Progress of Theoretical Physics Supplement No. 101, 1990

215

Distortion of Energy Surfaces by a Strain Field Katuhisa SUZUKI

Tokyo Metropolitan Institute of Technology, Hino 191 (Received July 24, 1990) Deformation of energy bands in solids in the presence of a strain field is discussed from a symmetry point of view. The very particular case of the conduction band of silicon in an orthorhombic strain is examined in some detail. The distortion of the energy surface in this case gives rise to an additional contribution to piezoresistance and predicts the correct sign and magnitude of the piezoresistance coefficient 11"44, which has been predicted to vanish by the conventional electron transfer theory.

§ 1.

Introduction

Deformation of electronic energy bands in the presence of a strain or stress field has provided abundant information in the study of solids. On the one hand, it determines the amount to which phonons scatter conduction electrons, which would otherwise have an infinite mean free path in a periodic structure. On the other hand, application of an external stress has enabled fine measurements of energy bands in solids. Determination of hole effective masses by cyclotron resonance in stressed silicon and germanium1 H> and determination of interband energies at numerous symmetry points in the Brillouin zone by piezoreftectance measurements5 > are among noteworthy examples. Historically, yet another remarkable role was played by a strain in solids, when Smith in 1954 discovered a surprisingly large piezoresistance (PR) effect in germanium and silicon of both n- and P-types. 6 > An analysis of the effect contributed greatly to establishing the now well-known band structures of the semiconductors. Many of the features of the PR effect observed by Smith in n-type materials were explained very successfully by Herring in terms of the strain-induced transfer of electrons between the conduction band valleys. 7>,s> In the three and half decades since Smith's discovery, the PR effect of semiconductors has found applications in a wide range of fields, but few significant progress9 > has been made as to its underlying mechanism, presumably because people thought that the effect was completely understood by Herring's theory. However, a very simple but important puzzle has remained unsolved: On the basis of the wellestablished band model, some of the PR coefficients are shown to vanish by the electron transfer theory, but this is not always what is observed.

216

K. Suzuki

§ 2.

Electron transfer theory

The deformation of energy bands by strain was systematically discussed by Whitfield, 10 > Bir and Pikus 11 > by using a deformation potential operator fJJap representing the response of electrons to a strain field eap and a general geometrical transformation from an undeformed to a deformed lattice. In a quantitative treatment of the strain effect, however, we resort to using the matrix elements of fJJ ap, which are determined in most cases empirically. In the following we will confine most of our discussions to the semiconductor silicon, which has so far been studied in greatest detail. In the conduction band edge Ll1 of silicon, we need only one matrix element (deformation potential) 8> (1) to describe the relative motion of the six valleys, whose energy surfaces are given by (2)

and similar expressions in the absence of strain, where ko=0.85 X (271:/a), a being the lattice constant, designates the position of the conduction band minimum. When only linear terms in strain are retained, the electron transfer theory gives, under simplifying assumptions on the relaxation time, the three PR coefficients of the cubic crystal: 8 > 7rH =

2Eu ( 3ks8 s11 -

Siz

) mj_ -mtt mj_ + 2mtt '

(3)

(4) (5)

where 8 is the absolute temperature, ks the Boltzmann constant, and Sij tqe cubic compliance constants. The reason for the vanishing of 1l:44 is straightforward: No electron transfer takes place when a uniaxial stress is applied along [111], because this stress direction makes the same angle to each of the six valleys and therefore causes no valley splitting. Observed data do not follow this prediction closely. We present here two sets of typical room-temperature data on the 1r's in 10- 11 Pa- 1: 7ru

Smith (1954) Matsuda, Kanda and SuzukF 2 > (1989) 6>

-102.2 -71.5

53.4 37.3

-13.6 -9.5

It is clear that the coefficient 71:44 differs from zero beyond experimental uncertainties. § 3.

Origin of 1t'44

Suppose a uniaxial stress of magnitude Tis applied along [11 0] and suppose the

Distortion of Energy Surfaces by a Strain Field

217

longitudinal (current i#[11 OJ) and transverse (i#[1 I OJ) resistivities are measured. The relevant resistivities are (6)

and Pt = Po(1 + ITtT),

(7)

where Po is the cubic (unstressed) resistivity and (8) (9)

ITt- ITt= IT44 .

(10)

The vanishing of IT44 is equivalent to ITt= ITt in this configuration, which again is readily expected if only the raising and the lowering of the valleys in energy are considered. The energy surfaces given by Eq. (2) are ellipsoids of revolution and the six valleys as a whole look the same whether looked along [11 0] (longitudinal) or along [1 I 0] (transverse). From a symmetry point of view the form (2) is required from the invariance of E(k) under the group of kat k=(O, 0, ko) in undeformed crystal, which consists of E, Czz, 26d, 26v and 2C4z. With an orthorhombic distortion exy=S44( T/2) that group now consists only of E, Czz and 26d, under which the general invariant expression quadratic in k is

- o+ -

where

e

oe, ko(e), 1

fi2

2mie)

[k - k (e)]2 +_____j£__j kx + ky )2 + z

0

~\

/2

fi2 ( kx- ky )2

2m2(e)

/2

'

(11)

mie) and ml.(e) can generally depend on the strain magnitude, and 1

mr(e)

m1. ( e )

1 m2(e)

1

+aexy,

(12)

aexy.

(13)

The energy surface (11) is no longer an ellipsoid of revolution, but an ellipsoid with three different radii along its principal axes. In particular it gives different effective masses (12) and (13) along [11 0] and [1 I O]. The energy band (11) immediately gives rise to a non-vanishing ITL-ITt and therefore a non-vanishing 1T44 as seen below. Under a rather general assumption again on the relaxation time r, the electron conductivity in cubic silicon is given by8 > (14)

218

K. Suzuki

where n is the electron concentration. The longitudinal and transverse conductivities are given under the same condition 2 1 ) az-ne -31( m111( e ) + ml_1( e ) +-(-) m1 e ,

1( 1

2 1 ) at-ne -3 m11 ( e ) + ml_1( e ) +-(-) m2 e ,

(15) (16)

where the electron transfer between the split valleys caused by the oe term in Eq. (11) has been neglected. (This transfer does not affect the linear PR coefficient Jr44.) From (14)~(16) we obtain

1

1

--mJe)-mTe)

2aexy

_1_+2-1m//

(17)

ml_

where m is the free electron mass.

On the other hand, from (6)~(10) we have (18)

and therefore 7r44=

(19)

Hensel, Hasegawa and Nakayama 13> were the first to introduce the expression (11). They carried out a low temperature electron cyclotron resonance experiment with great precision and analyzed the result on its basis. They determined the newly introduced coefficient a=(86.8±5.0)/m

(20)

as well as the zero stress effective masses m/1=0.9163 m,

(21)

ml_ =0.1905 m .

(22)

With the compliance S44=1.248X10- 11 Pa- 1 Eq. (19) gives

which is of the same sign and magnitude with the experimental values mentioned earlier. It should be noted that the electron transfer mechanism and the distorted energy surface mechanism make characteristically distinct contributions to the PR effect. The former contributions (3) and (4) are inversely proportional to absolute temperature, while the latter (19) is temperature independent. To date no experimental data have been available to examine this difference.

Distortion of Energy Surfaces by a Strain Field

§ 4.

219

Discussion

One may wonder why a hitherto forgotten slight distortion of the energy surface could lead to a Jl'44 which is by no means of negligible magnitude. The reason consists solely in the anomalously large magnitude of the constant a, the origin of which was closely examined in Ref. 13): The conduction band edge L11 of silicon is located close to the Brillouin zone boundary X, where L11 touches another band Llz' and an e;xy strain induces a strong mixing of the two Bloch functions. One may also wonder whether a similar effect for the kx and kY ellipsoids could lead to an effective mass change with stress. Symmetry arguments, however, show the absence of such a term and that conclusion was confirmed definitively by cyclotron resonance experiment. 13 > Similar symmetry arguments lead to the absence of a distorted energy surface contribution to Jl'u and Jl'12 of silicon, that is, Eqs. (3) and (4) remain valid as they stand. A final question is what happens in n-type germanium, which has also a manyvalley structure but whose band edges are located at the zone boundary L point (nondegenerate). The observed room-temperature PR coefficients are Jl'44= -140, Jru=-3, (Jru-Jl'lz)=(0.3~1.2)X10- 11 Pa- 1 • 6 > The electron transfer theory predicts Jru-mz=O in this case, which is rather well satisfied experimentally. Symmetry allows for the presence of an a-term in germanium also, but its coefficient is not likely to be as large as in silicon because of the difference in the band edge points. In conclusion, we have considered in a many-valley semiconductor two types of energy band deformation by strain: the shift of band edge energy and the distortion of energy surfaces. It is demonstrated by using the particular coefficient of n-type silicon as an example that the latter can be a primary source of PR when the well-known mechanism based on the former predicts a null effect from the band edge symmetry. A more straightforward numerical study done by Ohmura recently for the valence band of silicon also implies the important contributions of mass and .mobility changes to PR. 14 > Acknowledgements

It is my great pleasure and honor to dedicate this paper to Professor Junjiro Kanamori in celebration of his sixtieth birthday. For the many years since I came to know him, he has always been an immense source of my delight and enlightenment both professionally and personally. I wish also to thank Professor Yozo Kanda, who originally recognized the mystery in the silicon PR coefficient and contributed to its solution. References

1) 2) 3)

]. C. Hensel and G. Feher, Phys. Rev. 129 (1963), 1041. H. Hasegawa, Phys. Rev. 129 (1963), 1029. K. Suzuki and]. C. Hensel, Phys. Rev. 9 (1974), 4184.

220 4) 5)

K. Suzuki

}. C. Hensel and K. Suzuki, Phys. Rev. 9 (1974), 4219. M. Cardona, Modulation Spectroscopy, in Solid State Physics, ed. H. Ehrenreich, F. Seitz and D. Turnbull (Academic Press, New York, 1969), Suppl. 11. 6) C. S. Smith, Phys. Rev. 94 (1954), 42. 7) C. Herring, Bell System Tech. }. 34 (1955), 237. 8) C. Herring and E. Vogt, Phys. Rev. 101 (1956), 944. 9) K. Suzuki, H. Hasegawa and Y. Kanda, }pn. }. Appl. Phys. 23 (1984), L871. 10) G. D. Whitfield, Phys. Rev. 121 (1961), 720. 11) G. L. Bir and G. E. Pikus, Simmetriya i Deformatsionnye Ejfekty v Poluprovodnikakh (Nauka, Moscow, 1972). 12) K. Matsuda, Y. Kanda and K. Suzuki, }pn. }. Appl. Phys. 28 (1989), Ll676. 13) }. Hensel, H. Hasegawa and M. N akayania, Phys. Rev. 138 (1965), A225. 14) Y. Ohmura, talk at the 45th Annual Meeting of the Physical Society of Japan, Toyonaka, Spring 1990.