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MAIN ELECTRONIC FEATURES OF GRAIN
BOUNDARIES IN METALLIC AND NON-METALLIC
MATERIALS
H. Mataré
To cite this version:
MAIN ELECTRONIC FEATURES OF GRAIN BOUNDARIES
IN METALLIC AND NON-METALLIC MATERIALS
H. F. MATARE ISSEC
Post Office Box 49177
Los Angeles, California 90049, U.S.A.
Abstract. — After a short description of the grain boundary model in metals, the diffusion property of grain boundaries is explained and the energy difference between edge dislocations and twins considered. The dislocation-row is introduced and the electronic occupation according to the Read model is assessed.
The role of grain boundaries in ceramics is described considering specifically the model of a varistor. Grain boundaries in semiconductors represent a wide field of endeavor and only the outstanding features are described briefly.
1. Introduction. — A considerable wealth of information data and theoretical interpretations are available in the theory of metals concerning the built-up and influence of grain boundaries on the mechanical properties. Not much work has been done and little has been published about the electrical features of grain boundaries in metals.
This is due in part to the fact that interest in work-hardening centered attention on the
mechani-cal properties of grain boundaries. Also, solid state physics was not sufficiently advanced to contribute to the overall state of knowledge of crystal defects in solid state electronics. It is a fact, however, that knowledge concerning the electronic features of grain boundaries can help considerably in the understanding of the mechanical features.
We will look shortly at the models for grain boundaries in metals, the different intergrain struc-tural theories and their bearing on the physical-electrical properties of grain boundaries in metals and will advance a few propositions for measure-ments in a field in which very little has been published.
Grain boundaries in non-metallic material have not been studied as intensively with respect to their mechanical features, however, electronic data have been taken for ceramic intergranular structures.
Finally, we consider shortly the status in the field of grain boundaries in semiconductors. This field developed on the basis of the mechanical models known from earlier work on metals to which
electronic features were added gradually as more was known about the semiconductor itself.
2. Grain boundaries in metals. — The model for these structures has undergone numerous changes and propositions. As McLean [1] summarizes, there are three main models : (a) The Void Model: (Voids between two crystals) ; (b) The Distortion Model: Both lattices share the same atoms in the grain boundary zone ; (c) Amorphicity: The grain boundary zone is amorphous. There is another model described by N. F. Mott [2] as the island model.
According to Smoluchowski's model, the grain boundary forms increasing islands with increasing tilt angle. Figure 1 shows these different approa-ches and their importance for the electronic fea-tures.
The void model (a) has been abandoned as also the amorphous model (c). However, model (b) as well as model (d) have realistic features. It is correct that some lattice distortion takes place around dangling bonds. The distortion is smaller than usually assumed. The idea of Smoluchowski that increasing misfit causes increasing overlap is correct. But what overlaps here are not amorphous areas, but dangling bonds and stress. Electrically, case (a) would require a boundary conductivity much lower than the bulk conductivity, contrary to the experience. The transverse resistance would be higher than the bulk resistance. In case (b), overlap JOURNAL DE PHYSIQUE ColloqueCA, supplément au n° 10, Tome36, Octobre 1975, pageC4-447
Résumé. — On décrit brièvement le modèle des joints de grains dans les métaux, les
propriétés dans la diffusion des impuretés, et la différence entre l'état énergétique des dislocations de carne et des géniaux. La suite de dislocations est introduite et l'occupation électronique est évaluée d'après le modèle de Read.
On décrit alors le rôle des joints de grains dans les céramiques sous considération du modèle d'un varistor. Les joints de grains dans les semiconducteurs sont un domaine très vaste d'activité et seules les propriétés les plus remarquables sont revues brièvement.
of dangling bonds would cause a high parallel conductivity. Without further detail, the transverse ,resistance should be smaller than the bulk value.
a) Void C,,<Cb
R I > R I D,*u( o,>za o,<m D.<<?R
e---
C ) Amorphous d l Island
FIG. I . - Grain Boundary Models. a) Void model ; b) Distor- tion model ; c) Amorphous layer ; d) Island growth with tilt
angle 6.
In case (c), one expects a boundary conductivity smaller than u b u l k , the transversal resistance R,
should be larger than Rb,lk. We know that only model (b) matches the facts. The transverse resis- tance in metals may be smaller than the bulk value if no scattering occurs. There are complex ques- tions connected with this last electronic feature. In metals, the local strain can manifest itself in a scattering of the Bloch waves and reduce the mobility of carriers while in semiconductors the polar character of the free bonds is more pro- nounced and the space charge begins to play the deciiive role.
Smoluchowski's model (c) was introduced to resolve the status of grain boundaries in diffusion experiments. As the pipes in the lattice around dislocations get closer to each other with increasing misfit-angle, grain boundary diffusion increases. ~ a t e r the model was corrected as it neglects the fact that separated pipes have a wider stress field and enhance diffusion (Turnbull & Hoffman [3]). But at a closer look, it turned out that the different findings can be combined. Pipe diffusion D, decrea-
ses a s the angle 0 increases until finally D, 2 Dbulk
for complete bond overlap. In volume diffusion experiments, grain boundary diffusion is a compo- site of pipe and bulk diffusion with a maximum at
40" to 50" for 0, decreasing again for higher angles of misfit. Therefore, our model explains the low grain boundary diffusion constants at 0
<
lo0 and0
>
80" by the small density of pipes in the first case and the bond overlap (no pipes) in the second case. It was also measured that penetration per-pendicular to the grain boundary is always small compared to the case of diffusion parallel to the grain boundary plane. One has to mention that all diffusion experiments on bicrystals suffer from the fact that a crystal subject to surface diffusion enhances the penetration by micro-cracking or micro-cleavage.
It turned out later that the analysis of the measured diffusion constant in a bicrystal is high at the surface and reaches normal bulk values at greater depths (microcracks) confirming Smolu- chowski's results [4].
In experiments concerning dopant distribution and segregation, it is generally assumed that the grain boundary will accumulate the dopant due to the extra space available. We see that this is only so in well defined cases. For small tilt angle, pipe diffusion predominates. However, penetration is small as the pipes d o not overlap. With increasing tilt angle, penetration increases and reaches a maximum a t 0 = 45". From here on, D decreases as
rapidly as it increased from 0 = 10" to 0 = 45" and is equal to the bulk value at 0 = 70 t o 80" again [5, 61.
Conclusions with respect t o grain boundary properties on account of probable impurity accumu- lations, therefore, have to consider the misfit dependent diffusion mechanism.
In bicrystal growth experiments, it is often argued that the grain boundary plane as the zone of higher energy freezes last and thus receives impuri- ties by segregation. But it has to be considered that segregation works both ways at the liquidlsolid interface and that it depends on the type of impurity if depletion or accumulation takes place. Thus, copper cannot be responsible for grain boundary conduction, e . g. in Germanium when
0 30°.
The grain boundary energy is a well known mechanical quantity. It has a relatively minor effect on electronic features. The Shockley-Read form is :
Eo = constant, basically product of stress and Burgers vector ; 0 = angle of misfit ; A = parame-
ter of integration generally
<
1 .MAIN ELECTRONIC FEATURES OF GRAIN BOUNDARIES C4-449 Metal CU-G.B. Cu-Twin AI-G.B. Al-Twin Pb-G.B. Sn-G.B. Ag-G.B. E (ergs/cm2)
-
490-860 mean : 646 440 600 120 200 (315 "C) 100 (220 "C) 400 (900 "C) The grain boundary free energy is about five times higher than the twin free energy :Tensile stress of polycrystalline material is about this much higher than the value of single crystal material. We can see here some correlation to the electrical behavior. As disrupted bonds cause charge redistribution and increased local stress, a
random distribution of these vains of higher tensile stress causes hardening of the material.
. Some authors have considered the problem of electron scattering at lattice imperfections on a more mechanical basis. The localized defect is regarded as a spherical density variation and the scattering cross section for a Bloch wave with wave vector k is then proportional to
ao
D (Density variation)
and
(Rigidity variation)
G
A more realistic approach is the perturbation potential treatment of the free electron population around a dislocation. The dislocation is considered neutral, but causing a perturbation potential expres- sed by :
V(r) = Vo(r)
+
Vdr)+
x
Un(r)n
Vo(r) = unperturbed periodic lattice potential ;
VF(r) = free electron contribution ; U,(r) = per- turbing potential.
This approach yields the electric field gradient around the dislocation which has oscillatory character 181. Such perturbation methods and their use in Bloch-wave scattering yield very localized charge accumulations only a few Angstroms from the dislocation.
In reality, the charge redistribution is more extended even in metals like copper. Especially in the case of grain boundary planes separating two entire crystal facets, one has a much wider range of influence on the charge carrier distribution. Measu- rements on metals could be done at low tempera-
ture to enhance the effects. T o our knowledge, no work has ever been published about charge distri- bution, resistivity-anisotropy, mobility, etc. in grain boundaries in metals at low temperature.
FIG. 2. - a) Dangling bond, core region as scattering sphere at
low and high temperature. 6 ) Model of tilt boundary with Burgers vectors and grain boundary plane. Electronic features.
FIG. 3.
-
Band structure and energy levels at dislocation site.In a row of edge dislocations, the bonds are filled according to a minimum energy situation. If the dislocation energy level falls below the Fermi level
EF, electrons drop into the empty sites until so many levels are filled that no energy can be gained by a transition of more electrons into free levels. The close spacing of these levels and electrostatic repulsion will limit this process at a point where the electrostatic energy per added electron
~ , ( f l
a s a function of the fraction f of sites occupied satisfies the conditionW. T. Read has calculated E, for different statistics
and found that the Fermi statistics does not govern the occupation of closely spaced lines of acceptors on dislocations except if f, the fraction occupied, is very small (lod to In his analysis of the mobility, he concludes that the resistivity is much less affected for current flow parallel to the dislocations than orthagonal t o the dislocations.
In a simple scheme of the unit cell blocks, the unsaturated bonds U (Fig. 2b) ending at the grain boundary will represent scattering centers for current flow perpendicular to the g.b.-plane. Within -the plane, however, bond overlap will create a zone
of higher conductivity.
In metals, this situation could be tested at low temperature. It is known that the resistivity of most metals (Zn, Cu, Ag, Al, etc.) does not follow the Bloch-Griineisen formula at low temperatures. While a practially linear decrease over six decades in resistivity is predicted for a decrease of tempera- ture from 100 K to I K , a plateau i n resistivity is reached at around 20 K where resistivities remain near the 1 ohm cm range. This is undoubtedly due to dislocations. It will be a worthwhile task to carry out the measurements which have been perforved on dislocations in semiconductors also on metals, but at low temperature where these effects should be well in the range of measurable effects.
3 . Grain boundaries in ceramics. - Materials like
A1203, MgO, NaCl, B e 0 have been studied with respect to grain boundary growth and diffusion. A marked difference between metal grains and cera- mic grains is their thickness. In high purity metals, one measures boundary widths of the order of a
few atomic diameters, whereas in ionic materials, such boundaries have a width, orders-of-magnitude wider :
A1203 : A x 100
A
B e 0 : A % 200A
NaCl : A w lo4
A
M ~ O : A w 2.5 x 104
A
[lo1 The reason for this difference is the fact that in metals a dangling or broken bond does not cause a major disruption or change in the electron distribu- tion. The extension of the electronic disturbance depends on the availability or absence of a wave vector adjustment for the Laue-Bragg-condition. The Bloch lattice potential expressed by a Fourier series is :Us
=Uhk,
e2 ~ ~ ( ~ h k 1 . r ) hkl(1)
Uhkl = Fourier coefficients ; ahkl = vector in reci- procal lattice ; r = lattice vector.
Expressing the energy by a volume integral :
.
( C = normalization vector ; I,V = conjugate com-
plex of $) according to (1) leads t o the energy integral :
which is different from zero only if :
Thus, the wave energy is invariable for simulta- neous addition of a vector of the reciprocal lattice
ahkl t o k and k'.
As this vector is easily available in a metal, dangling bonds, states pulled out of the conduction band or metal-metal interfaces (heteroepitaxy) pro- duce little or no electronic disturbance. This explains the small width of such grain boundaries in metals as compared to ceramics (and semi- conductors) where strong changes in wave energy and wide extension of the interface are indicative for charge rearrangements and extended space charge domains (see semiconductors below). In MgO and NaCl, e. g., such grain boundary zones are a few microns thick which corresponds to the extension of the space charge in Germanium and Silicon grain boundaries (see [ 5 ] ) .
MAIN ELECTRONIC FEATURES O F GRAIN BOUNDARIES
FIG. 4. - Model of varistor microstructure. ZnO-grains of mean thickness d are surrounded by grains of thickness t.
lar material [1 I]. As mixtures of ZnO
+
Bi203, ZnO+
A1203, ZnO+
TiOz, and ZnO+
Sb203+
Biz03+
COO+
MnO+
Cr203 are prepared and sintered, it is altogether not clear what the exact composition of this intergranular material is. In general, one assumes that large ZnO-grains are the conducting portion (see Fig. 4) and that the grains of thickness t a r e responsible for the nonlinear characteristics of the type shown in figure 5 .VARISTOR CHARACTERISTIC
300 K
FIG. 5. - Varistor characteristic : Field E versus current density j.
From a field strength on of somewhat over
1 0 0 0 ' ~ / c m , a higher current is drawn. Here a breakdown sets in. Fowler-Nordheim tunneling, Schottky-field emission and Poole-Frenkel emission have been invoked for the effect across grain boundaries. As t (grain thickness) is assumed to lie in the 200 range, a field of F = lo3 V/cm results in a local grain boundary field of
fB = FB d/t (Fig. 4)
the usual breakdown field.
When we consider a symmetric tilt boundary for
an ionic crystal, the boundary may have strong electrostatic charge accumulation and a s such produces a local dielectric layer independent of additional segregation. The extraspace can be filled
FIG. 6. - Bicrystal growth and orientation.
FIG. 7.
-
Band structure model of grain boundary under bias condition.with other parts of the ceramic mixture, however, which would be acting as a dielectric interface.
Pictures of the strongly granular structures (see [ I l l ) suggest that the large platelets act as conducting paths while the grains act as dielectric layers.
FIG. 8. - Radius of space charge pipe at grain boundary as f ( O ) ,
f(AN), and f(Ve). D = dislocation spacing.
FIG. 9. - Grain boundary in Ge-bicrystal. 50x.
of dislocation orientation with respect to carrier flow have been considered :
Z
llD H I 1{ Z I D
In the last case, both current directions I are rectangular to the magnetic field H as the disloca- tions D define a plane (Fig. 6).
In artificial (defined) growth of grain boundaries,
-
CRT 20KV 0 . 0 2 p A HORIZ .5 ms/cm VERT 1 0 mV/cm A. C. COUPLINGFIG. 10. - X-scanning of Ge-bicrystal in scanning electron microscope in surface-current mode (bias applied, showing
barrier layer).
FIG. I I . - Back-to-back current-voltage characteristic of N-P-N bicrystal.
we have a precise alignment of all pipes in a plane (Fig. 6). Two basic current modes are possible :
111 D, I D. For tilt angles 8
>
5", no basic difference in electronic behavior can be found [ 5 ] .MAIN ELECTRONIC FEATURES O F GRAIN BOUNDARIES C4-453
FIG. 12.
-
Degenerate conductance (resistivity) of bicrystal compared to normal monocrystal.FIG. 13. - Photoelectric response of bicrystal.
boundary (1
<
0<
25O) is represented by a . cur- ving-up of valence and conduction band at the spot of the extra-levels due to the unfilled states. There100.0 I- Z P 10.0 a a 0 0 I- 0 x P W 1.0
2
0: .WITHOUT BIAS 0.1 A .5 .6 .7 .8 .9 1.0 1.1 PHOTON ENERGY-
eVFIG. 14. - Optical frequency response of bicrystal without and with bias applied.
is now a direct voltage dependence of the number of states on the bias Ve :
when cp = 7rq2/2 k N is the barrier height at equili-
brium. k = dielectric constant ; N = charge density (Fig. 7). The resultant space charge around the overlapping pipes is also a function of the transver- sal bias V, :
2
sin
812 1+
1+
e ~ q ) ~=
{
nb AN.
( 1+
exp[(ED - E F ) / k T ] )With the Fermi-function f = 0 . 1 for room tempera- ture, one has
0.1 2 sin 812 A N
I
~2 =
-
n b 1
+
( 1+
eve/q)lI2]sin 812
R = const
-
AN [ l+
( 1+
eVelq)li2]. In figure 8, this function is plotted. The space charge pipe radius R increases with 0 and decreas- ing doping. For A N = 1016 ~ m - ~ , we have also plotted the increase in R with the external voltageVe applied for increasing ratios of eV,lcp (where cp is assumed to be 1 eV). The dislocation spacing D is also plotted for comparison.
We see that the bias across the grain boundary widens the space charge considerably (consider that
scanner (Fig. 10). The basic characteristics of a quency response due to the particular band scheme bicrystal are [5] : (1) Rectification characteristic of (Fig. 14).
an N-P-N layer (Fig. 11) ; (2) High grain boun- These properties make grain boundaries in semi- dary conductance and degeneracy (Fig. 12) ; conductors one of the most interesting structures
(3) Photoelectric response with sign reversal a t the with extremely varied influence on carrier flow and grain boundary (Fig 13) ; (4) Extended optical fre- electronic properties.
References
[I] MCLEAN, Donald : Grain Boundaries in Metals, Oxford
(Clarendon Press), 1957.
[21 MOTT, N. F. : Proc. Phys. Soc. 60 (1948) 391. [3] TURNBULL, D. and HOFFMAN, R. E. : a T h e Effect of
Relative Crystal and Boundary Orientations on Grain Boundary Diffusion Rates >>, Acta Metall. 2 (1954) 4 19-426.
[4] QUEISSER, H. J . , HUBNER, K. and SHOCKLEY, W. :
.
Diffu- sion Along Small Angle Grain Boundaries in Silicon s, Phys. Rev. 123 (1961) 1245-1254.[5] MATARE, H. F. : Defect Electronics in Semiconductors
(Wiley Interscience), 1971, 395-404.
[6] COULING, S. R . L. and S M ~ L ~ C H ~ W ~ K I , R . : a Anisotropy of Diffusion in Grain Boundaries D, 3. Appl. Phys. 25 ( 1954) 1538- 1542.
171 Z I M A N , J. M. : Electrons and Phonons (Clarendon Press,
Oxford), 1960.
[8] OCURTANI, T. 0 . and HUCCINS, R. A. : a Theory of Electric
Field Gradient due t o Conduction Electron Charge Density Redistribution around Screw Dislocations in Metals >>, Phys. Stat. Sol. 24 (1967) 301.
[9] READ, W. T . : u Theory of Dislocations in Semiconduc- tors w , Phil. Mug. 45 (1954) 775-796 ; 45 (1954) 1 1 19- 1128 ; 46 (1955) 111-131.
[lo] MISTLER, R. E. and COBLE, R. L. : <<Grain Boundary Diffusion and Boundary Width in Metals and Cera- mics ., J. Appl. Phys. 45 (1974) 1507-1509 (1974). [I I] LIONEL M., LEVINSON and PHILIPP, H. R. : a The Physics of
Metal Oxide Varistors x, J. Appl. Phys. 46 (1975) 1332-1341.
DISCUSSION
D. WARRINGTON : TO a metallurgist interested boundary is really the beam probe diameter plus the in structural rather than electronic effects, the electron diffusion distance in the material ?
<< width >> of grain boundaries in non metallic or H . F. MATARE : Yes, I should have stressed the
