Hot́ hole anisotropic effect in silicon and germanium

Solid State Communications,Vol. 15, pp. 1213-1216, 1974.

Pergamon Press.

Printed in Great Britain

HO'r HOLE ANISOTROPIC EFFECT IN SILICON AND GERMANIUM* C. Canali, G. Ottaviani and G. Majni Laboratorio di Elettronica, Istituto di Fisica dell'Universitfi, Via Vivaldi n. 70, 4 ! 100 Modena, Italy

(Received 18 June 1974"by L. Hedin)

Anisotropic effect of the hole drift velocity in silicon and germanium has been investigated with the time of flight technique by applying the electric field parallel to the (100) and (111 ) crystallographic axis. The measurements were performed for electric fields ranging from 10 to 3 X 104Wcm and temperatures from 40 to 200°K. The results indicate that the anisotropic effect vn(100)/va(111) increases with decreasing temperature and increasing electric field, and reaches a saturation Value at high electric fields ( ) 10" V/cm). The maximum anisotropic effect for Ge is 1.25 at 40VK and for Si is 1.2 at 450K. A qualitative analysis of the experimental data indicates that the anisotropic effect is due to the warped heavy-valence-band shape.

SILICON and germanium are the most extensively studied semiconductor materials. However, several aspects of their transport properties are not fully established as for instance the longitudinal anisotropic effect of the drift velocity of holes. 1'2 The aim of this paper is to clarify this point by reporting experimental data of the hole drift velocity in a wide range of temperature (40-200°K) and of electric field (10 - 3 X 104V/cm) applied parallel to (111) and (100) crystallographic axis. The results suggest a strong correlation between anisotropic effect and the warping structure of the valence band in both materials. The measurements have been performed with the time of flight technique which has been extensively described in previous papers, a'4 High purity silicon (NA --ND = 1012cm -a) supplied from Wacker Chemitronie, and germanium (NA --ND = 1011cm -a) supplied from Lawrence Berkeley Laboratory and from General Electric, have been used. Si surface barriers diodes and high-purity Ge p+in+were used. The n + and p+ regions have been obtained by the solid state epitaxy regrowth at low temperature (300°C) in order to avoid contamination of the material, n's

Wafers were cut parallel to different crystallographic planes out of the same ingot. This allowed us to have samples differently oriented but with the same electrical characteristics. In order to cover the largest possible range of electric field and temperature we have used samples of different thickness mounted in such a way as to minimize thermal gradients. Pulsed bias voltages have been used in order to avoid heating effects. Details of sample preparation, experimental set up and measurement techniques have been described previously,a The measurements were performed in three different samples for each crystallographic direction investigated. The same samples have been used for measuring electron drift velocity and the measurements for electrons are in agreement with previous data either for the values of drift velocity or for the magnitude of the anisotropic effect. Errors in drift velocity measurements arising from electric field distortion due to the high density of created carriers have been minimized by reducing the injection level. The error due to the electronic apparatus was less than -+ 3 per cent and the spread of the data obtained in different samples was no more than + 5 per cent. Consequently We assume this number as the total experimental error

*This work has been partially supported by C.N.R. (Italy). 1213

1214

HOT HOLE ANISOTROPIC EFFECT IN SILICON AND GERMANIUM '

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Vol. 15, No. 7

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FIG. 1. Holes drift velocity in silicon as a function of

the electric field applied parallel to the ( l 11 ) and (100) crystallographic axis, at several temperatures. The dependence of hole drift velocity upon the electric field at three different temperatures is shown in Figs. l(a) and (b) for silicon and Figs. 2(a) and (b) for germanium. The electric field was applied parallel to the (111) and ( 100 ) crystallographic axis. For clarity only the average of the drift velocity values at any electric field strength is plotted in the figures. The main features of the results shown in Figs. l(a) and (b) and 2(a) and (b) are: (i) at the lowest applied electric field our values of mobility are in good agreement with published data; 7`s (ii) away from the ohmic region an anisotropic effect of va is observed for the electric field applied parallel to the (100) and (111) crystallographic axis. The Vd(100) is always higher than the Vd( l 11 ) in both materials. The magnitude of the anisotropic effect is comparable in germanium and silicon;

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E (V cm-') FIG. 2. Holes drift velocity in germanium as a function of the electric field applied parallel to the (111) and (100) crystallographic axis, at several temperatures. (iii) the electric field value where the Vd( i 11 ) starts to be different from the vd(lO0)decreases with decreasing temperature. For temperatures lower than 77°K the minimum electric field at which the anisotropic effect is observable is lower for germanium than for silicon. The anisotropic effect increases in both materials with decreasing temperature and the maximum vd( 100)[Va(111) ratio is 1.25 at 40°K for germanium and 1.2 at 45°K for silicon; (iv) a saturation value of the drift velocity is not reached in silicon at the highest applied electric field; In the case of germanium and at temperatures lower than 77°K the drift velocity tends to saturate at values different for each investigated direction. The saturation drift velocities of Ge at 77°K are 1.1 X 107 cm/sec f o r e II (100) and 0.9 X 10Tern/see f o r e II ( l i D .

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HOT HOLE ANISOTROPIC EFFECT IN SILICON AND GERMANIUM

A quantitative analysis of these experimental data is very difficult and out of the range of this paper. However, on the basis of the valence band structure of germaaium and silicon a qualitative interpretation is possible. Band structure calculations have quantitatively pictured the top of the degenerate valence band of Si and Ge.9 Results show that while the light-hole band is almost spherical the heavy-hole band. exhibits a pronounced warping.1° The transport properties of hot holes are expected to be determined mainly by the heavy hole band. xl Neglecting the light band should, in fact, introduce serious errors since the larger equilibrium density of states of the heavy band with respect to the light produces a higher population of holes in the heavy band. The carrier heating effect in non-ohmic regions should produce the net effect of further growing this population in favour of the other band. In the effective mass approximation it has been calculated that the effective mass of heavy holes is highly anisotropic, exhibiting 1° a maximum ratio m<~l~>/m~100>between the (100) and ( 111 ) crystallographic directions of 2.74 in silicon and of 1.83 in germanium. Accordingly, present experimental results support this picture by exhibiting a higher hole drift velocity in the (100) direction (characterized by a low effective mass) than in the (111) direction (characterized by a large effective mass). Also the differences

1215

between silicon and germanium in the value of ohmic mobility and in the appearance of the anisotropy at different electric fields should be in the first approximation due to the different values of the effective mass of the heavy hole band. The peculiar characteristic of a saturation of the anisotropic effect in the region of the highest applied field (E -- 104V/cm) may be attributed to the negligible importance of scattering mechanisms (such as h - h interaction) which should randomize the hole mean energy leading to a net lowering of the anisotropic effect. In conclusion we report clear experimental evidence of the anisotropic effect of holes in silicon and germanium. The experimental data suggest a strong correlation between the warped heavy-valenceband shape and anisotropic hole drift velocity. The high purity characteristic of the samples used permits to neglect the ionized impurity scattering contrary to the suggestions of Asche et al. 2 and Tschulena1 that anisotropic effects should be predominantly due to this scattering mechanism.

Acknowledgements - We like to thank Profs. Alessandro Alberigi Quaranta, L. Reggiani and C. Jacoboni for the fruitful discussions, and Mr. P. Cantoni and Mr. M. Bosi for the help in collecting the data. E.E. Hailer is acknowledged for the useful comments.

REFERENCES 1.

TSCHULENAG.R., J. Phys. Chem. Solids 33, 1219 (1972).

2.

ASCHEM., VON BORZESKOWSKIJ. and SARBEI O.G., Phys. Status Solidi 38, 357 (1970).

3.

CANALIC., OTTAVIANI G. and ALBERIGI QUARANTA A., J. Phys. Chem. Solids 32, 1707 (1971).

4.

MARTINIM., MAYER J.W. and ZANIO K., Advances in Solid State Science, Vol. 3 (edited by WOLFE R.) Academic Press, New York (1972).

5.

OTTAVIANIG., MARELLO V., MAYER J.W., NICOLET M.A. and CAYWOOD J.M., Appl. Phys. Lett. 20, 323 (1972).

6.

MARELLO V., CAYWOOD J.M., MAYER J.W. and NICOLET M.A., Phys. Status Solidi 13, 531 (1972).

7.

OTTAVIANIG., CANALI C., NAVA F. and MAYER J.W., J. Appl. Phys. 44, 2917 (1973).

8.

CANALIC., COSTATO M., OTTAVIANI G. and REGGIANI L., Phys. Rev. Lett. 31,536 (1973).

9.

KANE E.O., J. Phys. Chem. Solids l, 82 (1956).

10.

COSTATOM., GAGLIANI G., JACOBONI C. and REGGIANI L., (to be published in J. Phys. Chem. Solids).

11.

REGGIANI L. (private communication).

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HOT HOLE ANISOTROPIC EFFECT IN SILICON AND GERMANIUM

L'effet de l'anisotropie de la vitesse de d6rive des trous dans le silicium et darts le germanium a 6t6 6tudi6 avec la t6chnique du temps de vol an appliquent le champ 616ctrique parallel aux axis cristallographiques (100) et (111), Les mesures ont 6t6 execut6es pour des champs ~16ctriques entre 10 et 3 X l ~ V / c m aux temp6ratures de 40-200°K. Les r6sultats indiquent que l'effet de l'anisotropie v~( 100)/v,~( 11 l) croit avec la tel,~perature d6croissante et avec le champ electrique croissant. On a obtenu une valeur de saturation a des champs 61ev6s (1> 104 V/cm). L'effet de l'anisotropie maximum est de 1.25 a 40°K pur Ge et 1.2 a 45°K pour Si. Une analyse qualitative des dates experimentales indique que l'effet de l'anisotropie r6sulte de la forme (warped) de la bande de valence des trous lourds.

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