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Submitted on 1 Jan 1978
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DIFFUSION SIZE EFFECTS IN BISMUTH
J. Aubrey, K. Anagnostopoulos, Z. Hatzopoulos
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplkment au
no
8 , Tome 39, aoat 1978, page C6-1121DIFFUSION SIZE EFFECTS IN BISMUTH
J.E. Aubrey, K. Anagnostopoulos and 2. Hatzopoulos
Department'of Physics, EZectronics and EZectricaZ Engineering UWIST, Cathays Park, Cardiff CFI 3NU, U.K. ,
Rgsurn5.- Nous avons calcule des expressions pour la conductivite effective et le champ glectrique transversal moyen pour des lames minces de bismuth en presence d'une diffusion des porteurs par les faces de la lame. Une comparaison avecl'exp6rience donne des valeurs numeriques pour les temps de re- laxation entre les ellipso'ides 1 la ternpgrature d'helium liquide.
Abstract.- A theory of diffusion size effects has been worked out for bismuth, and expressions found for effective plate conductivity and mean transverse electric field. Comparison with experiment gi- ves numerical values for the inter-ellipsoid scattering times at liquid helium temperatures
2 H
Rashba/l/ has discussed conduction in thin DH
dn
-
3n yH2
dE-
(l nI II+nII1 -nH) +n = 0 (4)22
dz2 0 ZZ
plates of n-Ge using the following pair of equations dz Teh
Here, T and r are the characteristic times for for the electrons in each valley. ee eh
- -
I div ja + "Rfia-
Rag)at e -
B
an electron to scatter to another electron ellipsoid (l) and to the bole spheroid respectively.
a a
f
= en g*E
+ ega*vna (2) The four coupled equations for the excess car- Redistribution of electrons amongst the valleys oc- rier concentrations na can, as in Rashba/l/, be re- curs owing to the intervalley transition tenns R.. duced to three by applying the 'quasineutrality in equation (1) resulting from the non-vanishing " condition',a
of divf near the sample surfaces. The net result = O a = Is H (5)
a is that the effective plate conductivity is reduced from its bulk value. Also implicit in Rashba's cal- culation is the appearance of an enhanced transverse electric field across the thin plate dimension.
In an effort to understand the results of d.c. size effect experiments on bismuth (Aubrey and Creasey/2/ ; Aubrey and Barrell/3/), this approach has been applied to a model of the bismuth Fermi sur- face (see e.g. Aubrey and Chambers/4/) consisting of three electron ellipsoids (denoted by I, I1 and
111) and a hole spheroid (denoted by H).
Consider a thin crystal plate of bismuth and take the z-axis of a Cartesian coordinate system parallel to the thin dimension. Choose the X- and y-axes to coincide with the principal directions of
which is valid at all points other than within a Debye screening length of the sample surfaces, which we ignore. A further simplification results by igno- ring the small (approximately 6'141) tilt of the electron ellipsoids about the binary axes in L-spa- ce, and choosing plates whose normals lie in either the binary-trigonal or bisectrix-trigonal planes. In this case, two of the electron ellipsoids are equivalent to each other in all respects (ellipsoids I1 and 111, say), so we begin with two equations of the form (3), and equation (4). Use of (S), and eli-
dEz
mination of the term- then leads to a pair of dz
coupled equations,
the plate conductivity tensor. Under steady condi-
d2n1 d.n1'
+
g2n1+
y2n1'= 0
tions, equations (1) and (2) for each group of car-
dz7
+ a2 (7)riers lead to the following equations The coefficients here are functions of the mobility 2 1 I
e + n P 1
5 -
(-+-ln
and diffusion tensor elements of the various ellip-zr dz2 o zz dz H Tee Teh soids and of the characteristic times T and
T
ee eh'
+
L
(nII+nIII)-
L
L,v c ? I O (3) For a plate of thickness 2b with identical con- Lee 'eh ditions prevailing at its surfaces z = f b , equations for the electrons of ellipsoid I, similar equations
(6) and (7) have the solutions . . . .
for the electrons of ellipsoids I1 and 111, and the
nf = alsinhXlz+ a2sinhX2z following equation for the holes,
nl' = ualsinhXlr + va2sinhX2z (9)
where 'X1, +h2 are the roots of the fourth order equation obtained by combining equations (6) and (71, and U = _?+B1
-
_:+B2 9 CLIA:+YI "2~:+~2 (10) _ h 2 V =~ $ + B I
- alh?+y2 a2X$+ylConsider now the transverse ellipsoid currents
j
: (equation 2) which arise when an electric field
(Ex, 0, 0 ) is applied to the plate. Firstly, the constants a1 and a2 in equations (8) and (9) can be determined by imposing suitable boundary conditions on the transverse currents at the sample surfaces. Physically, the most plausible assumption appears to be
'" +b) = 0
JJ-
(11)corresponding to negligible transfer of carriers bet- ween ellipsoids by surface scattering. The drift and diffusion terms in the transverse ellipsoid cur- rents simply cancel each other out at the surfaces. Secondly, the condition (11) taken together with the continuity equation for the total charge and current densities leads to the result
fromwhich the transverse electric field E (2) can
be found using equation (2) for the individual elli- psoid currents. The result is
Kn0wir.g EZ and applying the condition (11) we find XI Ex X2Ex
a1 = ; a2 = ( 1 4 )
hlcoshhlb X2coshX2b
where XI and X2 are functions .of the mobility and diffusion tensor elements of the various groups of carriers (~nagnostopoulos/5/)
The measureable properties of the plate, ta- ken to be of unit width, can now be readily deter- mined. We find for the effective conductivity
-
r~~ziz)dz 'zx tanhXlb tanhh~b
-
= -m--Ex bEx uzz
B1r
- B 2 7 (16) The coefficients A1,2 and B1,2 here are again func-tions of the transport tensor elements and of the inter-ellipsoid scattering times.
Preliminary experimental work (Anagnostopou- los/5/), on sets of samples of several crystallo- graphic orientations have been analysed using the expressions (15) and (16). The transverse field re- sults are somewhat more convenient to deal, with than the effective conductivity results since in this case the bulk term is almost independent of temperature in the liquid helium range.
The chief quantities of interest are the cha- racteristic times T and since the transport
ee
tensor elements are quite well known from galvano- magnetic measurements (e.g. see Hartman161). Values of %IO-' and % IO-~S., respectively, are found for these at temperatures in the liquid helium range. The value for T~~ is in excellent agreement with the values found by Zitter/7/, Hattorif81 and Lopez/Y/, while no value for T appears to be available for
ee comparison in the literature.
It seems clear that the effects discussed here should occur not only is anisotropic'conductors other than bismuth and the related semimetals, but also in any metal which has a Fermi surface consis- ting of several distinct sheets so that redistribu- tion of carriers can occur between the sheets. This means that care should be taken in analysing the results of resistivity and resistivity anisotropy measurements carried out under conditions where the characteristic lengths corresponding to Xit2 are comparable with the sample dimensions, a condition which is likely to be satisfied relatively easily for pure samples at low temperatures. Finally, ex- periments of this kind, notably the measurement of
-
E (see Aubrey and Anagnostopoulos
1101)
should yield important information about umklapp scatte- ring times in metals.References
/I/ Rashba,E.I., Sov. Phys. JETP
2
(1965) 954 /2/ Aubrey,J .E. and Creasey ,C .J., S. Phys. C :Solid St. Phys.
2
(1969) 824/ 3 / Aubrey,J.E. and Barrell,A.J., J. Phys. F :
Metal Phys.
1
( 1 971) L36/4/ Aubrey,J.E. and Chambers,R.G., S. Phys. Chem. Solids
2
(1957) 128151 Anagnostopoulos,K., Ph. D. Thesis, U. of Wales (1976) unpublished
/6/ Hartman,R., Phys. Rev. (1969) 1070 /7/ Zitter,R.N., Phys. Rev.
139
(1965) 2021/8/ Hattori,T., J. Phys. Soc. Japan
3
(1967) 19/g/ Lopez,A.A., Phys. Rev. (1968) 823 /10/ Aubrey,J.E. and Anagnostopoulos,K., J.
