- Email: [email protected]
Electronic theory for impurity segregation at lattice defects in metals
PHYSICS LETTERS
Volume 107A. number 4
ELECTRONIC
THEORY FOR IMPURITY SEGREGATION
K. MASUDA-JINDO
28 January 1985
AT LATTICE DEFECTS IN METALS
’
Department of Materials Science and Engineering, Tokyo Institute of Technology, Nagatsuta, Midori-ku, Yokohama 227, Japan Received 8 June 1984 Revised manuscript received 4 October 1984
A tight-binding type electronic theory is used to study the impurity segregation at lattice defects in metals. It is shown results from a simple physical origin: It is roughly proportional to the difference in the that the heat of segregation Esegegr diagonal matrix elements (Vi - Vo) of impurity potentials, where Vi (Vo) is determined for atomic sites in the vicinity of
lattice defects (for a perfect lattice site). Applications to impurity segregation at cleaved surfaces, screw dislocations and tilt grain boundaries are discussed.
It is now generally recognized that the segregation of impurity or solute atoms at lattice defects (cleaved surfaces, interfaces, stacking faults, grain boundaries, dislocations, etc.) does drastically change the various physical properties of crystalline materials. For instance, it is known that the catalytic properties of alloys depend strongly on the surface composition [l41. Solid solution hardening of softening of metals is closely related to the interaction between the solute atom and the dislocation [5] (or stacking fault, ref. [6]). On the other hand, the segregation of solute atoms at grain boundaries influences many of the properties of metals and ceramics such as grain growth, creep, corrosion and fracture [7]. However, the effects of impurity or solute segregation at lattice defects are often difficult to observe reliably because of the extreme cleanliness required for the sample [ 31. For the systematic understanding of impurity segregation at lattice defects, it is very important to derive a simple and straightforward theoretical scheme to predict the segregation. In the present report, we are making an attempt to derive such a theoretical scheme (an explicit analytic formula) using the tight-binding (TB) electronic theory. As a typical example, we will 1 K. Masuda-Jindo was formerly known as K. Masuda.
0.3759601/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
here discuss the segregation of substitutional impurlties in d-band metals. This theory can provide physical insight without elaborate numerical calculations for a wide variety of segregation problems. When the host band structure is represented by TB d-bands the impurity potential is determined by its matrix elements Vo, between the d-orbitals ImO) where m labels the five d-orbitals (xy, yz, zx, x2 - y2 and 3z2 - r2) and 0 the atomic site: the matrix form of the impurity potential will be denoted by Fo. Since the screening in such narrow bands is well localized we assume that the impurity potential is localized in the impurity cell (single-site approximation). The localized impurity potential is assumed to be spherically symmetric so that Vom is m independent. It can be determined by the Frledel sum rule Z = -a
g
Arg( 1 - VOGO~)E=E~ ,
(1)
where Z is the atomic number difference between impurity and host, The diagonal elements Go, of the unperturbed Green function (matrix form co) are given by (mOIGo Im’O)= 6,,,.,~G,
=PJ
Porn (0 E-E’
dE’ - inPorn@),
185
Volume 107A, number 4
PHYSICS LETTERS
where P is the Cauchy principal part and porn(E) is the partial density of states (DOS) for the mth orbital. If we approximate the heat of dissolution Ed of a substitutional impurity by its band contribution, Ed can be given as [3,8] EF Ed=ZEF
-iJ
q(E)dE,
(3)
where q(E) is the phase shift function -v(E) = -Arg det ]I 1 - FOG0 (I .
(4)
Since the heat of segregation Esegr is the energy due to exchange of positions between a host atom located near the lattice defect and an impurity dissolved in the perfect lattice (far from the lattice defect), one can simply obtain these quantities (chemical interaction) as the difference of dissolution energies of the impurity at site R, and a perfect lattice site [3,4] EseBr(Rf) = Ed(Ri) - E&perfect
lattice) .
(5)
(When solvent and solute atom sizes differ too much, the lattice strain due to impurity atoms needs to be taken into account.) From eq. (1) one can obtain the relation between the impurity potential V. determined for the perfect lattice site and Vi for the atomic site Ri around the lattice defect F&‘o(EF)/]
28 January 1985
E, = ZE, + V,N, EF + lOV;s
pO(E)G$(E)dE+...,
(8)
where Nd represents the number of d-electrons. Clearly, this type of expansion works well for impurity atoms having weak impurity potentials. We do not consider the impurity atoms with sufficiently strong impurity potentials since the usual single-site approximation is invalid for them. For the Green’s function GjR(E) of the band region, we take the canonical form [9] = (E - ~11j)/2 II; j ,
G:(E)
(9)
where /.I& = /..Q~- /Lfi and pli (/L$ is the first (second) moment of the local DOS pi(E). This approximation of the Green’s function contains the essential feature of the usual function GiR(E) and is not crucial for the present problem (as will be shown below). One can then obtain the explicit expression, in a power series of Yo, for the dissolution energy Ed Ed = ZEF + VoNd + Vi( 1OMr - Nd/~l)/2&
+ . .. 00) where ccl, pi are for the perfect lattice and the M, (n = 1, 2,3 , . . .) are defined by EF
M,=j
Enpo(E)dE.
(11)
1 - F,G,R(E,)]
= Fipi(E~)/]l
- V,G,R(EF)l
,
(6)
where pi(E) (pa(E)) denotes the local DOS on the atomic site Ri (on the perfect lattice site). This relation on the impurity potentials is derived from the condition of charge conservation of the whole system. Here, we have assumed for simplicity that each d-subband is equivalent, and the suffx m has been omitted. Then the heat of dissolution Ed can be given by
A similar expression for Ed can also be given, in terms of the site dependent /Lri, p;i andM,ri, for the impurity atom located at_Ri around the lattice defect (imposing the condition of charge neutrality for each atom [3,4]). The heat of segregation Esepr(Ri) canthen be given from eqs. (5) and (10) as *r Rsegr(Ri) = (Vi - Vu)Nd + “f(l@Mli - I$( l&M, - N&)/2E”;
+ . .. .
- Ndpli)/2& (12)
EF Ed=ZEF-FJ
Arg(l-
VoGo)dE,
(7)
where the factor 10 is due to the degeneracy of the dband. In order to obtain a more explicit expression for Ed, we expand the second term of eq. (7) in terms of the impurity potential V. 186
We now apply eq. (12) to specific segregation problems. First, we consider impurity segregation at a
*t Strictly speaking, the first term of eq. (12) becomes ViNi - Vfl,~, where Ni denotes the number of d-electrons on the defect site i. However, it is well known that Ni w NO is a good approximation for usual lattice defect problems [3,9].
-0.117867 -0.207956 0.200831 0.106530 0.000978
-0.075215 -0.086974 0.056989 0.033781 0.000596
-0.402480 -0.148003
-0.268414 -0.153319
1 2 3 4 5
1 2 3 4 5
1 1
1 1
Ni(ll1)
Ni(ll1)
Ni(ll0) Ni(100)
Ni(ll0) Ni(100)
(a)
(b)
(c)
(d)
-0.163926 -0.065894
-0.207608 -0.047658
-0.028930 -0.070446 0.018818 0.024108 0.004105
-0.030343 -0.113091 0.040119 0.04495 1 0.007088
cr/-4
Fe/-2 0.025737 -0.049858 -0:005678 0.010312 0.003215 0.018302 -0.04068 1 -0.006038 0.007956 0.0025 99 -0.068836 -0.010199 -0.049249 -0.000107
Mn/-!3 0.007266 -0.076112 0.004239 0.021799 0.006072 0.000393 -0.056272 0.000723 0.014773 0.004330 -0.12375 1 -0.021949 -0.095691 -0.021301
0.0 0.0
-0.025756 -0.001034
0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.021301 -0.021526 -0.006121 0.003570 0.000721
-0.014596 0.007467
0.0 0.0 0.0 0.0 0.0
0.0255 12 -0.024600 -0.006063 0.004239 0.000808
a)
Ni/O
Co/-l
Zn/+2 -0.203613 0.028153 0.099847 -0.003704 -0.004390 -0.208700 0.026200 0.104600 -0.003200 -0.004600 -0.150579 -0.168006 -0.227870 -0.244189
Cu/+l -0.058806 0.021667 0.020266 -0.004286 -0.000197 -0.057479 0.02055 1 0.020919 -0.004036 -0.000221 -0.009767 -0.026818 -0.026630 -0.043385
and Ni(lOO) surfaces (in Ry). Cases (a) and (c) are calculated using the first term of eq. (12)
a) For almost the same Vo and Vi values, the present E segr calculations using eq. (12) (first term) should not be taken seriously but simply be taken to be Esqr a 0.
VI-5
Impurity/Z
Z
Plane
Case
Table 1 Segregation energies E segr versus impurity position for Ni(l1 l), Ni(ll0) while cases(b) and (d) are calculated from refs. [3] and 141.
Volume 107A. number 4 cleaved
crystal
surface.
PHYSICS LETTERS In particular,
we investigate
on the Ni( 11 l), Ni( 110) and Ni( 100) surfaces: The segregation of these systems has been studied extensively [ 3,4] and detailed information on the impurity potentials Vi and VO is available. We can check the validity of the expansion of eq. (12) for the segregation energies E segr_ In table 1 we present the calculated Esegr values using eq. (12), but retaining only the first term, and compare them with those obtained by much more elaborate numerical calculations [3,4]. Surprisingly enough, the approximate Esegr values calculated from the first term of eq. (12) can reproduce the “true” segregation energies of ref. [3] well, especially for impurities with lZ( < 2 (weak impurity potentials): Even for impurities with IZ I > 3 the sign of Esegr is correct and the order of magnitude reasonable. Therefore, E segr of eq. (12) can be used in general to predict impurity segregation at lattice defects in metals as well as to check the results ofEseg, obtained by much more elaborate numerical calculations. Next we consider the segregation at dislocations and grain boundaries in transition metals. These problems are quite important in fracture science and in metallurgy. We consider the impurity segregation in bee transition metals with nearly half-filled d-bands like &Fe, MO and W. When treating the weak impurity potentials (roughly speaking for impurities with IZ I < 2), one can use the relation Vopo(EF) = Vipi(EF), instead of eq. (6). This approximation becomes more reasonable for metals with nearly half-filled bands, since the GP(EF) are generally small Under this condition, one can rewrite eq. (12) as the segregation
E
Ndz
segr = mo(EF)
of transitional
elements
Ap(EF)Ih(EF) 1+ &(EF)IP~(EF)
+ o(vi”,
+),
where Ap(EF) denotes the change in the total DOS at the Fermi energy. This expression tells us that once the knowledge on the local DOS pi(E) is available one can immediately predict the impurity (or solute) segregation. Nowadays, it is not difficult to get information on the local DOS pi(E) in the vicinity of lattice defects in metals and semiconductors. For this purpose, the TB recursion method [lo] has widely been used for various problems. However, in view of the lack of theoretical calculations on the electronic structure for dislocations in metals, we have calculated the local 188
28 January 1985
LDOS
c
I
(a)
-0.2
0 E
0.2
(C)
-0.2
0
0.2
E
Fig. 1. Local d-electron DOS on atoms around the (a/2) (111) screw dislocation in or-Fe: Nearest-neighbour (a), next-nearest neighbour (b) and third nearest-neighbour (c) atomic sites. Also the DOS of a perfect lattice is shown in (d). Energies are given in Ry units.
DOS p&E) for a (a/2) ( 111) screw dislocation in bee transition metals using the recursion method and present the results for a-Fe, first, second and third nearest-neighbour positions around the dislocation center, in fig. 1. The d-band parameters are taken from ref. [ 1 l] and the atomic relaxations around the screw dislocation are calculated using the method outlined in ref. [12]. The non-degenerate type core of the screw dislocation has been obtained for bee transition metals. It is interesting to note that the calculated local DOS pj(E) around the screw dislocation quite resemble those of the amorphous Fe [ 13,141. The change in the local DOS at the Fermi energy can be estimated from fig. 1, to be A/J(EF)/PO(EF) = -0.4 to -0.5 (for Nd = 6.5-7). Thus the segregation energies are roughly estimated to be Esegr x -0.082 Ryd for impurities with IZI G 2, i.e., for Cr, Mn, Co and Ni. These Esegr values are comparable to those calculated for the Ni surfaces (see table l), but much larger than those estimated by using continuum elasticity theory [ 15,161; elastic (modulus or size) interaction energies between substitutional solute and screw dislocations in a-Fe have been estimated to be less than 0.01 Ryd (in absolute values). In view of the large deformation on the local DOS Pi(E) at the screw dislocation core (comparable to those observed for the surface DOS of Ni), the present Esegr values are considered to be quite reasonable. One can thus see that the “chemical interaction” plays a
Volume 107A, number 4
PHYSICS LETTERS
dominant role in determining the interaction (segregation) energy between the impurity and screw dislocation. This type of chemical interaction is convenient to discuss the solid solution hardening of softening of bee transition metals [ 171 but has been omitted in the previous theoretical treatments [ 161. Similar effects can also be expected for the solute segregation at gram boundaries in transition metals. As has been shown in ref. [ 181. for z1= 5 and X = 9 tilt grain boundaries in bee transition metals, A&E,)/ po(EF) are generally large and take values of the order of 0.1-0.7 for metals with IVd = 7-5. One can thus expect that the chemical interaction plays also a dominant role in determining the impurity segregation energies Esegr at tilt grain boundaries. Experimentally, it is well known that the substitutional impurities like Cr (2 = -2) Mn (Z = - 1) and P tend to segregate to gram boundaries in Fe or Fe based alloys [7] and affect strongly the fracture strength of gram boundaries. However, the importance of this chemical interaction has not been fully recognized. In conclusion, we have used a TB electronic theory and clarified the physical origin for impurity segregation metals. A simple analytic formula has been derived for the systematic understanding (or for the prediction) of impurity segregation. We have applied the theory to typical segregation problems (cleaved surfaces, dislocations and grain boundaries), and demonstrated the importance of the “chemical interaction”, i.e., band term contributions.
28 January 1985
References [l] [2] [ 31 [4]
V. Ponec, Surf. Sci 80 (1979) 352. Ph. Lambin and J.P. Gaspard, J. Phys. FlO (1980) 2413. R. Riedinger and H. Dreysse, Phys. Rev. B27 (1983) 2073. H. Dreysse and R. Riedinger, Phys. Rev. B28 (1983) 5669. [S] H. Suzuki, Proc. Symp. on Structural properties of crystal defects (Lib&, Czechoslovakia, 1983) p. 205. [6] H. Saka, Phil. Mag. 47A (1983) 131. [7] E.D. Hondros, J. Phys. (Paris) 36 (1975) C4-117. [8] M.A. Khan and C. Demangeat, Phys. Lett. 95A (1983) 499. [9] K. Masuda, Phys. Rev. B26 (1982) 5968. [lo] R. Haydock, V. Heine and M.J. Kelly, J. Phys. C5 (1972) 2845;C8 (1975) 2591. [ 111 K. Masuda, N. Hamada and K. Terakura, J. Phys. F14 (1984) 47. [12] K. Masuda and A. Sato, Phil. Mag. 37B (1978) 531; 44A (1981) 799. [13] S.N. Khanna and F. Cyrot-Lackmann, Phys. Rev. B21 (1980) 1412. [14] T. Fujiwara, J. Phys. F12 (1982) 661. [15] S. Takeuchi, J. Phys Sot. Japan 27 (1969) 929. [ 161 H. Suzuki, Dislocations in solids, ed. F.R.N. Nabarro, Vol. 4, Cl-l. 15 (1979) p. 191. [ 171 K. Masuda, IXth Yamada Conf. on Dislocations in solids (1984) to be published. [ 181 M. Hashimoto, Y. Ishida, R. Yamamoto, M. Doyama and T. Fujiwara, J. Phys. Fll(1984) L141; Proc. Vth Yamada Conf. on Point defects and defect interaction in metals (Kyoto, 1981) p. 776; M. Hashimoto, Doctoral thesis, University of Tokyo (1983).
189
Volume 107A. number 4
ELECTRONIC
THEORY FOR IMPURITY SEGREGATION
K. MASUDA-JINDO
28 January 1985
AT LATTICE DEFECTS IN METALS
’
Department of Materials Science and Engineering, Tokyo Institute of Technology, Nagatsuta, Midori-ku, Yokohama 227, Japan Received 8 June 1984 Revised manuscript received 4 October 1984
A tight-binding type electronic theory is used to study the impurity segregation at lattice defects in metals. It is shown results from a simple physical origin: It is roughly proportional to the difference in the that the heat of segregation Esegegr diagonal matrix elements (Vi - Vo) of impurity potentials, where Vi (Vo) is determined for atomic sites in the vicinity of
lattice defects (for a perfect lattice site). Applications to impurity segregation at cleaved surfaces, screw dislocations and tilt grain boundaries are discussed.
It is now generally recognized that the segregation of impurity or solute atoms at lattice defects (cleaved surfaces, interfaces, stacking faults, grain boundaries, dislocations, etc.) does drastically change the various physical properties of crystalline materials. For instance, it is known that the catalytic properties of alloys depend strongly on the surface composition [l41. Solid solution hardening of softening of metals is closely related to the interaction between the solute atom and the dislocation [5] (or stacking fault, ref. [6]). On the other hand, the segregation of solute atoms at grain boundaries influences many of the properties of metals and ceramics such as grain growth, creep, corrosion and fracture [7]. However, the effects of impurity or solute segregation at lattice defects are often difficult to observe reliably because of the extreme cleanliness required for the sample [ 31. For the systematic understanding of impurity segregation at lattice defects, it is very important to derive a simple and straightforward theoretical scheme to predict the segregation. In the present report, we are making an attempt to derive such a theoretical scheme (an explicit analytic formula) using the tight-binding (TB) electronic theory. As a typical example, we will 1 K. Masuda-Jindo was formerly known as K. Masuda.
0.3759601/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
here discuss the segregation of substitutional impurlties in d-band metals. This theory can provide physical insight without elaborate numerical calculations for a wide variety of segregation problems. When the host band structure is represented by TB d-bands the impurity potential is determined by its matrix elements Vo, between the d-orbitals ImO) where m labels the five d-orbitals (xy, yz, zx, x2 - y2 and 3z2 - r2) and 0 the atomic site: the matrix form of the impurity potential will be denoted by Fo. Since the screening in such narrow bands is well localized we assume that the impurity potential is localized in the impurity cell (single-site approximation). The localized impurity potential is assumed to be spherically symmetric so that Vom is m independent. It can be determined by the Frledel sum rule Z = -a
g
Arg( 1 - VOGO~)E=E~ ,
(1)
where Z is the atomic number difference between impurity and host, The diagonal elements Go, of the unperturbed Green function (matrix form co) are given by (mOIGo Im’O)= 6,,,.,~G,
=PJ
Porn (0 E-E’
dE’ - inPorn@),
185
Volume 107A, number 4
PHYSICS LETTERS
where P is the Cauchy principal part and porn(E) is the partial density of states (DOS) for the mth orbital. If we approximate the heat of dissolution Ed of a substitutional impurity by its band contribution, Ed can be given as [3,8] EF Ed=ZEF
-iJ
q(E)dE,
(3)
where q(E) is the phase shift function -v(E) = -Arg det ]I 1 - FOG0 (I .
(4)
Since the heat of segregation Esegr is the energy due to exchange of positions between a host atom located near the lattice defect and an impurity dissolved in the perfect lattice (far from the lattice defect), one can simply obtain these quantities (chemical interaction) as the difference of dissolution energies of the impurity at site R, and a perfect lattice site [3,4] EseBr(Rf) = Ed(Ri) - E&perfect
lattice) .
(5)
(When solvent and solute atom sizes differ too much, the lattice strain due to impurity atoms needs to be taken into account.) From eq. (1) one can obtain the relation between the impurity potential V. determined for the perfect lattice site and Vi for the atomic site Ri around the lattice defect F&‘o(EF)/]
28 January 1985
E, = ZE, + V,N, EF + lOV;s
pO(E)G$(E)dE+...,
(8)
where Nd represents the number of d-electrons. Clearly, this type of expansion works well for impurity atoms having weak impurity potentials. We do not consider the impurity atoms with sufficiently strong impurity potentials since the usual single-site approximation is invalid for them. For the Green’s function GjR(E) of the band region, we take the canonical form [9] = (E - ~11j)/2 II; j ,
G:(E)
(9)
where /.I& = /..Q~- /Lfi and pli (/L$ is the first (second) moment of the local DOS pi(E). This approximation of the Green’s function contains the essential feature of the usual function GiR(E) and is not crucial for the present problem (as will be shown below). One can then obtain the explicit expression, in a power series of Yo, for the dissolution energy Ed Ed = ZEF + VoNd + Vi( 1OMr - Nd/~l)/2&
+ . .. 00) where ccl, pi are for the perfect lattice and the M, (n = 1, 2,3 , . . .) are defined by EF
M,=j
Enpo(E)dE.
(11)
1 - F,G,R(E,)]
= Fipi(E~)/]l
- V,G,R(EF)l
,
(6)
where pi(E) (pa(E)) denotes the local DOS on the atomic site Ri (on the perfect lattice site). This relation on the impurity potentials is derived from the condition of charge conservation of the whole system. Here, we have assumed for simplicity that each d-subband is equivalent, and the suffx m has been omitted. Then the heat of dissolution Ed can be given by
A similar expression for Ed can also be given, in terms of the site dependent /Lri, p;i andM,ri, for the impurity atom located at_Ri around the lattice defect (imposing the condition of charge neutrality for each atom [3,4]). The heat of segregation Esepr(Ri) canthen be given from eqs. (5) and (10) as *r Rsegr(Ri) = (Vi - Vu)Nd + “f(l@Mli - I$( l&M, - N&)/2E”;
+ . .. .
- Ndpli)/2& (12)
EF Ed=ZEF-FJ
Arg(l-
VoGo)dE,
(7)
where the factor 10 is due to the degeneracy of the dband. In order to obtain a more explicit expression for Ed, we expand the second term of eq. (7) in terms of the impurity potential V. 186
We now apply eq. (12) to specific segregation problems. First, we consider impurity segregation at a
*t Strictly speaking, the first term of eq. (12) becomes ViNi - Vfl,~, where Ni denotes the number of d-electrons on the defect site i. However, it is well known that Ni w NO is a good approximation for usual lattice defect problems [3,9].
-0.117867 -0.207956 0.200831 0.106530 0.000978
-0.075215 -0.086974 0.056989 0.033781 0.000596
-0.402480 -0.148003
-0.268414 -0.153319
1 2 3 4 5
1 2 3 4 5
1 1
1 1
Ni(ll1)
Ni(ll1)
Ni(ll0) Ni(100)
Ni(ll0) Ni(100)
(a)
(b)
(c)
(d)
-0.163926 -0.065894
-0.207608 -0.047658
-0.028930 -0.070446 0.018818 0.024108 0.004105
-0.030343 -0.113091 0.040119 0.04495 1 0.007088
cr/-4
Fe/-2 0.025737 -0.049858 -0:005678 0.010312 0.003215 0.018302 -0.04068 1 -0.006038 0.007956 0.0025 99 -0.068836 -0.010199 -0.049249 -0.000107
Mn/-!3 0.007266 -0.076112 0.004239 0.021799 0.006072 0.000393 -0.056272 0.000723 0.014773 0.004330 -0.12375 1 -0.021949 -0.095691 -0.021301
0.0 0.0
-0.025756 -0.001034
0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.021301 -0.021526 -0.006121 0.003570 0.000721
-0.014596 0.007467
0.0 0.0 0.0 0.0 0.0
0.0255 12 -0.024600 -0.006063 0.004239 0.000808
a)
Ni/O
Co/-l
Zn/+2 -0.203613 0.028153 0.099847 -0.003704 -0.004390 -0.208700 0.026200 0.104600 -0.003200 -0.004600 -0.150579 -0.168006 -0.227870 -0.244189
Cu/+l -0.058806 0.021667 0.020266 -0.004286 -0.000197 -0.057479 0.02055 1 0.020919 -0.004036 -0.000221 -0.009767 -0.026818 -0.026630 -0.043385
and Ni(lOO) surfaces (in Ry). Cases (a) and (c) are calculated using the first term of eq. (12)
a) For almost the same Vo and Vi values, the present E segr calculations using eq. (12) (first term) should not be taken seriously but simply be taken to be Esqr a 0.
VI-5
Impurity/Z
Z
Plane
Case
Table 1 Segregation energies E segr versus impurity position for Ni(l1 l), Ni(ll0) while cases(b) and (d) are calculated from refs. [3] and 141.
Volume 107A. number 4 cleaved
crystal
surface.
PHYSICS LETTERS In particular,
we investigate
on the Ni( 11 l), Ni( 110) and Ni( 100) surfaces: The segregation of these systems has been studied extensively [ 3,4] and detailed information on the impurity potentials Vi and VO is available. We can check the validity of the expansion of eq. (12) for the segregation energies E segr_ In table 1 we present the calculated Esegr values using eq. (12), but retaining only the first term, and compare them with those obtained by much more elaborate numerical calculations [3,4]. Surprisingly enough, the approximate Esegr values calculated from the first term of eq. (12) can reproduce the “true” segregation energies of ref. [3] well, especially for impurities with lZ( < 2 (weak impurity potentials): Even for impurities with IZ I > 3 the sign of Esegr is correct and the order of magnitude reasonable. Therefore, E segr of eq. (12) can be used in general to predict impurity segregation at lattice defects in metals as well as to check the results ofEseg, obtained by much more elaborate numerical calculations. Next we consider the segregation at dislocations and grain boundaries in transition metals. These problems are quite important in fracture science and in metallurgy. We consider the impurity segregation in bee transition metals with nearly half-filled d-bands like &Fe, MO and W. When treating the weak impurity potentials (roughly speaking for impurities with IZ I < 2), one can use the relation Vopo(EF) = Vipi(EF), instead of eq. (6). This approximation becomes more reasonable for metals with nearly half-filled bands, since the GP(EF) are generally small Under this condition, one can rewrite eq. (12) as the segregation
E
Ndz
segr = mo(EF)
of transitional
elements
Ap(EF)Ih(EF) 1+ &(EF)IP~(EF)
+ o(vi”,
+),
where Ap(EF) denotes the change in the total DOS at the Fermi energy. This expression tells us that once the knowledge on the local DOS pi(E) is available one can immediately predict the impurity (or solute) segregation. Nowadays, it is not difficult to get information on the local DOS pi(E) in the vicinity of lattice defects in metals and semiconductors. For this purpose, the TB recursion method [lo] has widely been used for various problems. However, in view of the lack of theoretical calculations on the electronic structure for dislocations in metals, we have calculated the local 188
28 January 1985
LDOS
c
I
(a)
-0.2
0 E
0.2
(C)
-0.2
0
0.2
E
Fig. 1. Local d-electron DOS on atoms around the (a/2) (111) screw dislocation in or-Fe: Nearest-neighbour (a), next-nearest neighbour (b) and third nearest-neighbour (c) atomic sites. Also the DOS of a perfect lattice is shown in (d). Energies are given in Ry units.
DOS p&E) for a (a/2) ( 111) screw dislocation in bee transition metals using the recursion method and present the results for a-Fe, first, second and third nearest-neighbour positions around the dislocation center, in fig. 1. The d-band parameters are taken from ref. [ 1 l] and the atomic relaxations around the screw dislocation are calculated using the method outlined in ref. [12]. The non-degenerate type core of the screw dislocation has been obtained for bee transition metals. It is interesting to note that the calculated local DOS pj(E) around the screw dislocation quite resemble those of the amorphous Fe [ 13,141. The change in the local DOS at the Fermi energy can be estimated from fig. 1, to be A/J(EF)/PO(EF) = -0.4 to -0.5 (for Nd = 6.5-7). Thus the segregation energies are roughly estimated to be Esegr x -0.082 Ryd for impurities with IZI G 2, i.e., for Cr, Mn, Co and Ni. These Esegr values are comparable to those calculated for the Ni surfaces (see table l), but much larger than those estimated by using continuum elasticity theory [ 15,161; elastic (modulus or size) interaction energies between substitutional solute and screw dislocations in a-Fe have been estimated to be less than 0.01 Ryd (in absolute values). In view of the large deformation on the local DOS Pi(E) at the screw dislocation core (comparable to those observed for the surface DOS of Ni), the present Esegr values are considered to be quite reasonable. One can thus see that the “chemical interaction” plays a
Volume 107A, number 4
PHYSICS LETTERS
dominant role in determining the interaction (segregation) energy between the impurity and screw dislocation. This type of chemical interaction is convenient to discuss the solid solution hardening of softening of bee transition metals [ 171 but has been omitted in the previous theoretical treatments [ 161. Similar effects can also be expected for the solute segregation at gram boundaries in transition metals. As has been shown in ref. [ 181. for z1= 5 and X = 9 tilt grain boundaries in bee transition metals, A&E,)/ po(EF) are generally large and take values of the order of 0.1-0.7 for metals with IVd = 7-5. One can thus expect that the chemical interaction plays also a dominant role in determining the impurity segregation energies Esegr at tilt grain boundaries. Experimentally, it is well known that the substitutional impurities like Cr (2 = -2) Mn (Z = - 1) and P tend to segregate to gram boundaries in Fe or Fe based alloys [7] and affect strongly the fracture strength of gram boundaries. However, the importance of this chemical interaction has not been fully recognized. In conclusion, we have used a TB electronic theory and clarified the physical origin for impurity segregation metals. A simple analytic formula has been derived for the systematic understanding (or for the prediction) of impurity segregation. We have applied the theory to typical segregation problems (cleaved surfaces, dislocations and grain boundaries), and demonstrated the importance of the “chemical interaction”, i.e., band term contributions.
28 January 1985
References [l] [2] [ 31 [4]
V. Ponec, Surf. Sci 80 (1979) 352. Ph. Lambin and J.P. Gaspard, J. Phys. FlO (1980) 2413. R. Riedinger and H. Dreysse, Phys. Rev. B27 (1983) 2073. H. Dreysse and R. Riedinger, Phys. Rev. B28 (1983) 5669. [S] H. Suzuki, Proc. Symp. on Structural properties of crystal defects (Lib&, Czechoslovakia, 1983) p. 205. [6] H. Saka, Phil. Mag. 47A (1983) 131. [7] E.D. Hondros, J. Phys. (Paris) 36 (1975) C4-117. [8] M.A. Khan and C. Demangeat, Phys. Lett. 95A (1983) 499. [9] K. Masuda, Phys. Rev. B26 (1982) 5968. [lo] R. Haydock, V. Heine and M.J. Kelly, J. Phys. C5 (1972) 2845;C8 (1975) 2591. [ 111 K. Masuda, N. Hamada and K. Terakura, J. Phys. F14 (1984) 47. [12] K. Masuda and A. Sato, Phil. Mag. 37B (1978) 531; 44A (1981) 799. [13] S.N. Khanna and F. Cyrot-Lackmann, Phys. Rev. B21 (1980) 1412. [14] T. Fujiwara, J. Phys. F12 (1982) 661. [15] S. Takeuchi, J. Phys Sot. Japan 27 (1969) 929. [ 161 H. Suzuki, Dislocations in solids, ed. F.R.N. Nabarro, Vol. 4, Cl-l. 15 (1979) p. 191. [ 171 K. Masuda, IXth Yamada Conf. on Dislocations in solids (1984) to be published. [ 181 M. Hashimoto, Y. Ishida, R. Yamamoto, M. Doyama and T. Fujiwara, J. Phys. Fll(1984) L141; Proc. Vth Yamada Conf. on Point defects and defect interaction in metals (Kyoto, 1981) p. 776; M. Hashimoto, Doctoral thesis, University of Tokyo (1983).
189