- Email: [email protected]
Phonons in bcc transition-metal interfaces
Surface Science 209 (1989) 492-500 North-Holland, Amsterdam
492
PHONONS
IN bee TRANSITION-METAL
R.A. BRITO-ORTA*,
V.R. VELASCO
Institute de Ciencia de Mat&ales,
CSIC,
INTERFACES
and F. GARCiA-MOLINER
Serrano 123, 28006 Madrid, Spain
Received 28 June 1988; accepted for publication 24 November 1988
The surface Green function matching method is applied to calculate phonon bands and local densities of states for the W(001)/Mo(OO1) interfaces. A phenomenological lattice dynamical model including two-body forces up to third-nearest neighbours and three-body forces up to second-nearest neighbours is used. No Stoneley modes are found but localized modes lying between the gaps of the bulk bands are obtained.
1. Introduction Very little attention has been devoted to the theoretical study of phonons at interfaces between crystals. This is in part a consequence of the lack of experimental information on interface modes [l]. The first attempts to study interface phonons in descrete systems involved very simple models. Djafari-Rouhani, Masri and Dobrzynski [2] considered a simple cubic bicrystal with first- and second-nearest-neighbour interactions. They recovered the Stoneley waves of elasticity theory in the long-wave limit and predicted the existence of other localized and resonant short wave modes even in the absence of Stoneley waves. A related problem, phonons in crystals with stacking faults, was treated by Velasco and Yndurain [3] in fee crystals with nearest-neighbour interactions. In a more realistic fashion, Benedek and Velasco [4] studied the lattice dynamics of an interface with square symmetry between two cubic ionic crystals with commensurate lattice constant. This was done within the framework of the breathing shell model. They found strong resonant modes above the sagittal transverse band of the heavier crystal as the most important feature of the lattice dynamics of a LiF(OOl)/KF(OOl) interface. We know of no attempt to study the lattice dynamics of a realistic metallic interface. The existence of localized modes or the changes in the spatial behaviour of the bulk modes near the interface may influence the * Permanent address: Departamento de Fisica, Instituto de Ciencias, Universidad Audnoma Puebla, Apdo. J-48, Puebla, Pue. 72570, Mexico.
0039-6028/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
de
R.A. Brito-Orta
et al. / Phonons in bee transition-metal
interfaces
493
electron-phonon interaction and affect properties, like, for example, superconductivity in metallic systems with one or more interfaces. We apply here a recent generalization of the surface Green function matching (SGFM) method [6,7] and a realistic model of lattice dynamics [8] to study phonon bands and phonon local densities of states at an interface between two bee transition metals. We obtain results for the Mo(OOl)/W(OOl) interface assuming perfect interfaces with average interface interactions. We calculate also local densities of states for unrelaxed and unreconstructed (001) free surfaces of MO and W for the sake of comparison with the interface case. The method used here has been applied before to study the electronic structure of an interface between two semiconductors with tight binding models [9]. The generality of the method allows one to transfer straightaway the algorithms developed for the electronic case to study the lattice dynamical problem. It is worth noting that having solved the practical aspects of the application of the SGFM method to a single interface, the method may be readily applied to study systems with multiple interfaces. We describe the method and the lattice dynamical model in the next section (section 2). Results are given in section 3 and our conclusions in section 4.
2. Method and model A detailed account of the SGFM analysis may be found in refs. [6,7]. Some practical aspects arising from its application to the problem of electronic states in semiconductor interfaces are described in ref. [9]. We will given here only those results required for the present application. We consider a plane interface between two semi-infinite, perfectly matching, bee crystals A (left side) and B (right side). It is convenient now to use the concept of principal layer. By definition, a principal layer interacts only with itself and with its two nearest-neighbour principal layers, so that each one of them contains a number of atomic planes that depends on the range of the interatomic potential. We divide the system in principal layers labelled by an integer n which increases from left to right. The interface lies between the n = 0 and n = 1 principal layers. We assume that the direct coupling between the two semi-infinite crystals exists only between the n = 0 and n = 1 principal layers. These two layers form the interface domain. If we denote their projectors by I,., and I,, the projector for the interface domain will be 1= 1, + I,. A Fourier transform parallel to the interface makes all the quantities of interest, like the projections of the dynamical matrix or of the Green function, dependent on a wavevector K parallel to the interface. The dynamical matrix for the A/B interface system has the form H, = PAHAPA + P,H,P,
+ IHiI,
(1)
494
R.A. Brim-Orta
where P, (M = A, B) including the I, part finite crystal M with between A and B. The
et al. / Phonons in bee transition-metal
interfaces
is the unit or projector of the semi-infinite medium M, of I, P,H,P, the dynamical matrix for the semi-ina free ideal surface, and IH,I the interface coupling Green function of the system satisfies the relation
(f2-H,)G,=P,+P,. We obtain the dynamical matrix using the phenomenological model of Ramji Rao and Rajput [X] for the lattice dynamics of bee metals. To obtain expressions for the force constants within the model, one starts from the crystal potential $I written in terms of scalar products of interatomic vectors [lo] in the form
xl,, is the difference between the position vectors of atoms 1 and n, X,, is the same difference but between the atoms in their equilibrium positions, and the sums over 1, m, and n run over all the atoms in the crystal. The coefficient B,,(n) is taken as different from zero only for those terms related to two-body forces up to third-nearest neighbours and three-body forces up to secondnearest neighbours. One ends up with force constants which depend on the six adjustable parameters defined in table 1. When forming the dynamical matrix for a semi-infinite crystal with a free surface, two relevant points must be noticed. Firstly, due to the inclusion of three-body forces the force constants between two given atoms will be influenced by the existence of other atoms, which may or may not be located on the same atomic planes on which the given atoms lie. Secondly, the dynamical matrix contains one term, the “self-force constant” or diagonal term, which is Table 1 Definition of adjustable parameters used in the lattice dynamical model Parameter
Restrictions over atomic indices
a= 2B,,(n)
IislstNNofn
B = 2B,,(n)
I is 2nd NN of n
Y = 2B,,(n)
I is 3rd NN of n
0 = 2B,,(n)
I and m are both 1st NN of na)
6 = 2E,,(n)
I is 1st NN of n and m is 2nd NN of n or vice versa@
P = 2B,,(n)
I and m are both 2nd NN of na)
NN stands for “nearest neighbour(s)“. a) Take only those terms corresponding to the minimum possible angle in equilibrium.
R.A. Brito-Orta et al. / Phonons in bee transition-metal interfaces
495
the negative of the sum of all the nonvanishing force constants between one atom and all the other atoms in the system and, therefore, even in the absence of three-body forces, the diagonal projection of the dynamical matrix on a single atomic plane will depend on the existence or non-existence of other atomic planes. It follows that the interface coupling ZZZiZ has not only the off-diagonal terms ZAZZi I, + I, Hi Z,, as in the problem of tight binding electrons in a semiconductor interface [9], but also the diagonal terms ZAHiZA + Z,HiZ,. These replace, with different parameter values, the terms cut off when forming the dynamical matrices for the free-surface semi-infinite crystals and which do not arise directly from the projection of the dynamical matrix between two atomic planes belonging to different media. For a bee lattice with interactions up to third-nearest neighbours it is necessary to consider principal layers containing two atomic planes. The dynamical matrix and the Green function are (3 X 3) matrices. Their projections or “matrix elements” between principal layers are then (6 x 6) matrices. The inverse of the interface projection of the bicrystal Green function - the secular matrix - is [6,7]
where G, (M = A, B) is the bulk Green function for an infinite crystal M and Z,OZ, = M,w*Z, with M, the mass of one atom of medium M. Note that the first/second bracket on the right hand side of (4) is just the secular matrix for the semi-infinite crystal A/B with a free surface. is a 2 x 2 supermatrix. Using the principal The interface projection q’ layer labels it may be written as s’=
(MA~‘--HA)M)-HA.o--~GA.--~~G~.~~-H,.~~ - H,.oI - Ht.10 (M& -&I),, - HFwG,~,,G,:, - Hi.11’ (5)
where each element is a (6 x 6) matrix. This yields a (12 X 12) secular matrix. In practice the direct evaluation of the bulk Green functions is often a major numerical task but this is avoided by introducing transfer matrices TM (M = A, B) defined by Tr,, = GM,- I oG:,t,.
(6)
The secular matrix is then given by q’=
(M*W’-HHA)M)-HA.O-ITA-Hi.OO - Hi,,0
- Hi.01 ( M,a2 - H,),, - HB,IZTB- Hi.11 ’
(7)
and, since the transfer matrices can be readily evaluated by means of fastly convergent algorithms [ll] involving only projections of the dynamical matrix, there is no need to find the bulk Green functions.
496
R.A. Brito-Orta
The eigenvalues equation det %‘(K,
a’)
et al. / Phonons in bee transition-metal
of the interface
problem
interfaces
are the roots of the secular
= 0,
(8)
which contains no redundant information since every root corresponds to a true eigenvalue of the interface system. The secular matrices for the surface modes of semi-infinite crystals A and B are the diagonal terms of (7). Thus the lattice dynamics of the free surface systems is obtained as a byproduct of the calculation. In practice, the frequency of the interface modes and the bulk bands projected over the interface may be more easily obtained by examining the singularities and the regions of non-vanishing value of Im Tr c!?~(K,w2 + ie). This quantity is proportional to the local density of states at the interface. To obtain the local density of states in any principal layer we need the diagonal projections G,,,,, of the Green function of the interface system. On the B side this is GS,nn= G~.nn + G,,,,G,,:,(G,,,,
- GB,II)GB,IIG&.
Defining the transfer matrix S,
by
S, = G&G,,,,
(9)
(10)
7
which may be calculated by means of an algorithm similar to the one used for T,, we may write Gs,nn = G,J,
(11)
+ TB”(G,,,i - Gs,i&L
The diagonal bulk Green function G,,,, is easily obtained algebraically from the transfer matrices and projections of the dynamical matrix. Finally, the local density of states in a given layer n of the B side is D,(w)
= - 32M,w)
Im TrxG,,,,(tc, I
w2+ ie),
(12)
where the sum is extended over the two-dimensional Brillouin zone. There are similar expressions for the A side. The local density of states in one principal layer is the sum of the partial diagonal sums corresponding to the single atomic layers. These can be easily identified and evaluated separately.
3. Results We applied the previous formalism to study phonons in the W(OOl)/ Mo(001) interface. We chose this system because the corresponding bulk crystals have almost the same lattice constant (a = 3.16 A for W and 3.15 A for MO). We took a = 3.16 A for the W/MO system. Furthermore, the lattice
R.A. Brito-Orta
et al. / Phonons in bee transition-metal
491
interfaces
Fig. 1. Interface phonon bands for W(OO1)/Mo(OO1). The dashed lines are interface modes. Wavevector in (?r/a) units.
dynamical model fits very well the bulk dispersion curves of W and MO by using only four of the six adjustable parameters. Details of the fitting procedure and the values of the parameters are given by Ramji Rao and Rajput [7]. Phonon bands for the W(OOl)/Mo(OOl) interface are shown in fig. 1. Given the lack of experimental information about the interface coupling, we took the values of the coupling parameters at the interface as averages of the parameters of the two semi-infinite crystals (table 2). There are no Stoneley modes for these interfaces, but there are some interface high frequency modes appearing within the gaps of the bulk modes. No distinct resonant modes could be identified for these interfaces. The local density of states gives a direct picture in real space, layer by layer, of the effects of the surface or interface and one can follow its evolution until the bulk values are attained. This is complementary to the picture obtained from the dispersion relations of the phonon branches along symmetry directions of the surface Brillouin zone, which give no information about the spatial extent of the surface or interface effects. We used formula (12) to calculate local densities of states for the first four atomic planes at each side of the interface. The sum over the two-dimensional Brillouin zone was performed by using Cunningham’s method [12] with 36 special points. Fig. 2 shows the Table 2 Values of the coupling parameters (in (1/2a)x
10” dyn/cm’,
where a is the lattice constant in
cm)
W MO
15.00 11.61
8.45 6.46
0.45 0.61
- 0.75 - 2.02
0 0
0 0
498
R.A. Brim-Orta
et al. / Phonons in bee transition-metal
interfaces
MO
0.0
2.L
4.8
7.2
Fig. 2. Local density of phonon modes (in arbitrary units) versus frequency for the first four atomic planes at both sides of the W(OOl)/Mo(OOl) interface. The curves shown were truncated when the density of modes was greater than 4 units.
results for the W/MO system. For comparison, figs. 3 and 4 show the bulk and local densities of states for the first six atomic layers near the free surfaces of W and MO, respectively. We see that the modes belonging originally to one side of the interface penetrate somewhat into neighboring atomic layers on the other side. The local density of states tends, however, very quickly towards the bulk density as the distance from the interface increases. This may be seen very clearly by comparing the local density of states of atomic planes I = 4 and I= - 3 of fig. 2 with their corresponding bulk densities shown at the bottom of figs. 3 and 4. In the free surface case, there is an evident increase in the density of states at low frequencies and a corresponding decrease at high frequencies. The tendency towards the bulk density as one penetrates into the material occurs, in this case, more slowly. This is because cutting the links to obtain the free surface is a stronger perturbation than the one introduced by the interaction through average coupling with another medium.
R.A. Brim-Orta
et al. / Phonons in bee transition-metal
11 0.0
A,
, 2.b 42
interfaces
499
BULK
L.8 7.2 n(T Hz 1
Fig. 3. Local density of phonon modes (In arbitrary units) versus frequency for the first six atomic planes near an ideal W(OO1) surface. The curve labelled 15 = 1 is for the surface plane. The curves shown were truncated when the density of modes was greater than 4 units. MO
0.0
2.4
4.8 ‘-4217
7.2 (THz)
Fig. 4. As in fig. 3 but for Mo(001).
500
R.A. Brito-Orta
et al. / Phonons in bee transition-metal
interfaces
4. Conclusions We have demonstrated the practical application of the SGFM method to calculate phonon bands and local densities of states for surfaces and interfaces. Using a model with two-body interactions up to third neighbours and three-body interactions up to second neighbours, we obtained results for the W(OOl)/Mo(OOl) interface and also for the free (001) surfaces of W and MO. No Stoneley modes appear for these interface systems but the calculation yields localised interface modes at higher frequencies. Formulae (12) and (11) provide and expedient method to study the penetration of surface or interface effects. As expected, the former penetrate more into adjacent atomic layers. The study of the interface is interesting as a step towards the study of superlattices, which for these metals should be experimentaly feasable in view of the good lattice matching. The SGFM analysis can be readily extended to the case of superperiodic systems with repeated interfaces [6,13]. A preliminary application to mode1 superlattices was carried out in ref. [14]. The work described here can be used to study bee transition metal superlattices with realistic force constants models. Work on this problem is currently in progress.
Acknowledgements One of us (R.B.O.) is indebted to the Direction General de Investigation Cientifica y Ttcnica (Spanish Ministry of Education and Science) for a Postdoctoral Research Fellowship during which this work was carried out.
References [l] G. Benedek and V.R. Velasco, in: Dynamical Phenomena at Surfaces, Interfaces and Superlattices, Vol. 3 of Springer Series in Surface Science, Eds. F. Nizzoli, K.-H. Rieder and R.F. Willis (Springer, Berlin, 1985) p.66. [2] B. Djafari-Rouhani, P. Masri and L. Dobrzynski, Phys. Rev. B 15 (1977) 5690. [3] V.R. Velasco and F. Yndurain, Surface Sci. 85 (1979) 107. [4] G. Benedek and V.R. Velasco, Phys. Rev. B 23 (1981) 6691. [5] C.-H. Tsai, Physica B 126 (1984) 107. [6] F. Garcia-Moliner and V.R. Velasco, Progr. Surface Sci. 21 (1986) 93. [7] F. Garcia-Moliner and V.R. Velasco, Phys. Scripta 34 (1986) 257. [8] R. Ramji Rao and A. Rajput, Phys. Status Solidi (b) 106 (1981) 393. [9] M.C. Muiioz, V.R. Velasco and F. Garcia-Moliner, Phys. Scripta 35 (1987) 504. [lo] P.N. Keating, Phys. Rev. 145 (1966) 637. [ll] M.P. Lopez Sancho, J.M. Lopez Sancho and J. Rubio, J. Phys. F (Metal Phys.) 14 (1984) 1205; 15 (1985) 851. [12] S.L. Cunningham, Phys. Rev. B 10 (1974) 4988. [13] F. Garcia-Moliner and V.R. Velasco, Phys. Scripta 34 (1986) 252. 1141 R. Brito-Orta, V.R. Velasco and F. Garcia-Moliner, Phys. Scripta 37 (1988) 131.
492
PHONONS
IN bee TRANSITION-METAL
R.A. BRITO-ORTA*,
V.R. VELASCO
Institute de Ciencia de Mat&ales,
CSIC,
INTERFACES
and F. GARCiA-MOLINER
Serrano 123, 28006 Madrid, Spain
Received 28 June 1988; accepted for publication 24 November 1988
The surface Green function matching method is applied to calculate phonon bands and local densities of states for the W(001)/Mo(OO1) interfaces. A phenomenological lattice dynamical model including two-body forces up to third-nearest neighbours and three-body forces up to second-nearest neighbours is used. No Stoneley modes are found but localized modes lying between the gaps of the bulk bands are obtained.
1. Introduction Very little attention has been devoted to the theoretical study of phonons at interfaces between crystals. This is in part a consequence of the lack of experimental information on interface modes [l]. The first attempts to study interface phonons in descrete systems involved very simple models. Djafari-Rouhani, Masri and Dobrzynski [2] considered a simple cubic bicrystal with first- and second-nearest-neighbour interactions. They recovered the Stoneley waves of elasticity theory in the long-wave limit and predicted the existence of other localized and resonant short wave modes even in the absence of Stoneley waves. A related problem, phonons in crystals with stacking faults, was treated by Velasco and Yndurain [3] in fee crystals with nearest-neighbour interactions. In a more realistic fashion, Benedek and Velasco [4] studied the lattice dynamics of an interface with square symmetry between two cubic ionic crystals with commensurate lattice constant. This was done within the framework of the breathing shell model. They found strong resonant modes above the sagittal transverse band of the heavier crystal as the most important feature of the lattice dynamics of a LiF(OOl)/KF(OOl) interface. We know of no attempt to study the lattice dynamics of a realistic metallic interface. The existence of localized modes or the changes in the spatial behaviour of the bulk modes near the interface may influence the * Permanent address: Departamento de Fisica, Instituto de Ciencias, Universidad Audnoma Puebla, Apdo. J-48, Puebla, Pue. 72570, Mexico.
0039-6028/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
de
R.A. Brito-Orta
et al. / Phonons in bee transition-metal
interfaces
493
electron-phonon interaction and affect properties, like, for example, superconductivity in metallic systems with one or more interfaces. We apply here a recent generalization of the surface Green function matching (SGFM) method [6,7] and a realistic model of lattice dynamics [8] to study phonon bands and phonon local densities of states at an interface between two bee transition metals. We obtain results for the Mo(OOl)/W(OOl) interface assuming perfect interfaces with average interface interactions. We calculate also local densities of states for unrelaxed and unreconstructed (001) free surfaces of MO and W for the sake of comparison with the interface case. The method used here has been applied before to study the electronic structure of an interface between two semiconductors with tight binding models [9]. The generality of the method allows one to transfer straightaway the algorithms developed for the electronic case to study the lattice dynamical problem. It is worth noting that having solved the practical aspects of the application of the SGFM method to a single interface, the method may be readily applied to study systems with multiple interfaces. We describe the method and the lattice dynamical model in the next section (section 2). Results are given in section 3 and our conclusions in section 4.
2. Method and model A detailed account of the SGFM analysis may be found in refs. [6,7]. Some practical aspects arising from its application to the problem of electronic states in semiconductor interfaces are described in ref. [9]. We will given here only those results required for the present application. We consider a plane interface between two semi-infinite, perfectly matching, bee crystals A (left side) and B (right side). It is convenient now to use the concept of principal layer. By definition, a principal layer interacts only with itself and with its two nearest-neighbour principal layers, so that each one of them contains a number of atomic planes that depends on the range of the interatomic potential. We divide the system in principal layers labelled by an integer n which increases from left to right. The interface lies between the n = 0 and n = 1 principal layers. We assume that the direct coupling between the two semi-infinite crystals exists only between the n = 0 and n = 1 principal layers. These two layers form the interface domain. If we denote their projectors by I,., and I,, the projector for the interface domain will be 1= 1, + I,. A Fourier transform parallel to the interface makes all the quantities of interest, like the projections of the dynamical matrix or of the Green function, dependent on a wavevector K parallel to the interface. The dynamical matrix for the A/B interface system has the form H, = PAHAPA + P,H,P,
+ IHiI,
(1)
494
R.A. Brim-Orta
where P, (M = A, B) including the I, part finite crystal M with between A and B. The
et al. / Phonons in bee transition-metal
interfaces
is the unit or projector of the semi-infinite medium M, of I, P,H,P, the dynamical matrix for the semi-ina free ideal surface, and IH,I the interface coupling Green function of the system satisfies the relation
(f2-H,)G,=P,+P,. We obtain the dynamical matrix using the phenomenological model of Ramji Rao and Rajput [X] for the lattice dynamics of bee metals. To obtain expressions for the force constants within the model, one starts from the crystal potential $I written in terms of scalar products of interatomic vectors [lo] in the form
xl,, is the difference between the position vectors of atoms 1 and n, X,, is the same difference but between the atoms in their equilibrium positions, and the sums over 1, m, and n run over all the atoms in the crystal. The coefficient B,,(n) is taken as different from zero only for those terms related to two-body forces up to third-nearest neighbours and three-body forces up to secondnearest neighbours. One ends up with force constants which depend on the six adjustable parameters defined in table 1. When forming the dynamical matrix for a semi-infinite crystal with a free surface, two relevant points must be noticed. Firstly, due to the inclusion of three-body forces the force constants between two given atoms will be influenced by the existence of other atoms, which may or may not be located on the same atomic planes on which the given atoms lie. Secondly, the dynamical matrix contains one term, the “self-force constant” or diagonal term, which is Table 1 Definition of adjustable parameters used in the lattice dynamical model Parameter
Restrictions over atomic indices
a= 2B,,(n)
IislstNNofn
B = 2B,,(n)
I is 2nd NN of n
Y = 2B,,(n)
I is 3rd NN of n
0 = 2B,,(n)
I and m are both 1st NN of na)
6 = 2E,,(n)
I is 1st NN of n and m is 2nd NN of n or vice versa@
P = 2B,,(n)
I and m are both 2nd NN of na)
NN stands for “nearest neighbour(s)“. a) Take only those terms corresponding to the minimum possible angle in equilibrium.
R.A. Brito-Orta et al. / Phonons in bee transition-metal interfaces
495
the negative of the sum of all the nonvanishing force constants between one atom and all the other atoms in the system and, therefore, even in the absence of three-body forces, the diagonal projection of the dynamical matrix on a single atomic plane will depend on the existence or non-existence of other atomic planes. It follows that the interface coupling ZZZiZ has not only the off-diagonal terms ZAZZi I, + I, Hi Z,, as in the problem of tight binding electrons in a semiconductor interface [9], but also the diagonal terms ZAHiZA + Z,HiZ,. These replace, with different parameter values, the terms cut off when forming the dynamical matrices for the free-surface semi-infinite crystals and which do not arise directly from the projection of the dynamical matrix between two atomic planes belonging to different media. For a bee lattice with interactions up to third-nearest neighbours it is necessary to consider principal layers containing two atomic planes. The dynamical matrix and the Green function are (3 X 3) matrices. Their projections or “matrix elements” between principal layers are then (6 x 6) matrices. The inverse of the interface projection of the bicrystal Green function - the secular matrix - is [6,7]
where G, (M = A, B) is the bulk Green function for an infinite crystal M and Z,OZ, = M,w*Z, with M, the mass of one atom of medium M. Note that the first/second bracket on the right hand side of (4) is just the secular matrix for the semi-infinite crystal A/B with a free surface. is a 2 x 2 supermatrix. Using the principal The interface projection q’ layer labels it may be written as s’=
(MA~‘--HA)M)-HA.o--~GA.--~~G~.~~-H,.~~ - H,.oI - Ht.10 (M& -&I),, - HFwG,~,,G,:, - Hi.11’ (5)
where each element is a (6 x 6) matrix. This yields a (12 X 12) secular matrix. In practice the direct evaluation of the bulk Green functions is often a major numerical task but this is avoided by introducing transfer matrices TM (M = A, B) defined by Tr,, = GM,- I oG:,t,.
(6)
The secular matrix is then given by q’=
(M*W’-HHA)M)-HA.O-ITA-Hi.OO - Hi,,0
- Hi.01 ( M,a2 - H,),, - HB,IZTB- Hi.11 ’
(7)
and, since the transfer matrices can be readily evaluated by means of fastly convergent algorithms [ll] involving only projections of the dynamical matrix, there is no need to find the bulk Green functions.
496
R.A. Brito-Orta
The eigenvalues equation det %‘(K,
a’)
et al. / Phonons in bee transition-metal
of the interface
problem
interfaces
are the roots of the secular
= 0,
(8)
which contains no redundant information since every root corresponds to a true eigenvalue of the interface system. The secular matrices for the surface modes of semi-infinite crystals A and B are the diagonal terms of (7). Thus the lattice dynamics of the free surface systems is obtained as a byproduct of the calculation. In practice, the frequency of the interface modes and the bulk bands projected over the interface may be more easily obtained by examining the singularities and the regions of non-vanishing value of Im Tr c!?~(K,w2 + ie). This quantity is proportional to the local density of states at the interface. To obtain the local density of states in any principal layer we need the diagonal projections G,,,,, of the Green function of the interface system. On the B side this is GS,nn= G~.nn + G,,,,G,,:,(G,,,,
- GB,II)GB,IIG&.
Defining the transfer matrix S,
by
S, = G&G,,,,
(9)
(10)
7
which may be calculated by means of an algorithm similar to the one used for T,, we may write Gs,nn = G,J,
(11)
+ TB”(G,,,i - Gs,i&L
The diagonal bulk Green function G,,,, is easily obtained algebraically from the transfer matrices and projections of the dynamical matrix. Finally, the local density of states in a given layer n of the B side is D,(w)
= - 32M,w)
Im TrxG,,,,(tc, I
w2+ ie),
(12)
where the sum is extended over the two-dimensional Brillouin zone. There are similar expressions for the A side. The local density of states in one principal layer is the sum of the partial diagonal sums corresponding to the single atomic layers. These can be easily identified and evaluated separately.
3. Results We applied the previous formalism to study phonons in the W(OOl)/ Mo(001) interface. We chose this system because the corresponding bulk crystals have almost the same lattice constant (a = 3.16 A for W and 3.15 A for MO). We took a = 3.16 A for the W/MO system. Furthermore, the lattice
R.A. Brito-Orta
et al. / Phonons in bee transition-metal
491
interfaces
Fig. 1. Interface phonon bands for W(OO1)/Mo(OO1). The dashed lines are interface modes. Wavevector in (?r/a) units.
dynamical model fits very well the bulk dispersion curves of W and MO by using only four of the six adjustable parameters. Details of the fitting procedure and the values of the parameters are given by Ramji Rao and Rajput [7]. Phonon bands for the W(OOl)/Mo(OOl) interface are shown in fig. 1. Given the lack of experimental information about the interface coupling, we took the values of the coupling parameters at the interface as averages of the parameters of the two semi-infinite crystals (table 2). There are no Stoneley modes for these interfaces, but there are some interface high frequency modes appearing within the gaps of the bulk modes. No distinct resonant modes could be identified for these interfaces. The local density of states gives a direct picture in real space, layer by layer, of the effects of the surface or interface and one can follow its evolution until the bulk values are attained. This is complementary to the picture obtained from the dispersion relations of the phonon branches along symmetry directions of the surface Brillouin zone, which give no information about the spatial extent of the surface or interface effects. We used formula (12) to calculate local densities of states for the first four atomic planes at each side of the interface. The sum over the two-dimensional Brillouin zone was performed by using Cunningham’s method [12] with 36 special points. Fig. 2 shows the Table 2 Values of the coupling parameters (in (1/2a)x
10” dyn/cm’,
where a is the lattice constant in
cm)
W MO
15.00 11.61
8.45 6.46
0.45 0.61
- 0.75 - 2.02
0 0
0 0
498
R.A. Brim-Orta
et al. / Phonons in bee transition-metal
interfaces
MO
0.0
2.L
4.8
7.2
Fig. 2. Local density of phonon modes (in arbitrary units) versus frequency for the first four atomic planes at both sides of the W(OOl)/Mo(OOl) interface. The curves shown were truncated when the density of modes was greater than 4 units.
results for the W/MO system. For comparison, figs. 3 and 4 show the bulk and local densities of states for the first six atomic layers near the free surfaces of W and MO, respectively. We see that the modes belonging originally to one side of the interface penetrate somewhat into neighboring atomic layers on the other side. The local density of states tends, however, very quickly towards the bulk density as the distance from the interface increases. This may be seen very clearly by comparing the local density of states of atomic planes I = 4 and I= - 3 of fig. 2 with their corresponding bulk densities shown at the bottom of figs. 3 and 4. In the free surface case, there is an evident increase in the density of states at low frequencies and a corresponding decrease at high frequencies. The tendency towards the bulk density as one penetrates into the material occurs, in this case, more slowly. This is because cutting the links to obtain the free surface is a stronger perturbation than the one introduced by the interaction through average coupling with another medium.
R.A. Brim-Orta
et al. / Phonons in bee transition-metal
11 0.0
A,
, 2.b 42
interfaces
499
BULK
L.8 7.2 n(T Hz 1
Fig. 3. Local density of phonon modes (In arbitrary units) versus frequency for the first six atomic planes near an ideal W(OO1) surface. The curve labelled 15 = 1 is for the surface plane. The curves shown were truncated when the density of modes was greater than 4 units. MO
0.0
2.4
4.8 ‘-4217
7.2 (THz)
Fig. 4. As in fig. 3 but for Mo(001).
500
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et al. / Phonons in bee transition-metal
interfaces
4. Conclusions We have demonstrated the practical application of the SGFM method to calculate phonon bands and local densities of states for surfaces and interfaces. Using a model with two-body interactions up to third neighbours and three-body interactions up to second neighbours, we obtained results for the W(OOl)/Mo(OOl) interface and also for the free (001) surfaces of W and MO. No Stoneley modes appear for these interface systems but the calculation yields localised interface modes at higher frequencies. Formulae (12) and (11) provide and expedient method to study the penetration of surface or interface effects. As expected, the former penetrate more into adjacent atomic layers. The study of the interface is interesting as a step towards the study of superlattices, which for these metals should be experimentaly feasable in view of the good lattice matching. The SGFM analysis can be readily extended to the case of superperiodic systems with repeated interfaces [6,13]. A preliminary application to mode1 superlattices was carried out in ref. [14]. The work described here can be used to study bee transition metal superlattices with realistic force constants models. Work on this problem is currently in progress.
Acknowledgements One of us (R.B.O.) is indebted to the Direction General de Investigation Cientifica y Ttcnica (Spanish Ministry of Education and Science) for a Postdoctoral Research Fellowship during which this work was carried out.
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