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Size dependence of electrical resistivity in 1D-layered nanostructures
NanoStrudured Materials, Vol. 7, No. 7. pp. 741-748.1996 Elsevier Science. Ltd Copyright 0 1996 Acta Metallurgica Inc. Printed in the USA. All rights reserved 0965-9773196 $15.00 + .oO
Pergamon
PI1 S~65-9773(9~0~47-5
SIZE DEPENDENCE OF ELECTRICAL RESISTIVITY IN lD-LAYERED NANOSTR~CTURES R.S. Qin’ and B.L. Zhou1v2 t Institute of Metal Research, Academia Sinica, Shenyang 110015, P.R. China 21nt~mation~ Center for Material Physics, Academia Sinica, Shenyang 110015, P.R. China (Accepted May 1996)
Abstract - The Landauer electrical resistivity R of one-dimensional (IL)) layered nanostru~tures is ~a~~~ated. The analysis of the iogarit~ of the resistance as af~ction of grain size reveals that there is a minimum value of resistivityR minwhen the grain size takes a value of dC,where dCdepends on the barrier height and excess volume of interface. R increases in the region of dCt > d > dC. where d is the grain size, and det is another critical value of grain size. The mechanism of the anomalous resistivity behavior is discussed briefly. I. INTRODUCTION One-dimensional nanocrystalline materials are polycrystals with the grain size of a few (typically I- 100) nanometers in only one dimension. They can consist of a lamella structure, and will be termed a one-dimensional layered nanostructure (ID-LN). The ID-LN is intended for electronic application because of a series of unusual electronic and electric properties. For example, the phenomenon of giant magneto~sis~ce (decrease of electrical resistance of materials when exposed to a magnetic field) has been reported in a number of 1D-LN systems (1). The development of quench technology such as vapor deposition (2) andelectrodeposition (3) give the opportunity to study one-dimensional nanocrystalline materials in all sizes. Actual measurements of electrical resistivity on the nanocrystalline materials have been carried out only recently. The results about ID-LN are not published to our knowledge, although there are several published works about 30 n~~~s~line materials (4-6). In the present paper, we will investigate the electrical resistivity of ID-LN theoretically. We first give the model and theoretical analysis (section II), then present the numerical calculation and discussion (section III). A brief summary is given in section IV. II. THE MODEL AND THEORETICAL
INVESTIGATION
Consider a unit of one-dimensional layered structure which consists of N-atoms. There is only one grain with n-atoms in a unit. The structural differences between grain and matrix are due to the arrangement or properties of atoms. The interface between grain and matrix plays an 744
742
RS
QIN
AND
BL
&OU
important part in lD-LN. Assuming every interface consists of m-atoms (nl = 3-5 for metal and 5-10 for ceramic), the number of atoms within matrix in a unit is N-n-2nl. In the coarse grain materials which correspond to nl -CCn c N and n is larger than the mean free path of electrons, the effect of interface can be neglected and the lD-layered structure can be modeled in two phases in a series. The resistivity is (7) R = (l--h&
+ ?R,
PI
where R is the resistivity of one-dimensional layered materials. his the volume fraction of grain R, andRB are the electrical resistivity of matrix and grain, respectively. In our consideration, n is comparable to the mean free path of an electron, and quantum effects must be taken into account. In this case eq. [l] is invalid due to quantum interference. We use the Landauer theory to calculate the electrical resistivity of ID-LN. The interaction between the conducting electron and atoms is approximated by a &function barrier with height Uo. The structures of grain, interface and matrix are represented by different arrangements of barriers. For example, the grain is described by a periodic arrangement of barriers with the period of u (a is lattice constant), and the interface is described by a dilated arrangement of barriers with a dilatant parameter AV (AV is the excess volume of interface). AV is satisfied
KF
(i = 1,2,...nt)
w.-J---=AV xi+l -xi
where Xi (i=1,2,...nl) is the spatial coordinate of i-th barriers in interfaCe,fAV is the distribution function, and the bar denotes the averaging value. The size of each part is corresponding to the number of barriers included in each part. Assuming the considered system is made up of uniform clusters (the cluster may be an atom for pure material or a chemical unit for compound material), each cluster is represented by a b-function potential, and all the &function potential are uniform. According to Landauer’s formulas (8) the electrical resistivity of a one-dimensional system is connected to its transmissivity by R = T’ -1, where R is the electrical resistivity and T is the transmissivity. The Schr&linger equation is: ‘pm+ k2-V$6(X_Xi) [
i=l
1
Cp=O
where k is the wave vector of the electron which is given by k = (2m,E / ti2)1’2, where m, and E are the mass and Fermi energy of the conducting electrons respectively, V = 2mJJ, / h2 is the reduced potential barrier height. Xi is the spatial coordinate of i-th barrier, and x is the spatial coordinate. In the region Ofxi-1 < x < xi (i = 2,3, ...N). an arbitrary solution to the Schrodinger equation for wave vector k can be expressed in the form (9)
cpi= exp(G;)sin[k(x - xi-t) + +i]
[5]
SIZE DEPENDENCE OF ELECTRICAL RESISTIVIT~ IN 1 D-LAYEREDNANOSTRUCNAES
743
Considering the continuum of wave function and discontinuation of its slope at the position of i-th &potential barriers, we obtain the recurrence formula for Gi and $i
Gi-Gi_l
=In
l/2
V2
1+Hsin2[k(x,-xi_l)+k_1]+3Sin2[k(xi-xi_l)+~i-1]
pi = arcsin[exp(Gi_l -Gi)Sin(kri + @ii-t)]
[61
[71
(where i = 2,3, ...N). Let
RJI
Ci
The transmissivity T takes the following form:
r91
It is very important to note that if one neglects multiple reflections of the wave, the transmissivity will take the form: T’=fi i=l
[( l_
2
s
I
I101 )I
It is obvious that T’ + T. It may be due to the resonance or interference, and makes the electric wave more localized or more expended. This result is helpful for making nanometer resonant tunneling devices. However, there is a free parameter $0, which is a periodic function. Performing an integration of this function in one period which correspond to multi-channel averaging, we get the mean transmissivity: 2 nci($O,~l(~O),...,~i(~O))-l
I
NO
nCi(~O,Ol(~O)r.~.~~ii(~O))+l
II111
Equation [ll] cannot be solved analytically because $0, $1, k, $3...& have different definition regions. For example, if expGt=OS and Q. E [-z / 2,7t / 21, than $1 E [-n / 6,a /6], etc.
744
RS
III. NUMERICAL
QIN AND
BL
ZHOU
CALCULATrON
AND DISCUSSION
Consider a case of ID-LN with matrix of disorder arrangement and grain of ordered arrangement. We use the dilated crystal model to describe the structure of the interface. The considered includes three kinds of arrangements corresponding to matrix, grain, and interface, respectively. The spatial ~~gement of barriers is as follows: grain: Xi- Xi-1= ff
]l2]
matrix: xi - xi-1 = a+ a sin(OS)cos(i+OS)
[I31
interface: Xi-Xi-l= (I+AV)a
1141
wherei=1,2,3,. . N,forNatomssystems. AVistbeexcessvolumeofinterface. aislatticeconstant and assuming a = 21dk. JZqn.[131is not a periodic function here because the definition region of i is integer, and the periodicity of triangle function is 2~. Using equation [ 1l] , we calculated Landauer’s resistance of 1D-LN. The unit consists of 120 atoms. Figure 1illustrates the results where V/k = 0.1,0.7, and 1S, respectively. It in~cates that if V/k is not very small the electrical resistance (ER hereafter) would decrease in a region of 0 < d c de; ER takes the minimum value Rmi,,when the grain size is dC;ER is increasing when grain size is larger than dC. Experiments tell us that when grain size d is larger than dCl (4-6), the ER will decrease. We did not find dCl because the considered unit is too small. dCdepends on V/k (see Figure 2). It is shown in Figure 2 that the larger V/K is the larger dCwill be found. For example
1
40.00
AV=9.14yo
0
8
16
24
32
40
48
56
64
72
80
Ordered grain size (number of atoms) Figure 1. Variation of Landauer resistance during grain growth. Horizontal axis is the number of atoms in an ordered grain, where the excess volume of interface is 9. 14%.
SIZE DEPENDENCE OF ELECTRICAL RESISTIVITY IN 1D-LWERED NANOSTRUCTURES
50
745
0
40
G-
0 30 q 20
10
0
0.00
0.40
0.80
1.20
1.60
2.00
Strength of potential (V/K) Figure 2. Variation of critical grain size (where minimum electrical resistivity appeared) vs. strength of barrier.
22
11
0
0
40
80
120
160
Ordered grain size (number of atoms) Figure 3. Landauer resistivity when excess volume of interface is zero.
200
RS QINANDBL ZHOU
746
ordered grain size 0 (atoms)
ordered grain size 40 (atoms)
2
ordered grain size 80 (atoms) 1
0
-1
-2
Position of atoms Figure 4. Variation of phase angle when the initial value is x/2. Grain sizes are 0,40 atoms, and 80 atoms, respectively.
SIZE DEPENDENCE OF ELECTRICAL RESISTIVITV IN 1 D-LAYEREKI NANOSTRUCNRES
747
V/K = 1.2 co~s~n~g to dCof 35 atoms. V/k = 1.5 co~s~n~g to the c&48 atoms. When V/K is very small, which maybe exist in alkali materials, dc is very, very small and cannot be
realized in experiments. The size-dependent electrical resistivity is a well-known attractive phenomenon. Experiments show that the ER of the nanocrystalline semiconductor Si:H is only 1/20of the ERof amorphous Si:H (12), and at grain size of a few nanometers some semiconductors such as Si change into conduc~rs (13), which is evidence of the existence of dc. For metal, empirical data showed that some go& conductors such as Cu and Ag change into insulators when the grain size is very small (13), which is evidence for the existence of dcr (dcl = 14 nm for Ag). We believe that dc also exists for metals, although there is no such experimental report up to now, which is our theoretical prediction. However, excess interface volume plays an important role in superfine systems. Let AV= 0, rn~e~ng the interface with ~~hous-lye or crystal-like phase, ele&rical resistivity of the system satisfies equation [l]. Figure 3 is the numerical calculation of Eqn. [ 111,where V/k= 0.7 and 0.5 respectively, and AV = 0.0%. The relationship between electrical resistivity and excess volume are given elsewhere (14). The phase angle @ is a very important quantity in a supetime system. It makes those electric properties path-dependent. Although the integral of & makes the ER independent of spatial position of the ordered grain, Qliis obviously different in each part of the sample. Figure 4 shows the variation of phase angle with grain size. The existence of Rmin is useful for electric rheological technique, which can be used to determine the nucleation rate in solidification when pulse electric discharges are used. According to our calculation (15), the change of chemical potential A& due to passing electric current depends on the electrical conductivity of formed grain. The relationship is as follows:
WI where kis aconstantjis the density of passing electric current, a2 andol areconductivity of matrix and grain respectively, If 02 > cfl the nucleation rate will increase; otherwise, it will decrease. According to the results of this paper, pulse electric current prefer the formation of grain size dc to dcI . IV. SUMMARY
The size-dependent electrical resistivity of lD-LN cannot be explained by scattering at the interface only, ~thou~ interface plays an irn~~t role. The effect of grain size must be taken into account also. Small grain size would induce quantum resonance, and makes the electronic wave more localized or more expended. The existence of Rmjn is a helpful electric rheological technique, which can be used to determine the nucleation rate in solidification when pulse electric discharges are used. The reported results of superfine metal as an insulator and semiconductor as a conductor can be explained qualitatively in our calculation. REFERENCE
1.
M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff. l? Eitenne, G. Creuzet, A. Friderich, and J. Chazelas, Phys. Rev. L&t. a,2472 (1988).
748
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
RS
QIN
AND
BL
&IOU
R.I. Bickerdike, D. Clark, J.N. Easterbrook, G. Hughes, W.N. Mair, P.G. Pariridge, and H.C. Ranson, Int. J. Rapid. Solidf. 1, 305 (1984-85). G. Pahmbo, S.J. Thorpe, and K.T. Aust, Scripta Metall. a, 1347 (1987). K. Lu, J.T. Wang, and W.D. Wei, J. Phys. D, Appl. Phys. 25,808 (1992). I. Bakonyi, E. Toth-Kadar, T. Tamoczi, L.Varga, A. Cziraki, I. Gerocs, and B. Fogarassy, Nanostruct. Mater. 1, 155 (1993). Y Z. Wang, G.W. Qiao,. X.D. Liu, B.Z. Ding, and Z.Q. Hu, Mater. Lett. n, 152 (1993). D.R. Poirier and G.H. Geiger, Transport Phenomena in Materials Processing, TMS, Warrendale, PA, p. 210 (1994). R. Landauer, Philos. Mag. m, 863 (1970). M.Ya. Azbel, and M. Rubinstein, Phys. Rev. m, 6530 (1983). M. Wagner, Phys. Rev. m, 635 (1992). K.Lu, Phys. Rev. B51,18 (1995). R. Kakkad, G. Liu, and S.J. Fonash, J. Non-Crystal. Solids 66, 115 (1989). L.D. Zhang and J.M. Mao, Nanocrystalline Materials, Liaoning Sci. & Tech. Publishing, p. 19 (1994) (in Chinese). R.S. Qin, S.X. Su, J.D. Guo and B.L. Zhou, to be published elsewhere. R.S. Qin, H.C. Yan, G.H. He, and B.L. Zhou, Chin. J. Mater. Res. 2,219 (1995).
Pergamon
PI1 S~65-9773(9~0~47-5
SIZE DEPENDENCE OF ELECTRICAL RESISTIVITY IN lD-LAYERED NANOSTR~CTURES R.S. Qin’ and B.L. Zhou1v2 t Institute of Metal Research, Academia Sinica, Shenyang 110015, P.R. China 21nt~mation~ Center for Material Physics, Academia Sinica, Shenyang 110015, P.R. China (Accepted May 1996)
Abstract - The Landauer electrical resistivity R of one-dimensional (IL)) layered nanostru~tures is ~a~~~ated. The analysis of the iogarit~ of the resistance as af~ction of grain size reveals that there is a minimum value of resistivityR minwhen the grain size takes a value of dC,where dCdepends on the barrier height and excess volume of interface. R increases in the region of dCt > d > dC. where d is the grain size, and det is another critical value of grain size. The mechanism of the anomalous resistivity behavior is discussed briefly. I. INTRODUCTION One-dimensional nanocrystalline materials are polycrystals with the grain size of a few (typically I- 100) nanometers in only one dimension. They can consist of a lamella structure, and will be termed a one-dimensional layered nanostructure (ID-LN). The ID-LN is intended for electronic application because of a series of unusual electronic and electric properties. For example, the phenomenon of giant magneto~sis~ce (decrease of electrical resistance of materials when exposed to a magnetic field) has been reported in a number of 1D-LN systems (1). The development of quench technology such as vapor deposition (2) andelectrodeposition (3) give the opportunity to study one-dimensional nanocrystalline materials in all sizes. Actual measurements of electrical resistivity on the nanocrystalline materials have been carried out only recently. The results about ID-LN are not published to our knowledge, although there are several published works about 30 n~~~s~line materials (4-6). In the present paper, we will investigate the electrical resistivity of ID-LN theoretically. We first give the model and theoretical analysis (section II), then present the numerical calculation and discussion (section III). A brief summary is given in section IV. II. THE MODEL AND THEORETICAL
INVESTIGATION
Consider a unit of one-dimensional layered structure which consists of N-atoms. There is only one grain with n-atoms in a unit. The structural differences between grain and matrix are due to the arrangement or properties of atoms. The interface between grain and matrix plays an 744
742
RS
QIN
AND
BL
&OU
important part in lD-LN. Assuming every interface consists of m-atoms (nl = 3-5 for metal and 5-10 for ceramic), the number of atoms within matrix in a unit is N-n-2nl. In the coarse grain materials which correspond to nl -CCn c N and n is larger than the mean free path of electrons, the effect of interface can be neglected and the lD-layered structure can be modeled in two phases in a series. The resistivity is (7) R = (l--h&
+ ?R,
PI
where R is the resistivity of one-dimensional layered materials. his the volume fraction of grain R, andRB are the electrical resistivity of matrix and grain, respectively. In our consideration, n is comparable to the mean free path of an electron, and quantum effects must be taken into account. In this case eq. [l] is invalid due to quantum interference. We use the Landauer theory to calculate the electrical resistivity of ID-LN. The interaction between the conducting electron and atoms is approximated by a &function barrier with height Uo. The structures of grain, interface and matrix are represented by different arrangements of barriers. For example, the grain is described by a periodic arrangement of barriers with the period of u (a is lattice constant), and the interface is described by a dilated arrangement of barriers with a dilatant parameter AV (AV is the excess volume of interface). AV is satisfied
KF
(i = 1,2,...nt)
w.-J---=AV xi+l -xi
where Xi (i=1,2,...nl) is the spatial coordinate of i-th barriers in interfaCe,fAV is the distribution function, and the bar denotes the averaging value. The size of each part is corresponding to the number of barriers included in each part. Assuming the considered system is made up of uniform clusters (the cluster may be an atom for pure material or a chemical unit for compound material), each cluster is represented by a b-function potential, and all the &function potential are uniform. According to Landauer’s formulas (8) the electrical resistivity of a one-dimensional system is connected to its transmissivity by R = T’ -1, where R is the electrical resistivity and T is the transmissivity. The Schr&linger equation is: ‘pm+ k2-V$6(X_Xi) [
i=l
1
Cp=O
where k is the wave vector of the electron which is given by k = (2m,E / ti2)1’2, where m, and E are the mass and Fermi energy of the conducting electrons respectively, V = 2mJJ, / h2 is the reduced potential barrier height. Xi is the spatial coordinate of i-th barrier, and x is the spatial coordinate. In the region Ofxi-1 < x < xi (i = 2,3, ...N). an arbitrary solution to the Schrodinger equation for wave vector k can be expressed in the form (9)
cpi= exp(G;)sin[k(x - xi-t) + +i]
[5]
SIZE DEPENDENCE OF ELECTRICAL RESISTIVIT~ IN 1 D-LAYEREDNANOSTRUCNAES
743
Considering the continuum of wave function and discontinuation of its slope at the position of i-th &potential barriers, we obtain the recurrence formula for Gi and $i
Gi-Gi_l
=In
l/2
V2
1+Hsin2[k(x,-xi_l)+k_1]+3Sin2[k(xi-xi_l)+~i-1]
pi = arcsin[exp(Gi_l -Gi)Sin(kri + @ii-t)]
[61
[71
(where i = 2,3, ...N). Let
RJI
Ci
The transmissivity T takes the following form:
r91
It is very important to note that if one neglects multiple reflections of the wave, the transmissivity will take the form: T’=fi i=l
[( l_
2
s
I
I101 )I
It is obvious that T’ + T. It may be due to the resonance or interference, and makes the electric wave more localized or more expended. This result is helpful for making nanometer resonant tunneling devices. However, there is a free parameter $0, which is a periodic function. Performing an integration of this function in one period which correspond to multi-channel averaging, we get the mean transmissivity: 2 nci($O,~l(~O),...,~i(~O))-l
I
NO
nCi(~O,Ol(~O)r.~.~~ii(~O))+l
II111
Equation [ll] cannot be solved analytically because $0, $1, k, $3...& have different definition regions. For example, if expGt=OS and Q. E [-z / 2,7t / 21, than $1 E [-n / 6,a /6], etc.
744
RS
III. NUMERICAL
QIN AND
BL
ZHOU
CALCULATrON
AND DISCUSSION
Consider a case of ID-LN with matrix of disorder arrangement and grain of ordered arrangement. We use the dilated crystal model to describe the structure of the interface. The considered includes three kinds of arrangements corresponding to matrix, grain, and interface, respectively. The spatial ~~gement of barriers is as follows: grain: Xi- Xi-1= ff
]l2]
matrix: xi - xi-1 = a+ a sin(OS)cos(i+OS)
[I31
interface: Xi-Xi-l= (I+AV)a
1141
wherei=1,2,3,. . N,forNatomssystems. AVistbeexcessvolumeofinterface. aislatticeconstant and assuming a = 21dk. JZqn.[131is not a periodic function here because the definition region of i is integer, and the periodicity of triangle function is 2~. Using equation [ 1l] , we calculated Landauer’s resistance of 1D-LN. The unit consists of 120 atoms. Figure 1illustrates the results where V/k = 0.1,0.7, and 1S, respectively. It in~cates that if V/k is not very small the electrical resistance (ER hereafter) would decrease in a region of 0 < d c de; ER takes the minimum value Rmi,,when the grain size is dC;ER is increasing when grain size is larger than dC. Experiments tell us that when grain size d is larger than dCl (4-6), the ER will decrease. We did not find dCl because the considered unit is too small. dCdepends on V/k (see Figure 2). It is shown in Figure 2 that the larger V/K is the larger dCwill be found. For example
1
40.00
AV=9.14yo
0
8
16
24
32
40
48
56
64
72
80
Ordered grain size (number of atoms) Figure 1. Variation of Landauer resistance during grain growth. Horizontal axis is the number of atoms in an ordered grain, where the excess volume of interface is 9. 14%.
SIZE DEPENDENCE OF ELECTRICAL RESISTIVITY IN 1D-LWERED NANOSTRUCTURES
50
745
0
40
G-
0 30 q 20
10
0
0.00
0.40
0.80
1.20
1.60
2.00
Strength of potential (V/K) Figure 2. Variation of critical grain size (where minimum electrical resistivity appeared) vs. strength of barrier.
22
11
0
0
40
80
120
160
Ordered grain size (number of atoms) Figure 3. Landauer resistivity when excess volume of interface is zero.
200
RS QINANDBL ZHOU
746
ordered grain size 0 (atoms)
ordered grain size 40 (atoms)
2
ordered grain size 80 (atoms) 1
0
-1
-2
Position of atoms Figure 4. Variation of phase angle when the initial value is x/2. Grain sizes are 0,40 atoms, and 80 atoms, respectively.
SIZE DEPENDENCE OF ELECTRICAL RESISTIVITV IN 1 D-LAYEREKI NANOSTRUCNRES
747
V/K = 1.2 co~s~n~g to dCof 35 atoms. V/k = 1.5 co~s~n~g to the c&48 atoms. When V/K is very small, which maybe exist in alkali materials, dc is very, very small and cannot be
realized in experiments. The size-dependent electrical resistivity is a well-known attractive phenomenon. Experiments show that the ER of the nanocrystalline semiconductor Si:H is only 1/20of the ERof amorphous Si:H (12), and at grain size of a few nanometers some semiconductors such as Si change into conduc~rs (13), which is evidence of the existence of dc. For metal, empirical data showed that some go& conductors such as Cu and Ag change into insulators when the grain size is very small (13), which is evidence for the existence of dcr (dcl = 14 nm for Ag). We believe that dc also exists for metals, although there is no such experimental report up to now, which is our theoretical prediction. However, excess interface volume plays an important role in superfine systems. Let AV= 0, rn~e~ng the interface with ~~hous-lye or crystal-like phase, ele&rical resistivity of the system satisfies equation [l]. Figure 3 is the numerical calculation of Eqn. [ 111,where V/k= 0.7 and 0.5 respectively, and AV = 0.0%. The relationship between electrical resistivity and excess volume are given elsewhere (14). The phase angle @ is a very important quantity in a supetime system. It makes those electric properties path-dependent. Although the integral of & makes the ER independent of spatial position of the ordered grain, Qliis obviously different in each part of the sample. Figure 4 shows the variation of phase angle with grain size. The existence of Rmin is useful for electric rheological technique, which can be used to determine the nucleation rate in solidification when pulse electric discharges are used. According to our calculation (15), the change of chemical potential A& due to passing electric current depends on the electrical conductivity of formed grain. The relationship is as follows:
WI where kis aconstantjis the density of passing electric current, a2 andol areconductivity of matrix and grain respectively, If 02 > cfl the nucleation rate will increase; otherwise, it will decrease. According to the results of this paper, pulse electric current prefer the formation of grain size dc to dcI . IV. SUMMARY
The size-dependent electrical resistivity of lD-LN cannot be explained by scattering at the interface only, ~thou~ interface plays an irn~~t role. The effect of grain size must be taken into account also. Small grain size would induce quantum resonance, and makes the electronic wave more localized or more expended. The existence of Rmjn is a helpful electric rheological technique, which can be used to determine the nucleation rate in solidification when pulse electric discharges are used. The reported results of superfine metal as an insulator and semiconductor as a conductor can be explained qualitatively in our calculation. REFERENCE
1.
M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff. l? Eitenne, G. Creuzet, A. Friderich, and J. Chazelas, Phys. Rev. L&t. a,2472 (1988).
748
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
RS
QIN
AND
BL
&IOU
R.I. Bickerdike, D. Clark, J.N. Easterbrook, G. Hughes, W.N. Mair, P.G. Pariridge, and H.C. Ranson, Int. J. Rapid. Solidf. 1, 305 (1984-85). G. Pahmbo, S.J. Thorpe, and K.T. Aust, Scripta Metall. a, 1347 (1987). K. Lu, J.T. Wang, and W.D. Wei, J. Phys. D, Appl. Phys. 25,808 (1992). I. Bakonyi, E. Toth-Kadar, T. Tamoczi, L.Varga, A. Cziraki, I. Gerocs, and B. Fogarassy, Nanostruct. Mater. 1, 155 (1993). Y Z. Wang, G.W. Qiao,. X.D. Liu, B.Z. Ding, and Z.Q. Hu, Mater. Lett. n, 152 (1993). D.R. Poirier and G.H. Geiger, Transport Phenomena in Materials Processing, TMS, Warrendale, PA, p. 210 (1994). R. Landauer, Philos. Mag. m, 863 (1970). M.Ya. Azbel, and M. Rubinstein, Phys. Rev. m, 6530 (1983). M. Wagner, Phys. Rev. m, 635 (1992). K.Lu, Phys. Rev. B51,18 (1995). R. Kakkad, G. Liu, and S.J. Fonash, J. Non-Crystal. Solids 66, 115 (1989). L.D. Zhang and J.M. Mao, Nanocrystalline Materials, Liaoning Sci. & Tech. Publishing, p. 19 (1994) (in Chinese). R.S. Qin, S.X. Su, J.D. Guo and B.L. Zhou, to be published elsewhere. R.S. Qin, H.C. Yan, G.H. He, and B.L. Zhou, Chin. J. Mater. Res. 2,219 (1995).