a comparison of fig. 4 with fig. 5 isolates as the dominant feature of the antiferromagnetic band structure. In accord with accepted theories one concludes that it is this gap which stabilizes the magnetic structure of chromium. The calculated bulk modulus is given in the third column of table 1; it is in reasonable agreement with the experimental value. The calculates value of the magnetic moment, po, (like all other values of table 1 it is the value at the theoretical lattice constant) is given in the fourth column of table 1. It is compared with a value of no quoted by Bacon [8] as the maximum moment, compare eq. (1). The agreement may be fortuitous.,In the last column of table 1 the relative pressure derivative of the magnetic moment is given and compared with an experimental value [34]

Table 2 Band structure data of paramagnetic (P-0) and antiferromagnetic (AF-Cr) at the theoretical%ttice constant: nS+p+f - summed s, p and f valence dlectrons, nd - d-valence electrons, NSip+f - summed 8, p and f density of states at EF (in eV ), Nd - ddensity of states at EF (in eV-‘), Ntot - total density of states at EF (in eV_‘), EG - antiferromagnetic band gap (in eV), (~/EG) dE~/dPlo pressure derivative of antiferromagnetic band gap (in kbar-I), Q - nesting wave vector (in 2x/a), (l/Q) dQ/d?‘lo - pressure derivative of nesting wave vector (in kbar-l), EG

(l/Q> dQ/dplo

Q

(~/EG) ~Gl@iO

nS+p+f

“d

Ns+p+f

Nd

Ntot

N%

1.427

4.573

0.104

0.637

0.741

0.984 a) -

-

0.950 (talc.)

0.952 +0.2 x 10-4 (exp.) ‘1 (talc.)

t 0.714 AF- Cr J 0.709

1.991

0.054

0.180 0.641

0.678 b, 0.51

-0.45 x1O-2

-

-

2.586

0.047

0.360

P-Cr

a) Cheng et al. [37]. b, Huhn and Blaugher [ 361. ‘1 Umebayashi et al. [ 341.

-

-6.4 x 1O-4 (exp.) ‘1

J. Kiibler /Spin-density functional calculationsfor 0

gap is seen to occur in fig. 5 along the Z axis. At the calculated lattice constant its value is 0.40 eV. Experimentally, Lind and Stanford [39] find two energy gaps which disappea.r at the Neel temperature, one at 0.124 eV, the other at 0.45 eV. Again, because of the simplifying assymptions, underlying the present calculations, a more detailed discussion of experimental and calculated values does not seem appropriate here. The wave vector, Q, of the spin-density wave is usually estimated from the mismatch of the ,I’-electron and H-hole Fermi surface of P-Cr which was discussed together with fig. 4a. The value so obtained and its pressure derivative are listed in the last two columns of table 2. They are compared with experimental data which were obtained by neutron diffraction experiments in the SDW-state [34]. While Q agrees both with Asano’s and Yamashita’s value [ 141 and with the experimental one, its pressure derivative does not agree with experiment. Previous band calculations for non-magnetic chromium at different lattice constants by Fletcher and Osborne [40] also gave pressure changes of Q that were too small to account for the experimental values. The wave vector Q under discussion is the nesting vector which translates one-half of the F-electron Fermi surface in the non-magnetic state to nest onto the corresponding half of the larger H-hole Fermi surface. Although the non-magnetic state is unstable with respect to formation of a SDW having this wave vector Q, it not necessarily means that the condensate (the SDW) indeed possesses this wave vector. Fenton [41,42] supplies theoretical evidence that the nesting wave vector and the wave vector of the SDW are basically two different entities. Especially their pressure dependences can be vastly different, so that the discrepancy between the calculated and measured values in the last column of table 2 can neither be regarded as a failure of accepted SDW-theory [41], nor as a failure of band theory. Finally, the energy gap EG and the spin polarization, An = ~(o/~(u,can be used to define an effective exchange integral, J, through J=EGIAn. The value one obtains is J = 0.86 eV with a (small) pressure derivative of (1 /J) dJ/dP = 0.05 X lo-* kbar-‘. The order of magnitude of J is in agreement with estimates by Asano and Yamashita [ 141.

283

The present calculation can be summarized as follows: Assuming a ferromagnetic ground state for Cr results in a vanishing magnetic moment and hence in an electronic structure that is identical to the nonmagnetic one. In view of the rather small density of states at the Fermi energy this is not surprising. The self-consistent non-magnetic electronic structure shows features needed by accepted theories to explain the spin density wave ground state. Antiferromagnetic chromium has a calculated magnetic moment of 0.59~~ and a lower ground-state energy than nonmagnetic chromium. This energy gain can be seen to arise from the formation of an energy gap at the Fermi energy.

Acknowledgement I am indebted to A.J. Freeman for helpful discussions and for initiating the present calculations.

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284

J. Kiibler /Spin-dens@ fiuwtional calculationsfor Cr

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