- Email: [email protected]
A model for bimetallic interfaces
Solid State Commumcatlons, Vol 58, No 1, pp 29-31, 1986 Printed m Great Britain.
0038-1098/86 $3.00 + 00 Pergamon Press Ltd
A MODEL FOR BIMETALLIC INTERFACES M P Des* and N Nafani" International Centre for Theoretical Physics, Mlramare, Trieste, Italy
(Recetved 12 September 1985 by S Lundqvzst) We have developed a theoretical model by considering the charge rearrangement m the region of a btmetalhc interface By using the density functional formahsm we have calculated variatlonally mterfaclal energies due to pairs of semi-mf'nute jelha m contact We refer that the electronic mteracUon has an important role in the blmetalhc adhesion INTERFACIAL PHENOMENA in metals and alloys have many important technological apphcatlons such as deposition of metalhc films, electrical contacts, soldering m sohd state devices, friction, wear, gram boundary and fractures etc Therefore, there have been theoretical and experimental interests to understand the physics of the btmetalhc interfaces [1, 2] Durmg the past eighteen years many attempts have been made mamly to study the adhesive properties of two metal surfaces [3-14] Most of these theories are based on the density functional formahsm for the mhomogeneous electron gas [ 15 ] When two semi-infinite metal surfaces are brought together from mfmtte separation, the force of attraction between them is mainly due to the daspersion forces But at small separation the electronic densities spdhng out of the surfaces overlap, as a result of which the metalhc bond sets m Tins idea is basic to most of the models [ 3 - 8 ] . When two disstrnflar metals are in contact it has been suggested [2, 16] that a strong adhesion is possible ff one of the metals is an electron-donor and the other is an electron-acceptor For the simple metals we know from the experiment as well as from the theory [17, 18] that the work function of a tugh denmy metal is larger than that of a low density metal From the consideration of the energetlcs it is clear that some charges will flow from the low denmy region to that of the lugh density m order to equalaze the chemical potential of the combined system To our knowledge none of the models incorporated the charge rearrangements exphcltly as pointed out above. In order to get a quantitative account of how the adhesion is influenced by the charge rearrangement m dissimilar alkah mterfaces we have developed a model for the electron density m the jelhum approximation. * Department of Physics, Sambalpur Umverslty, Jyotl Vthar Sambalpnr 768019, Indm t Department of Physics, Faculty of Science, A1-Fateh Umverslty, Tnpoh, Labya 29
The electron density along the chrectlon perpendacular to the interface is modelled as pA
pB
.4
a(x) = 1 + e -ax + 1a +~ e -
-
cosh ~l(x + Xo)
A' cosh/3'1(x -- x~)
•
(1)
PAtB) iS the bulk electron density of metal A(B). PA > PB The first two terms in equaUon (1) represent the usual quantum mechanical leakage of the density as considered m most of the models [ 3 - 8 ] . The tlurd and the fourth term represent the scooping of electrons from the low density metal and the pflmg of electrons on the tugh density region respectively, A and/31(.4', ~'1) are the depth (height), and the reverse width of the charge loss (gain) respectively Xo(X'o) is the location of the depth (height) of the loss (gaul) of the charge Without loss of generahty we choose (3=/31 = ~] and A = A' to reduce the number of variational parameters and further, since the charge rearrangement takes place m the mterfaclal region (wittun 20 A) we have taken the reflection point of all the four terms to he on the interface This criterion gives rise to cosh (~x0 --- V ~ and the equation (1) is stmphfled to
1 p(x) = pa
PB/PA
1 + e -~x + 1 + e a--------~
+ PA(1 + cosh 2 x) ]"
(2)
Tim model should satisfy (a) the charge neutrahty condmon (b) Ltx--, ±**(x)-~ PAtm and (c) at the raterface x = 0 p(x) = (,oA + PB)/2 The condition (c) may be relaxed m a more reahsUc model. Now m order to determine the energy we adopt the density functional theory. The energy density is wntten as [6].
30
Vol 58, No 1
A MODEL FOR BIMETALLIC INTERFACES
Table I Interfaces
Pn]PA
13
A/PA
G.~B
E,~d
La Cs La Rb LI K Na Cs Na Rb Na K
0 198 0247 0 289 0 360 0 444 0 521
0 630 0639 0 646 0 594 0 603 0 610
0 0 0 0 0 0
31 20 13 19 12 7
227 245 264 160 176 194
213 120 185 130 153 131
* Energies are m ergs cm -2
1
Z,Z,
E[p] = f u(r--)p(~)dY + ~ ,~, [.~_R,~-I + Ftp]
(3)
Here
1 F[p] = -~
d~d~' + T[p] + Exc[P] ff p(r)p(U) I-~-~'1
(4)
The first two terms m equatmn (3) are the electron-ion and ion-ran interaction energies respectively The functmnal Fro] contains the classical Coulomb, kinetic and exchange-correlation energtes respectively We have chosen
T[p] : ~(3r?) =/3 f P(F)S/adF+ f f(Pff))l VP(~)l=d~,
(5) where f(p(~)) m random phase approxtmation as gwen by f ( p (~)) = 1/72p (T) and
Exc[P] = --
f P(r)4/3dr
/-
o(~) ':3
- - 0 056 30.079 + o r , ( ? ) d~
(6)
blmetalhc system of two jelha with respect to two parameters /3 and A introduced m our model density We solve the following two equations stmultaneously aE --=0
and
a3
~E -- =0
aA
We calculate the adhesion energy given by
Ead = rE(oo) -- E(0)] ]2 = [(o.4 + oB) -- oaB] ]2 (7) E(oo) and E(0) are the total energies of the systems when the surfaces are mfimtely apart and are m contact respectively oan xs the mterfaclal energy, and Oa(m IS the surface energy of metal A(B) In Table 1 the results of our calculations are presented PA]PB lS the raUo of the bulk densities of two metals /3 is given m reverse length (atomxc units) A[pa is the height of the scooping parameter dwlded by the tugher bulk density of the metals oan and EAd are the mterfacial and adhesxve energies respectwely gwen in units of ergs cm -2 Table 1 shows that as the difference between the bulk densities of the metals increases, the scooping coefficient A increases and 13 becomes smaller Therefore more charges flow from low to tugh density region and the lnterfaclal thickness (~13-1)increases Tlus gwes nse to better adhesion The density p r o n e of L1-Cs interface (dashed hne) is shown m Fig 1 The dotted hne is the sum of first two terms of equation (2) corresponding to t t ~ usual quantum mechanxcal leakage The continuous lane showsthe effect of scooping and piling of charges m the respective regimes The charge transfer for various combmatmns of interfaces is consistent with our argument concermng the difference of the work functmns Tlus idea is further supported by the recent self-consistent calculation of Ferrante and Smith [14] In Fig 2 the total density p r o n e of some
We mlmm~ze the energy corresponding to our II~4
IC
12
09
/'~
I0 oo
08
_
i/
06 -
~ ~
.I
0 7 --
/
o4 - o2
I"
,, :)
-
/
06
/
I
/ "/ / I l l
,,I C]
05 O4
0
03~_
......
-02 O2
-0 4 -06 I I °630-504-378-252
I
I
-126
I
0
I
126
I
252
I
378
I
504
I
630
X-axis Fig 1 The density p r o n e for La-Cs mterface The dashed hne is the full density proNe the dotted line represents the usual quantum mechamcal leakage and the full lane represents the charge rearrangement term as m equation (2) The horizontal axis IS in units of/3
o- 3 2 5 0 - 2 5 ~I- - 1 9 3 ~ [
I
I
-1292 -0646
I
0
I
064.6
I
1292
I
1938
[
2584
!
3 230
X-QXIS
Fig 2 Density profiles for three interfaces The full line IS for L1-Cs the dashed line for I J - K and the dotted line for N a - K systems The plots are normahzed to the bulk density of the lugh densxty metal The horizontal axis is In umts of corresponding 13
Vol 58, No 1
A MODEL FOR BIMETALLIC INTERFACES
30--
31
regton and thereby an increase m the adhesive energy Details of these results wtll be reported later
25 20
Acknowledgements - The authors are grateful to Professors Norman H March and Stlg Lundqvlst for many stimulating discussions They are thankful to Professor Abdus Salam International Atomic Energy Agency and UNESCO for the hospltahty at the International Centre for Theoretical Physics Trieste
15 LL I.I.I ,{...) 121
I0
~ ~
05 o -05 -I 0
REFERENCES
-15 -20 I -3230-25~4
[
I
[
-1938-1Lx32-0646
I
0
I
0646
I
1292
I
1938
I
2584
I
3230
X-axis
Fig 3 Denuty different plot for the same systems as mF~g 2
1 2 3 4
chosen combinations as La-Cs, L1-K and N a - K are presented The density oscillations winch are exhibited m the simple overlap model of Ferrante and Smith seem to be smoothed out in their recent self-consistent calculations [14] Therefore our model density may not be too far from the reahstlc one Figure 3 shows the density difference as given by ~p(x) =
5 6 7 8 9
[ p ( x ) - p a o ( x ) - p B o ( - x)]
pAO(x) + pAO(-x)
where O(x) Is the unit step function For the same combinations as in Fig 2 the change in ther mterfaclal energies may be understood in view of the density differences The results discussed above are based on the calculations with the local denstty approxamatlon for the exchange-correlation potential (m equation 6) We have performed a prehmmary calculation for L1-Na interface by using a nonlocal exchange-correlation as suggested by Yamasluta and Ichlmaru [19] The results mdlcate more charge rearrangement m the mterfaclal
10 11 12 13 14 15 16 17 18 19
L.E Murr, Interracial Phenomena m Metals and Alloys, Addison and Wesley (1975) D Tabor, Surface Phystcs of MaterLals, (Edited by J M Blakely), 2,490 (1975) L Bennett & C B Duke, Phys Rev 160, 541 (1967),Phys Rev 162,578 (1967) M D Rouham & R Schuttler, Surf Sct 38, 503 (1973) J.R Smith & J Ferrante, Surf Scl 38, 77 (1973), Sohd State Commun 21, 1059 (1976) R Mehrotra, M M Pant & M P Das, Sohd State Commun 18, 199 (1976) J P Muscat&G Allan, J Phys F7,999 (1977) J N Swmgler & J C Inkson, J Phys ClO, 573 (1977) J Helnrexch & N Kumar, Phys Rev B12, 802 (1975) A Yanw, Phys Rev B17, 3904 (1978) R M Nlemmen, J Phys F7, 375 (1977) J Inglesfield, J Phys F6,687 (1976) J H Rose, J R Smith & J Ferrante, Phys Rev Lett 47, 675 (1981) J Ferrante & J R Smith, Phys Rev B31, 3427 (1985) See for example, Theory oflnhomogeneous Electron Gas (Edited by S Lundqvlst and N H March), Plenum (1983) H Czlchos,J Phys D5, 1890 (1972) N D Lang in [15] MM. Pant & M P Das, J Phys F5, 1301 (1975) I Yamashlta & S Ictumaru, Phys Rev B29, 673 (1984)
0038-1098/86 $3.00 + 00 Pergamon Press Ltd
A MODEL FOR BIMETALLIC INTERFACES M P Des* and N Nafani" International Centre for Theoretical Physics, Mlramare, Trieste, Italy
(Recetved 12 September 1985 by S Lundqvzst) We have developed a theoretical model by considering the charge rearrangement m the region of a btmetalhc interface By using the density functional formahsm we have calculated variatlonally mterfaclal energies due to pairs of semi-mf'nute jelha m contact We refer that the electronic mteracUon has an important role in the blmetalhc adhesion INTERFACIAL PHENOMENA in metals and alloys have many important technological apphcatlons such as deposition of metalhc films, electrical contacts, soldering m sohd state devices, friction, wear, gram boundary and fractures etc Therefore, there have been theoretical and experimental interests to understand the physics of the btmetalhc interfaces [1, 2] Durmg the past eighteen years many attempts have been made mamly to study the adhesive properties of two metal surfaces [3-14] Most of these theories are based on the density functional formahsm for the mhomogeneous electron gas [ 15 ] When two semi-infinite metal surfaces are brought together from mfmtte separation, the force of attraction between them is mainly due to the daspersion forces But at small separation the electronic densities spdhng out of the surfaces overlap, as a result of which the metalhc bond sets m Tins idea is basic to most of the models [ 3 - 8 ] . When two disstrnflar metals are in contact it has been suggested [2, 16] that a strong adhesion is possible ff one of the metals is an electron-donor and the other is an electron-acceptor For the simple metals we know from the experiment as well as from the theory [17, 18] that the work function of a tugh denmy metal is larger than that of a low density metal From the consideration of the energetlcs it is clear that some charges will flow from the low denmy region to that of the lugh density m order to equalaze the chemical potential of the combined system To our knowledge none of the models incorporated the charge rearrangements exphcltly as pointed out above. In order to get a quantitative account of how the adhesion is influenced by the charge rearrangement m dissimilar alkah mterfaces we have developed a model for the electron density m the jelhum approximation. * Department of Physics, Sambalpur Umverslty, Jyotl Vthar Sambalpnr 768019, Indm t Department of Physics, Faculty of Science, A1-Fateh Umverslty, Tnpoh, Labya 29
The electron density along the chrectlon perpendacular to the interface is modelled as pA
pB
.4
a(x) = 1 + e -ax + 1a +~ e -
-
cosh ~l(x + Xo)
A' cosh/3'1(x -- x~)
•
(1)
PAtB) iS the bulk electron density of metal A(B). PA > PB The first two terms in equaUon (1) represent the usual quantum mechanical leakage of the density as considered m most of the models [ 3 - 8 ] . The tlurd and the fourth term represent the scooping of electrons from the low density metal and the pflmg of electrons on the tugh density region respectively, A and/31(.4', ~'1) are the depth (height), and the reverse width of the charge loss (gain) respectively Xo(X'o) is the location of the depth (height) of the loss (gaul) of the charge Without loss of generahty we choose (3=/31 = ~] and A = A' to reduce the number of variational parameters and further, since the charge rearrangement takes place m the mterfaclal region (wittun 20 A) we have taken the reflection point of all the four terms to he on the interface This criterion gives rise to cosh (~x0 --- V ~ and the equation (1) is stmphfled to
1 p(x) = pa
PB/PA
1 + e -~x + 1 + e a--------~
+ PA(1 + cosh 2 x) ]"
(2)
Tim model should satisfy (a) the charge neutrahty condmon (b) Ltx--, ±**(x)-~ PAtm and (c) at the raterface x = 0 p(x) = (,oA + PB)/2 The condition (c) may be relaxed m a more reahsUc model. Now m order to determine the energy we adopt the density functional theory. The energy density is wntten as [6].
30
Vol 58, No 1
A MODEL FOR BIMETALLIC INTERFACES
Table I Interfaces
Pn]PA
13
A/PA
G.~B
E,~d
La Cs La Rb LI K Na Cs Na Rb Na K
0 198 0247 0 289 0 360 0 444 0 521
0 630 0639 0 646 0 594 0 603 0 610
0 0 0 0 0 0
31 20 13 19 12 7
227 245 264 160 176 194
213 120 185 130 153 131
* Energies are m ergs cm -2
1
Z,Z,
E[p] = f u(r--)p(~)dY + ~ ,~, [.~_R,~-I + Ftp]
(3)
Here
1 F[p] = -~
d~d~' + T[p] + Exc[P] ff p(r)p(U) I-~-~'1
(4)
The first two terms m equatmn (3) are the electron-ion and ion-ran interaction energies respectively The functmnal Fro] contains the classical Coulomb, kinetic and exchange-correlation energtes respectively We have chosen
T[p] : ~(3r?) =/3 f P(F)S/adF+ f f(Pff))l VP(~)l=d~,
(5) where f(p(~)) m random phase approxtmation as gwen by f ( p (~)) = 1/72p (T) and
Exc[P] = --
f P(r)4/3dr
/-
o(~) ':3
- - 0 056 30.079 + o r , ( ? ) d~
(6)
blmetalhc system of two jelha with respect to two parameters /3 and A introduced m our model density We solve the following two equations stmultaneously aE --=0
and
a3
~E -- =0
aA
We calculate the adhesion energy given by
Ead = rE(oo) -- E(0)] ]2 = [(o.4 + oB) -- oaB] ]2 (7) E(oo) and E(0) are the total energies of the systems when the surfaces are mfimtely apart and are m contact respectively oan xs the mterfaclal energy, and Oa(m IS the surface energy of metal A(B) In Table 1 the results of our calculations are presented PA]PB lS the raUo of the bulk densities of two metals /3 is given m reverse length (atomxc units) A[pa is the height of the scooping parameter dwlded by the tugher bulk density of the metals oan and EAd are the mterfacial and adhesxve energies respectwely gwen in units of ergs cm -2 Table 1 shows that as the difference between the bulk densities of the metals increases, the scooping coefficient A increases and 13 becomes smaller Therefore more charges flow from low to tugh density region and the lnterfaclal thickness (~13-1)increases Tlus gwes nse to better adhesion The density p r o n e of L1-Cs interface (dashed hne) is shown m Fig 1 The dotted hne is the sum of first two terms of equation (2) corresponding to t t ~ usual quantum mechanxcal leakage The continuous lane showsthe effect of scooping and piling of charges m the respective regimes The charge transfer for various combmatmns of interfaces is consistent with our argument concermng the difference of the work functmns Tlus idea is further supported by the recent self-consistent calculation of Ferrante and Smith [14] In Fig 2 the total density p r o n e of some
We mlmm~ze the energy corresponding to our II~4
IC
12
09
/'~
I0 oo
08
_
i/
06 -
~ ~
.I
0 7 --
/
o4 - o2
I"
,, :)
-
/
06
/
I
/ "/ / I l l
,,I C]
05 O4
0
03~_
......
-02 O2
-0 4 -06 I I °630-504-378-252
I
I
-126
I
0
I
126
I
252
I
378
I
504
I
630
X-axis Fig 1 The density p r o n e for La-Cs mterface The dashed hne is the full density proNe the dotted line represents the usual quantum mechamcal leakage and the full lane represents the charge rearrangement term as m equation (2) The horizontal axis IS in units of/3
o- 3 2 5 0 - 2 5 ~I- - 1 9 3 ~ [
I
I
-1292 -0646
I
0
I
064.6
I
1292
I
1938
[
2584
!
3 230
X-QXIS
Fig 2 Density profiles for three interfaces The full line IS for L1-Cs the dashed line for I J - K and the dotted line for N a - K systems The plots are normahzed to the bulk density of the lugh densxty metal The horizontal axis is In umts of corresponding 13
Vol 58, No 1
A MODEL FOR BIMETALLIC INTERFACES
30--
31
regton and thereby an increase m the adhesive energy Details of these results wtll be reported later
25 20
Acknowledgements - The authors are grateful to Professors Norman H March and Stlg Lundqvlst for many stimulating discussions They are thankful to Professor Abdus Salam International Atomic Energy Agency and UNESCO for the hospltahty at the International Centre for Theoretical Physics Trieste
15 LL I.I.I ,{...) 121
I0
~ ~
05 o -05 -I 0
REFERENCES
-15 -20 I -3230-25~4
[
I
[
-1938-1Lx32-0646
I
0
I
0646
I
1292
I
1938
I
2584
I
3230
X-axis
Fig 3 Denuty different plot for the same systems as mF~g 2
1 2 3 4
chosen combinations as La-Cs, L1-K and N a - K are presented The density oscillations winch are exhibited m the simple overlap model of Ferrante and Smith seem to be smoothed out in their recent self-consistent calculations [14] Therefore our model density may not be too far from the reahstlc one Figure 3 shows the density difference as given by ~p(x) =
5 6 7 8 9
[ p ( x ) - p a o ( x ) - p B o ( - x)]
pAO(x) + pAO(-x)
where O(x) Is the unit step function For the same combinations as in Fig 2 the change in ther mterfaclal energies may be understood in view of the density differences The results discussed above are based on the calculations with the local denstty approxamatlon for the exchange-correlation potential (m equation 6) We have performed a prehmmary calculation for L1-Na interface by using a nonlocal exchange-correlation as suggested by Yamasluta and Ichlmaru [19] The results mdlcate more charge rearrangement m the mterfaclal
10 11 12 13 14 15 16 17 18 19
L.E Murr, Interracial Phenomena m Metals and Alloys, Addison and Wesley (1975) D Tabor, Surface Phystcs of MaterLals, (Edited by J M Blakely), 2,490 (1975) L Bennett & C B Duke, Phys Rev 160, 541 (1967),Phys Rev 162,578 (1967) M D Rouham & R Schuttler, Surf Sct 38, 503 (1973) J.R Smith & J Ferrante, Surf Scl 38, 77 (1973), Sohd State Commun 21, 1059 (1976) R Mehrotra, M M Pant & M P Das, Sohd State Commun 18, 199 (1976) J P Muscat&G Allan, J Phys F7,999 (1977) J N Swmgler & J C Inkson, J Phys ClO, 573 (1977) J Helnrexch & N Kumar, Phys Rev B12, 802 (1975) A Yanw, Phys Rev B17, 3904 (1978) R M Nlemmen, J Phys F7, 375 (1977) J Inglesfield, J Phys F6,687 (1976) J H Rose, J R Smith & J Ferrante, Phys Rev Lett 47, 675 (1981) J Ferrante & J R Smith, Phys Rev B31, 3427 (1985) See for example, Theory oflnhomogeneous Electron Gas (Edited by S Lundqvlst and N H March), Plenum (1983) H Czlchos,J Phys D5, 1890 (1972) N D Lang in [15] MM. Pant & M P Das, J Phys F5, 1301 (1975) I Yamashlta & S Ictumaru, Phys Rev B29, 673 (1984)