Electric conductivity in a narrow strip of semimetal bismuth

J. Phys. Chem. Solids Vol. 53, No. 1, pp. 4550, Rimed in Great Britain.

ELECTRIC

1992

OOZZ-3697/92 $5.00 + 0.00 Q 1991 F%Q,amon Pm8 plc

CONDUCTIVITY IN A NARROW OF SEMIMETAL BISMUTH

STRIP

H. T. CHU and TIAN LI Department of Physics, The University of Akron, Akron, OH 44325, U.S.A. (Receiued 6 March 1991; accepted 11 June 1991) Abstract-In an ultrathin film of semimetal bismuth, the charge carriers would perform two-dimensional motions due to the quantum size effect. The film turns itself into a narrow strip when the width is reduced while the length is kept fixed. The continuous two-dimensional energy spectrum would be subject to another size quantization and divided into sub-bands. A numerical study has been carried out on the electric conductivity at T = 0 in narrow strips of semimetal bismuth. The conductivity generally increases with increasing width and exhibits oscillatory variations that can be identiCed as a consequence of the quantum size effect. Both the vanishing-wave-function and the vanishing-gradient boundary conditions have been studied. It has been shown that the conductivity is not sensitive to the boundary conditions. The major difference resulted from the different boundary conditions is that the conductivity drops to zero with widths 5 140 A and the semimetallic strip becomes semiconductive if the vanishing-wave-function boundary condition is applied; or the conductivity tends to be nearly flat for widths 5 100 A if the vanishing-gradient boundary condition is applied. Keywords: Bismuth narrow strip, electric conductivity, quantum size effect.

Vanishing-wave-function

1. INTRODUCTION

boundary

condition:

electrons:

In a previous work [l], the Fermi energy and the density of charge carriers have been numerically evaluated in narrow strips of semimetal bismuth. The film plane of the strips was normal to the trigonal axis of the bismuth structure and the the film thickness was ultrathin so that the electrons and holes would assume 2D motions in the film plane at low temperatures [2-41. A narrow strip was defined as the width of an ultrathin film was reduced down to the range

W +&I&) = (~*/2NaIlaz - a~2)/a221~~ 2 +

s=l,2,3,...;

(h2/2b22

holes: 2 E,,

for another quantum size effect while the length of the film was kept fixed. In our study, the length was along a binary axis and the width of the strips was along one of the bisectrix axes. In this communication, we shall present results of a numerical study on the electric conductivity in such narrow strips of bismuth. The conductivity has been evaluated as a function of the variable width, at zero degree of absolute temperature. The electron scattering by impurity atoms has been taken as the main mechanism that results in the electric resistivity at zero degrees of temperature.

=

(h*/2M,)k; + (h2/2M2)

, s = 1,2,3,.

Vanishing-gradient

boundary

. . . (2)

condition:

electrons: eqn (l), plus E,(l + EJE,)

= (h2/2)a,,kf;

(la)

holes: eqn (2), plus Eh = (h*/2M,)k:,

2. THEORETICAL

(1)

(2a)

where d is the strip width, k1 is the component wave vector along the strip length (binary axis), and the numerical values of the effective masses Mt = M2, of the inverse effective mass tensor elements a,, , a,2, a=, and of the energy gap EB can be found in [l]. The three electron bands consist of a non-degenerate band (the first band) and the doubly degenerate second and third bands; the inverse effective mass tensors

BACKGROUND

Semimetal bismuth consists of three equivalent conduction bands of electrons and one overlapping valence band of holes. The energy spectra in a narrow strip of bismuth (the length being along a binary axis and the film normal to the trigonal axis) have been summarized in [l] as follows: 45

46

H. T. Cnn and TUN LI

are different in the non-degenerate band and in the doubly degenerate bands [l]. The Boltzman transport equation [5] for electrons or holes with charge e in an electric field i is given by

e

(-g)

F.i=g(k)/z,

(3)

where f, is the Fermi function or the equilibrium dis~bution function in i = 0, g(k) is the correction to the dist~bution function in non-zero fields, E is the energy, B the velocity, and z is the relaxation time. In a narrow strip, the otherwise continuous 2D energy spectra of electrons and holes are sizequantized into discrete sub-bands given in eqns (1) and (2). Each sub-band represents a 1D energy spectrum continuously varying with the component wave vector k, along the strip length. For a given sub-band, there is a corresponding g(/c,), and the contribution to the electric current by the carriers in that sub-band is given by

i =2e

s

ug(k,)z,

(4)

where the factor 2 counts the spin degeneracy. The electric resistance R = EL/i, where Lis the electric field and L is the length of the strip. Defining resistivity p in a 2D film by R = pL[d, where d is the width of the strip, and the electric conductivity e by Q = p -‘, we obtain from eqns (3) and (4) the contribution to the conductivity by the carriers in that sub-band:

The relaxation time r in eqn (5) has been studied by evaluating the electron scatterings by fixed impurity atoms. The Bol~ann equation taking scattering into consideration can be given by e

g(k) l’e = y = c [g(k) - g(k’)]Q(k, k’), (3’) k

where Q(k, k’) is the probability of state transition per unit time from state k to state k’ or from state k’ to state k. The elastic transition within a given sub-band requires that k’ = -k. Thus, using eqn (3), we obtain l/r = 2Q(k, -k).

Assuming Ni is the number of stationary impurity atoms per unit area in the 2D film, the transition probability per unit time is then Q(k, -k) = NiuFda,

where u, is the carrier velocity at the Fermi level. In eqn (5), we have used the feature that ( - af,/W) behaves like a 6 -function at zero temperature and the fact that v = h-LaE/~k,. The electric conductivity of the electrons in each of the three conduction bands is the sum of the wn~butions in eqn (5) over the sub-bands in that condution band, the band bottom of each of the sub-bands included in the summation must be below the Fermi level, The addition of the wnductivities of the three electron bands gives the electric conductivity of all the electrons; the total wnductivity is the sum of the contributions of the electrons and the holes, respectively.

(7)

where da is the differential cross-section of scattering, which in a 2D film carries the dimension of a length instead of an area as in a 3D scattering. The differential cross-section can be evaluated using Fermi’s Golden Rule [6]: (ar/Wde

= (2~/~lg~~~~l~~l~l-~>12,

(8)

where g(E,) is the density of states at the energy Ek and V is the interaction potential between an electron (or a hole) and the scatterer. The integral in eqn (8) can not be evaluated unless the interaction potential is given. We shall now use for a qualitative discussion a &-function for the potential: V = V, S(x -x0) S(y - y,), where V, is a constant and (x,, y,) is the location of the scatterer. In the case of vanishing-wave-function boundary condition, the wave function of an electron in a 2D strip of area Ld is given by $(x, y) = (2/U)L’ze”1” sin

2&J =-Q+?, r&d

(6)

,

and thus the integral

The average of I(k, Iv ( - k,)J over the locations (x0, yJ of the impurity atoms in the area Ld is simply (VO/U), and thus on the average l<~,/~I-W12

= (v,/td)2.

(9)

It must be pointed out that the integral in eqn (9) should have the unit of energy (Joule). The unit of a 2D area did not appear explicitly when integrated with the 6-function; we may, for simplicity, consider the unit of V, being Joule (meter)‘.

47

Electric conductivity in semimetal bismuth

Combining eqns (6)-(g), an expression can be obtained for the relaxation time. Substituting in eqn (5), the contribution to the conductivity by the electrons in the first of the three conduction bands is given by the following, if vising-wave-f~c~on boundary condition is applied

whem EF is the Fermi energy and is a function of the width d [I]. For a given d, EF is known, and thus the number of sub-bands to be included in the summation can be determined. The conductivities a2 and b3 by the electrons in the other two conduction bands are equal and have a similar expression [eqn (IO)], but the numerical values of the inverse effective mass elements are different. The conductivity by the holes has a simple expression:

where the hoIe Fermi energy Er equals E,, - EFr where EF is the (electron) Fermi energy and E, is the energy overlap of the conduction bands (electrons) and the valence band (holes). The numerical value of E, can again be found in [I]. If vanishing-gradient boundary condition is applied along the width of the strip, there is an

1

3

5

f

9

additional sub-band in each of the three conduction bands and in the valence band, giving rise to a different value of the Fermi energy Er for each given value of the width d [I]. The expressions of eI, a,, a3 and a, are also modified by an additional term corresponding to the additional sub-band.

The conductivities cl, CT*(= 03) of the electrons and cu of the holes have been evaluated individually. Since no numerical values have been assigned to Ni and V, 1 only aNi Vz, where c = 6,) c2, 6) or on, have been numerically worked out, with results being directly proportional to the crs. We shall denote arv, V: by e *, where d can be u,, a,=a,+rs,+a,, cli7 u = a, + cru (the total conductivity), etc. Each e* has been expressed as a function of the variable width. e* carries a unit of (ohm)-’ (Joule)’ (meter)2. Figures 1 and 2 show the conductivities vs the strip width when the vanishing-wave-function boundary condition is applied. In Fig. 1, the details of the electron conductivity a, and the hole conductivity en are shown. The comparison between a, and the cond~ti~ty c1 of the first electron band shows that e, is much larger than u2 and c3 of the other two electron bands. In Fig. 2, the total conductivity is shown along with the predominant contribution of electrons and the much smaller contribution of holes.

11

13

15

f7

19

d(lCYA)

Fig.

1.

Electron/hole conductivities vs width in narrow strips of semimetal bismuth (boundary condition: vanishing-wave-function; u* proportional to conductivity).

H. T. CHU and TIAN LI

1

3

5

7

9

11

13

15

17

19

d(lO*A)

Fig. 2. Electron, hole, and total conductivities vs width in narrow strips of semimetal bismuth (boundary condition: vanishing-wave-function; c* proportional to conductivity).

The electron conductivity is more than 20 times larger than the hole conductivity (Figs 1 and 2). There have been few experimental results available for a close comparison with the theoretical calculations. However, some experimental work regarding the current-voltage characteristics of bismuth whiskers at helium temperatures provides more or less indirect comparisons. For instance, in the work by Bogod et al. [A, the conductivities in the whiskers are consistently increasing with the diameter of the samples. This is in good agreement with the calculated results. The calculated conductivities generally increase with the strip width, as shown in the figures, superimposed by the oscillatory variations or slope discontinuities which are the results of the quantum size effect on the energy spectra as explained below. The energy bottom of each sub-band is determined by setting k, = 0 and s = 1 in either eqn (1) or eqn (2) and is increasing with decreasing width. At zero degree of temperature, a sub-band whose bottom is beyond the Fermi level is not an occupied band. Thus the number of occupied sub-bands well depends on the strip width [l]. In Table 1, the width dependence of the number of occupied sub-bands (vanishingwave-function boundary conditions are applied) is given for the first electron band (non-degenerate), the second and third electron bands (doubly-degenerate), and the hole band; the numbers are denoted by S1, S, and S,,, respectively. As the strip width varies continuously, the numbers SI, S, and S, would undergo quantum jumps (change by unity) at their respective values of the width as seen in Table 1. When one of the numbers undergoes such a quantum jump, a discontinuity in the slope would be seen in the conductivity; some may be more pronounced

while some may be less pronounced. For instance, the two most pronounced ones occur at d - 650 A and d - 1550 A, where S, changes from zero to one and from one to two, respectively. Less pronounced slope discontinuities like those at d - 360 A, MO& and Table 1. Number of occupied energy sub-bands in a narrow strip of bismuth at T = 0 (vanishing-wave-function boundary condition) Strip width 140 160 180 240 360 380 560 580 640 660 700 750 900 1050 1200 1550 1600 1650 1700 2000

1 1 2 3 5 6 9 9 10 10 10 11 12 14 16 21 21 22 22 26

0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 2 2 2 2

1 1 1 1 1 2 2 3 3 3 3 4 5 6 7 9 9 9 10 12

tPartia1 listing. S, : number of sub-bands in the nondegenerate electron band. S,: number of sub-bands in the doubly degenerate electron bands. S,, : number of sub-bands in the hole band.

Electric conductivity in semimetal bismuth

49

18

171615-

‘& 10 22

9

b

8

1

0-_1-0 0

200

400

600

800

1000

1200

1400

1600

1800

2000

d(A) Fig. 3. Electron, hole, and total conductivities vs width in narrow strips of semimetal bismutb (boundary condition: vanis~ng-sadist; (r * proportional to ~nductivity).

1650 A correspond to the quantum jumps of S,. Thus, it is clear that the quantum size effect interprets the slope discontinuities in the conductivity vs width curves of narrow strips of semimetal bismuth. When the strip width is reduced below N 140 A, the last occupied electron and hole sub-bands cross over each other and the semimetal turns itself into a

Table 2. Number of occupied energy sub-bands in a narrow strip of bismuth at T = 0 (vanishing-gradient boundary condition) Strip width (4t

S,

s,

:

21

1

1

160 180 340 360 560 580

3

: 2 2 3 3

1;

1 1 1 1 1 1

z 700

11 11

21 2

9 4 4

750 900 1000 1050 1150 1200 1350 1400 1550 1600

12 13 14 15 16 17 19 20 22 22

2 2 2 2 2 2 2 2 3

: 6 I I 8

1700 1650 2ouO

23 27

3 3

: 6

tPartial listing.

sh

; 10 10

t

Y

13

semiconductor. The electric conductivity drops to zero as expected. Numerical results reveal that the electric conductivity in a narrow strip of bismuth is not sensitive to the boundary conditions applied to the strip width. In Fig. 3, shown are the conductivities a,, +, and CTevaluated under the vanishing-cadent boundary condition. The magnitudes and the oscillatory structures are similar to those in Figs 1 and 2. The conductivity is again predominantly contributed by the electrons over the holes; and the conductivity of the electrons in the non-degenerate conduction band is much larger than the conductivity of the electrons in the doubly degenerate conduction bands. The sIope di~ontinuities are again the results of the quantum size effect; the quantum jumps of the numbers S, , S, and Sh listed in Table 2 can again be used to identify the discontinuities. It is interesting to compare the structures on the curves in Fig. 3 with the quantum jumps given in Table 2. For instance, the slope discontinuities at d - 660 A and 1550 A correspond to the quantum changes of S,; those at d N 160 A, 360& 58O.& lOOO& 1200.&, 1350A and I7OOA correspond to the jumps of S,, and that at d - 70 A corresponds to the change of S, from 1 to 2, etc. When the vanishing-gradient boundary condition is applied, however, the lowest sub-band in each of the electron energy bands or in the hole band has a fixed band bottom at which the energy is simply zero [eqns (la) and (2a)]. Thus, there would be no semimetal-semiconductor transition and the values of S,, S, and S,, for a given width are generally one unit higher than those under the vanishingwave-function boundary condition. Differences in the conductivities under the two different boundary conditions are expected, however, to be more appreciable

50

N. T. Cuu and TUN Lt

in the region of smaller widths. In addition to a couple of more oscillatory structures under the vanishinggradient boundary condition, the ~onducti~ty does not tend to zero as d -P 0. It had been thought [1] that the conductivity would become increasingly large as d -+ 0, since the carrier density increases rather rapidly with decreasing d under the vanishing gradient boundary condition. It has now been shown that the relaxation time or collision time decreases with decreasing width and the conductivity seems to become somewhat flat as d + 0.

conducting when the vanishing-wave-fimction boundary condition is appkd. The cmductiviiy becomes somewhat stationary as d + 0 when the vanishinggradient boundary condition is applied. Although the evaluatian of the relaxation time is crude, and the same interaction potential has been used for both the electrons and the holes, we believe the results and the conclusions can well be used for qualitative purposes.

T,, Tian L. and Guo I-I., J. Phyz. C/rem, S&k 51, 1145 (1990). 2. Chu H. T., J. i’hys. Chem. So&& 49, 1191 (1988). 3. Komnik Yu F. and Andrievskii V. V.. Soviet J. Low Temp. Phys. 1, 51 (1975). 4. Chu I-L T., Henriksen P. N. and Alexander J., Phys. Rev. 837, 3900 (1988). 5. Ziman J. M., Prinfipes ofthe Theory ofSo&& chapter I. Cbu H.

4. SUMMARY

The electric conductivity in narrow strips of semimetal bismuth is predominantly contributed by the electrons over the holes. The conductivity increases with increasing width of the strips. Size quantizations to the energy spectra of the electrons and the holes result in the oscillatory variations of the slope in the conductivity vs width curves. The conductivity drops to zero at d - 140 ,& and the strip becomes semi-

7. Cambridge LJniversity Press, London (1972). 6. 7. Bo&d Yu A., Valeev i.. G., bitsu D. V. and &o& A. D., So&r J. Low Temp. Phys. 8, 54 (1982).