A. A. REREZIN et al. : Conductivity Mechanism of @-Rhombohedra1 Boron 447
phys. stat. sol. (a) 20, 447 (1973)
Subject classification: 14.3 and 14.3.4; 22.1
A.F. Ioffe Phyeico-Technical Institute, Academy of Sciences of the USSR, Leningrad
Studies of a Conductivity Mechanism of B-Rhombohedra1 Boron in a Strong Electric Field
BY A. A. BEREZIN, 0. A. GOLIKOVA, M. M. KAZANIN, E. N. TKALENKO, and
V. K. ZAITSEV
The dependence of electroconductivity of pure @-rhombohedra1 boron on the temperature and the intensity of a n electric field is presented. The results are discussed using three models: I) the Poole-Frenkel model for an isolated trapping centre and for a screened Coulomb centre, 11) the hopping models of Mott and Shklovskii, and 111) the small polaron model. The results of the electric field measurements may be satisfactorily interpreted in both the screened Poole-Frenkel and the hopping model giving for the dependence of the conductivity u on the electric field intensity E the relations (u/uo)1/2 In (o/uo) - E and In (u/uo) - E--114, respectively (uo is the conductivity in zero field). The model I11 alone gives an unreasonable result for the hopping length but nevertheless the essentiality of polaronic effect for @-boron seems sufficient from both the experimental and the theoretical point of view. It is supposed that the very complex crystal lattice of @-rhombohedra1 boron has a mixed conductivity of a polaron-Mott type. The magnetoconductivity of boron in the hopping region is also discussed.
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The nature of the electric conductivity mechanism of p-rhombohedra1 boron is not known with definite clearness at present. There is strong evidence that the electroconductivity of P-boron is mainly determined by hopping processes [l] to . The high density of trapping levels 14, 51 and low values of mobilities make
448 ,4. A. BEREZIN et al.
the usual theory of band-type conduction unapplicable to boron [7, 81 and lead to the supposition about hopping conduction . Two principal types of hop- ping conduction, polaronic and impurity-like, seem to have some connections with boron [9, lOJ but up to now there is no unambiguous picture of hopping of carriers in the crystal lattice of P-boron.
In the present paper we represent the results of conductivity measurements of pure @-boron in strong electric fields. The measurement of current-voltage ( I - U ) characteristics in strong electric fields is one of the most effective toolr for the investigation of materials with hopping conduction [11 to 131. Here we study the I-U characteristics of boron at 77, 113, 205, and 293 OK in fields up to 60 kV/cm. As we know I-U characteristics of boron were studied by Pruden- ziati et al.  (P-rhombohedra1 boron) and Moorjani and Feldman  (amor- phous boron). But in  and  the measurements of I-U characteristics were carried out for lorn fields only, where according to [a] and , the formation of I-U characteristics is determined primarily by space-charge-limited currents.
We shall try to demonstrate the possibility that the hopping nature of electric conductivity of P-boron is connected with the peculiarities of the structurc of its complex crystal lattice.
2. Experiment For thc measurements we used single crystals of pure P-rhombohedra1 boron
The I-U characteristics were measured in direct current, becausc for 7' which were prepared by zone melting.
300 OK thc resistivity of pure boron is large enough to allow sample heating to be neglected. The samples for the measurements were prepared in the form of plates with dimensions 0.1 x 4 x 4 mm3. At T = 300 OK their resistivity was about 4 x lo6 Qcm. The area of the silver contacts was 1 x 1 mm2.
To reduce the surface current1) the surface of the sample was etched in concen- trated nitric acid. After etching the surface current was reduced to 1 to 2% of the full current.
The I-U characteristics of pure zone-molten boron for 77, 113, 205, and 293 OK are shown on Fig. 1. As one can see from Fig. 1, the I-U characteristics of p-rhombohedra1 boron are essentially non-ohmic. In Section 3 we shall dis- cuss the possible ways to iiitmpret these results.
Fig. 1. Current-voltage characteristics of pure zone- molten boron for 77, 113, 205, and 293 OK. For the
sample geometry see the text
') The surface current depends on the pre-history of the sample; for our case its value was up to 40% of the full current through the sample.
Studies of a Conductivity Mechanism of p-Rhombohedra1 Boron
3. Theoretical Models
Several possible physical mechanisms of the non-ohmic conductivity of semi-
The most familiar of these mechanisms are : 1. the Poole-Frenkel effect [15 to 171, 2. the specific non-activated hopping conduction in sufficiently strong elec-
3. the electric field dependence of the mobility of the small polaron considered
4. the space-charge-limited currents [5, 141. According to  and [la] the space-charge-limited currents play an essential
role for the formation of I-U characteristics of @- and amorphous boron in the region of comparatively small electric fields only. In the present paper we shall consider the applicability of the first three above-mentioned models to the case of @-boron only.
conductors in strong electric field are known a t present.
tric fields considered by Mott [l l , 121 and Shklovskii ,
by Efros ,
4. The Pooh-Frcnkel Model 4.1 The Poole-Frenkel effect for a single trapping centre
When the conductivity of a system is determined by thermal ionization of Coulomb-like trapping centres the applied electric field leads to the reduction of a potential barrier and consequently to the increase of carrier concentration in the conduction band. Simple statistical arguments give the following depend- ence of the conductivity a = a(E) versus the electric field E  :
u = a, exp (T) . 7 JIE
I n (1) a, = o(0) is the static conductivity of a semiconductor with trapping centres in zero electric field. The parameter q is determined by the effective charge of a Coulomb centre Z and the static dielectric constant x
If one takes into account the induced space anisotropy of the thermal ioni- zation probability of a trapping centre then the dependence (1) becomes more complex r201:
When the potential of a trap has the form of a spherical rectangular poten- tial well of radius b one can get the relation
instead of (1) and correspondingly the relation
- -_- a '' [exp(e$)-~]+; a, 2 e E b
instead of (3).
450 A. A. BEREZIN et al.
t 3.0 - 25 Q
I 50 J 60
E ihllcm-'J - Fig. 2 Fig. 3
Fig. 2. Dependence of Ig [u(E)/a(O)] on the first power of field intensity (Poole law) for P-rhombohedra1 boron a t 77 and 113 OK, For both temperatures three calculated curves corresponding to b = 200, 250, and 300 A are given. The left triplet of calculated curves corresponds to T = 77 OK, the right one to T = 113 OK. The curves are calculated by (5 ) , i.e. by taking into account the field-induced anisotropy of the probability of thermal
Fig. 3. Left side: I-U Characteristics of p-boron in Poole coordinates for 77, 113, 205, and 293 OK (see subscript to Fig. 2). Right side: dependence of Ig (um/uo - 1) on the inverse temperature. The experimental points correspond to 77, 113, 205, 220, 248, and 293 O K
The relation (1) with (Fin the exponent is known as Frenkel law  and (4) with E instead of d z i s known as Poole's law .
Dependences corresponding to the Poole law in P-boron are shown in Fig. 2 and 3 (left side). Comparing the theoretical curves cr = o ( E , b ) with experimen- tal ones for T = 77 and 113 O K (Fig. 2) we found that b = 250 to 270 n, which seems to be too large because the linear dimensions of the unit cell of P-rhombo- hedral boron are 25 x 10 x 10 A3.
For the ease of the Frenkel law (1) similar considerations give that for T = 77 and 113 OK the value of 7 is near to 0.75 at. units. Then (2) gives for the effec- tive charge 2 (for P-boron ~t x 10) the estimate 2 = q2 x/e3 = (5 to 6) which also seems to be unrealistically high in order to be the charge of a single Coulomb trapping centre.
Thus the Frenkel-Poole model for both types of isolated trapping centres (i.e. for the spherical rectangular well and the single Coulomb centre) seems to be insufficient for the interpretation of I-U characteristics of P-boron.
4.2 The Poole-Frenkel effect for the Coulomb centre screened b~ free carriers
If one takes into account the shielding of the Coulomb-like potential of a happing centre by free carriers the dependence cr = o ( E ) is described approxi- mately by a relation [15, 161 formally similar to the Poole law (4) :
Studies of a Conductivity Wxhanism of p-Rhombohedra1 Boron 451
77 1.82 x 1014 3.06 x 10-9 113 1-70 x 1015 3.06 X lo-" 205 1.49 x 10l6 3.77 x 10-5 293 5.96 x 10le 3.50 x 10-5
428 16.7 4.55 2.86
E IhV cm4--
Fig. 4. Dependence of (cr/cro)1/2 lg o/n0 on E (screened Poole-Frenkel law) for P-rhombohedra1 boron for a) T = 77 OK and b) T = 113 OK
I n (6) E is a value of the order of unity which weakly (logarithmically) de- pends on the parameters of the problem and As is the inverse screening length. For 1, we take the Debye-Huckel approximation:
(n is the carrier concentration) and neglect the influence of the field on the mo- bility of the carriers. Then we find the following form of a(E)  :
where the expression under the square root does not depend on E. In (8) n,,(T) is the concentration of the charge carriers on the current level, i.e. n,(T) = = a,(T)/e p ( T ) (p is the drift mobility).
The dependence corresponding to screened Poole-Frenkel law (8) is shown in Fig. 4a and b. These dependences are nearly linear and hence we can estimate n,,(T) and the corresponding values of the mobilities ,u( T ) = a,(T)/e no(?'') (see Table 1).
The increase of p( T) with temperature could be described by the activation law
,u( T ) N exp (- &) (9)
452 A. 8. BEREZIN et al.
At sufficiently strong fields before breakdown the I-U characteristics of p-boron show a tendency to saturation, i.e.
( T ( E ) I E + ~ + coo x const,
where the limited value om is only a function of temperature. The values of urn( T)/oo( T ) are given in the right column of Table 1 . The value of oaS/a0 is rapidly reduced when the temperature increases.
One may consider this saturation as a consequence of the depletion of the trapping levels by the electric field. Then we can estimate the value of the ac- tivation energy of the current level, i.e. the binding energy of the carrier on the trapping centre ( A ) , the concentration of traps ( N t r ) , and the concentration of current states ( N v ) .
Let nt,( T ) be the number of localized carriers, i.e. the carriers bound on trap- ping cent,res, and A the energy separation between the bound (currentless) and current level. In case of Boltzmann statistics we have
Writing (11) we imply that the growth of (T in the electric field is caused by the
Plotting lg (am/ao - 1) as a function of l jkT (Fig. 3, right side) we find the growth of carrier concentration only and not by the growth of mobilities.
following estimates for A and N,/Nt , in @-boron:
- (2 to 8) . A x 0 . 0 5 e V , - - ( 1 3 ) NC Ntr The second value of (13) and N t , = no am/oo give the following estimate for
the concentration of trapping levels :
Nt, = (10'' to cm-3. (14) Kote that this activation energy A does not coincide with the activation ener-
gy of drift mobility A, given above.
5. The Hopping Model
In this section we shall consider the zero-field and high-field conductivity of P-rhombohedra1 boron from the point of view of the theory of hopping conduc- tion. This interpretation is alternative to the Poole-Frenkel interpretation de- veloped above. Some hints on the possible integration of these two approaches will be given in thc discussion (Section 8).
5.1 The tempevntzcre dependent Mott law
The large unit cell of P-rhombohedra1 boron and the non-equivalence of its lattice atoms lead to some analogy of boron with amorphous substances .
Studies of a Conductivity Mechanism of P-Rhombohedra1 Boron 4 3
One may suppose that in boron as well as in amorphous or doped semiconductors the local (or quasilocal) trapping states have energy levels which are not degenerate but have different positions within a finite energy interval of width W . Thus it is natural to expect that the conduction of boron at comparatively low temperatures should have some similarities with the hopping conduc- .
-' ?7 -I3
u N exp [ - (37, which is known as Mott T-ll4 law [23 to 251. Ig o versus T-lI4 for $-boron is shown in Fig. 5. The slope of the curve corre- sponds to the value To = 109 OK. The corresponding value of kT ( ~ 0 . 0 4 eV) is probably near to the width of energy interval of local states W (see  where at least two trapping levels with energy separation 0.04 eV were reported for p-boron from thermally stimulated current measurements).
The characteristic temperature To in the Mott law (15) is connected with go, the density of localized states on the Fermi level. In the case of properly dis- ordered systems (e.g. doped or amorphous semiconductors) theory gives 
v @.3 90
I n (16) a is the decay constant of the wave function y of the localized carrier
For oc the following estimation may be offered: (y IV ecUr) and v a dimensionless constant. According to  v = 16.
(17) &-1 - - ( 2 to 5) A . (17) agrees with the results of the calculations of the wave function of the lo- calized polaronic car...