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The mobility, diffusion constant, and lifetime of minority carriers in heavily dislocated germanium
J. Phys. Chem. Solids
F.2
Pergamon
THE MOBILITY, OF MINORITY
Press 1959. Vol. 8. pp. 147-149.
Printed in Great Britain
AND LIFETIME DISLOCATED
DIFFUSION CONSTANT, CARRIERS IN HEAVILY GERMANIUM A.F. GIBSON Royal Radar Establishment,
Malvem,
1. INTRODUCTION
England
Z.EXPERIMENTALRESULTS The experimental results summarized in Table 1.
BELL and HOGARTH have shown that the diffusion length, L, of minority carriers in germanium and silicon, measured by the travelling-light-spot technique, may be anisotropic if the crystal contains a high density of parallel edge dislocations. The diffusion length measured parallel to the dislocation array was typically a factor of two or three greater than that measured perpendicular to the array. It is well known that dislocations in germanium or silicon crystals act as efficient recombination centres (WERTHEIM and PEARSON(~)). AS BELL and HOGARTH point out, however, a significant anisotropy in carrier lifetime, assuming the diffusion constant to be isotropic, can be obtained only if two additional assumptions are made, namely: (a) The dislocations are largely polygonized into “walls”, and (b) the dislocations are surrounded by a potential barrier which tends to exclude minority carriers from the high recombination region. The object of the work to be described in the present paper was, primarily, to check the validity of the assumption that the diffusion constant, D, is isotropic and equal to the normal value in heavily dislocated material. Two approaches to the measurement of D have been made, namely drift-mobility measurements and the simultaneous measurement of phase and amplitude in the travelling-light-spot experiment. We shall show that in n-type germanium D is not isotropic. In p-type germanium, on the other hand, D is isotropic and all the anisotropy appears to reside in the lifetime, 7. In addition measurements of drift mobility at high electric fields will be described, the results obtained giving valuable quantitative support for the model proposed.
for germanium
Table 1.
are
-Results
Experimental Method
Travelling spot
light
Drift mobility at low fields (E < 100 V/cm) Drift mobility at high fields (100 V/cm < E < 5000 V/cm) Electrical resistance
Electrons in p-type Ge
Holes in n-type Ge
L, >Ls
L, >Lz D, >D,=Do 1 Tf 2 72
D, =Dz=Do 711> 72
/J( = pz = PO D,, >Ds=Do
/JLz-=CFu = PO
PO = kkz= PO
Isotropic
(a) Travelling-light-spot experiments By the simultaneous measurement of phase and amplitude, using “chopped” light, the diffusion constant and lifetime of injected carriers can be obtained separately in this experiment. As shown in Table 1, the diffusion constant of holes in n-type germanium was found to be about two or three times greater parallel to the dislocation array than across it, the latter having the normal diffusion constant. No comparable effect was observed in high-resistivity p-type germanium, all the anisotropy in diffusion length being accounted 147
148
SESSION
F:
DISLOCATIONS
for by an anisotropy in effective electron lifetime. (b) Low-jield drift-mobility experiments In view of the above results, the drift mobility of holes in n-type germanium was measured by the HAYNES~HOCKLEY@) technique in filaments cut parallel and perpendicular to the dislocation array. In the same experiment the diffusion constant could be deduced from the broadening of the hole pulse arriving at the collector. As shown in Table 1, the diffusion constant was found to be anisotropic, in agreement with the travellinglight-spot data, but the drift mobility, at least at low fields, was isotropic. It would appear, therefore, that the Einstein relation between D and p is not applicable to heavily dislocated n-type material.
4. DISCUSSION OF RESULTS SHOCKLEY@) has suggested that the dangling bonds associated with edge dislocations produce a row of acceptor levels. If a significant fraction of these levels are occupied, the potential energy of electrons is raised locally and the dislocation may be considered as a thread of p-type material embedded in the crystal. We shall now show, qualitatively, that this model can account for our results. (a) N-type germanium As a p-type region in an n-type bulk crystal represents a potential minimum for holes, injected carriers will be captured by the dislocations where they will become majority carriers. For space-charge compensation, another hole will be
7
I
400
600 Applied
field
800 in volts
1000
1200
1400
cmm’
FIG. 1. Drift velocity of electrons parallel (curve A) and perpendicular (curve B) to dislocation arrays in germanium.
At high electric fields (E > 100 V cm-l), some anisotropy in the drift mobility of holes was observed. The drift mobility was measured by the technique described by GIBSON and GMN~ILLE@) and some typical results are shown in Fig. 1. The mobility parallel to the dislocation array (curve A) is- the same as that in undislocated material, whereas that across the dislocation array (curve B) is significantly less.
emitted somewhere along the line.* Release of a hole will occur with equal probability on either side of the point of capture, so that, in the absence of an external field, the mean position of the hole is unaffected. It follows therefore that dislocations provide a mechanism for enhanced diffusion of * Compare the propagation “floating”
p
WEBSTER@)).
region
on
n
of a hole through a material (MOORE and
SESSION
F:
holes along the dislocations without any increase in hole mobility, so that the Einstein relation is not applicable to this system. Clearly the dislocations cannot assist the diffusion of holes perpendicular to the dislocation array. In as much as the dislocations act as traps for holes, the drift mobility in n-type germanium will be reduced by the dislocations. A trap of finite dimensions is effective, however, only if an untrapped carrier drifts in the applied field a distance greater than the dimensions of the trap in the mean escape time of the trap. This condition will be achieved at relatively low fields for drift across the dislocations, and is the basis of the interpretation of the results shown in Fig. 1. The data given in this figure indicate that the dislocations have an effective diameter of about 0.8 x 10-a cm and that, in a sample containing 106 dislocations per ems, a hole spends about half its total life trapped at a dislocation. The above interpretation of enhanced diffusion has been developed quantitatively by GIBSON and PAICE. The dislocation in n-type material will be surrounded by a p-n junction and may be treated as a coaxial line formed from iterative elements of the form shown in Fig. 2. The capacity
c C
1 T
1 G
FIG. 2. Equivalent circuit for one element of a transmission line analogue of a dislocation in n-type germanium (see text).
C and conductance G per unit length of the p-n junction may be calculated from p-n junction theory in cylindrical co-ordinates, the diameter of the space-charge cylinder being known from the high-field drift-mobility data. The resistance per unit length of the inner conductor is calculated on the assumption that the occupation density of the acceptor levels is about 10 per cent (REAI~).
149
DISLOCATIONS
The results indicate that, with a dislocation density of about 10s cm-z, the diffusion constant of holes parallel to the dislocations will be about 100-200 cmT2 set-1, in adequate agreement with the experimental data. If polygonization of the dislocations is assumed, two additional adjustable parameters become available and a much larger anisotropy in diffusion constant can be accounted for. (b) P-type germanium Inp-type material the dislocations are, of course, still p-type. However, the negative charge density along the dislocation may be significantly higher than the acceptor density in the high-resistivity bulk material, so that there is still a space-charge region in which the electron potential energy is increased. The presence of this barrier will impede the capture of minority carriers (electrons) by the dislocations and, following the argument given by BELL and HOGARTH, provides a mechanism by which 7 may be anisotropic. The dislocations will clearly have no significant effect on the diffusion constant of the electrons, however, and no anisotropy of D can be expected. Acknowledgments-In preparing this contribution I have drawn on the results of various colleagues at R.R.E.. notably Mr. J. B. ARTHUR, Dr. J. W. GRANVIL.LE,and Dr. E. G. S. PAIGE.We are in turn indebted to Mr. P. J. HOYLAND, who supplied the dislocated crystals. The paper is published by permission of the H.B.M. Stationery Office, London.
REFERENCES 1. BELL R. L. and HOGARTHC. A., J. Electronics and Control 3, 455 (1957). 2. WERTHEIM G. K. and PEAFISONG. L., Phys. Rev. 107, 694 (1957). 3. HAYNES J. R. and SHOCKLEY W., Phys. Rev. 81, 835 (1951). 4. GIBSONA. F. and GRANVILLEJ. W., J. Electronics 2, 259 (1956). 5. SHOCKLEYW., Phys. Rev., 91. 228 11953). 6. MOORE A. R. aid WEBSTER- W. M., &oc. Inst. Radio Enars. 43. 427 11955). 7. GIBSON A. F. and PA~GE‘E. G: S., Phil. Mug. (to be published). 8. READ W. T., Phil. Mug. 45, 775 (1954).
F.2
Pergamon
THE MOBILITY, OF MINORITY
Press 1959. Vol. 8. pp. 147-149.
Printed in Great Britain
AND LIFETIME DISLOCATED
DIFFUSION CONSTANT, CARRIERS IN HEAVILY GERMANIUM A.F. GIBSON Royal Radar Establishment,
Malvem,
1. INTRODUCTION
England
Z.EXPERIMENTALRESULTS The experimental results summarized in Table 1.
BELL and HOGARTH have shown that the diffusion length, L, of minority carriers in germanium and silicon, measured by the travelling-light-spot technique, may be anisotropic if the crystal contains a high density of parallel edge dislocations. The diffusion length measured parallel to the dislocation array was typically a factor of two or three greater than that measured perpendicular to the array. It is well known that dislocations in germanium or silicon crystals act as efficient recombination centres (WERTHEIM and PEARSON(~)). AS BELL and HOGARTH point out, however, a significant anisotropy in carrier lifetime, assuming the diffusion constant to be isotropic, can be obtained only if two additional assumptions are made, namely: (a) The dislocations are largely polygonized into “walls”, and (b) the dislocations are surrounded by a potential barrier which tends to exclude minority carriers from the high recombination region. The object of the work to be described in the present paper was, primarily, to check the validity of the assumption that the diffusion constant, D, is isotropic and equal to the normal value in heavily dislocated material. Two approaches to the measurement of D have been made, namely drift-mobility measurements and the simultaneous measurement of phase and amplitude in the travelling-light-spot experiment. We shall show that in n-type germanium D is not isotropic. In p-type germanium, on the other hand, D is isotropic and all the anisotropy appears to reside in the lifetime, 7. In addition measurements of drift mobility at high electric fields will be described, the results obtained giving valuable quantitative support for the model proposed.
for germanium
Table 1.
are
-Results
Experimental Method
Travelling spot
light
Drift mobility at low fields (E < 100 V/cm) Drift mobility at high fields (100 V/cm < E < 5000 V/cm) Electrical resistance
Electrons in p-type Ge
Holes in n-type Ge
L, >Ls
L, >Lz D, >D,=Do 1 Tf 2 72
D, =Dz=Do 711> 72
/J( = pz = PO D,, >Ds=Do
/JLz-=CFu = PO
PO = kkz= PO
Isotropic
(a) Travelling-light-spot experiments By the simultaneous measurement of phase and amplitude, using “chopped” light, the diffusion constant and lifetime of injected carriers can be obtained separately in this experiment. As shown in Table 1, the diffusion constant of holes in n-type germanium was found to be about two or three times greater parallel to the dislocation array than across it, the latter having the normal diffusion constant. No comparable effect was observed in high-resistivity p-type germanium, all the anisotropy in diffusion length being accounted 147
148
SESSION
F:
DISLOCATIONS
for by an anisotropy in effective electron lifetime. (b) Low-jield drift-mobility experiments In view of the above results, the drift mobility of holes in n-type germanium was measured by the HAYNES~HOCKLEY@) technique in filaments cut parallel and perpendicular to the dislocation array. In the same experiment the diffusion constant could be deduced from the broadening of the hole pulse arriving at the collector. As shown in Table 1, the diffusion constant was found to be anisotropic, in agreement with the travellinglight-spot data, but the drift mobility, at least at low fields, was isotropic. It would appear, therefore, that the Einstein relation between D and p is not applicable to heavily dislocated n-type material.
4. DISCUSSION OF RESULTS SHOCKLEY@) has suggested that the dangling bonds associated with edge dislocations produce a row of acceptor levels. If a significant fraction of these levels are occupied, the potential energy of electrons is raised locally and the dislocation may be considered as a thread of p-type material embedded in the crystal. We shall now show, qualitatively, that this model can account for our results. (a) N-type germanium As a p-type region in an n-type bulk crystal represents a potential minimum for holes, injected carriers will be captured by the dislocations where they will become majority carriers. For space-charge compensation, another hole will be
7
I
400
600 Applied
field
800 in volts
1000
1200
1400
cmm’
FIG. 1. Drift velocity of electrons parallel (curve A) and perpendicular (curve B) to dislocation arrays in germanium.
At high electric fields (E > 100 V cm-l), some anisotropy in the drift mobility of holes was observed. The drift mobility was measured by the technique described by GIBSON and GMN~ILLE@) and some typical results are shown in Fig. 1. The mobility parallel to the dislocation array (curve A) is- the same as that in undislocated material, whereas that across the dislocation array (curve B) is significantly less.
emitted somewhere along the line.* Release of a hole will occur with equal probability on either side of the point of capture, so that, in the absence of an external field, the mean position of the hole is unaffected. It follows therefore that dislocations provide a mechanism for enhanced diffusion of * Compare the propagation “floating”
p
WEBSTER@)).
region
on
n
of a hole through a material (MOORE and
SESSION
F:
holes along the dislocations without any increase in hole mobility, so that the Einstein relation is not applicable to this system. Clearly the dislocations cannot assist the diffusion of holes perpendicular to the dislocation array. In as much as the dislocations act as traps for holes, the drift mobility in n-type germanium will be reduced by the dislocations. A trap of finite dimensions is effective, however, only if an untrapped carrier drifts in the applied field a distance greater than the dimensions of the trap in the mean escape time of the trap. This condition will be achieved at relatively low fields for drift across the dislocations, and is the basis of the interpretation of the results shown in Fig. 1. The data given in this figure indicate that the dislocations have an effective diameter of about 0.8 x 10-a cm and that, in a sample containing 106 dislocations per ems, a hole spends about half its total life trapped at a dislocation. The above interpretation of enhanced diffusion has been developed quantitatively by GIBSON and PAICE. The dislocation in n-type material will be surrounded by a p-n junction and may be treated as a coaxial line formed from iterative elements of the form shown in Fig. 2. The capacity
c C
1 T
1 G
FIG. 2. Equivalent circuit for one element of a transmission line analogue of a dislocation in n-type germanium (see text).
C and conductance G per unit length of the p-n junction may be calculated from p-n junction theory in cylindrical co-ordinates, the diameter of the space-charge cylinder being known from the high-field drift-mobility data. The resistance per unit length of the inner conductor is calculated on the assumption that the occupation density of the acceptor levels is about 10 per cent (REAI~).
149
DISLOCATIONS
The results indicate that, with a dislocation density of about 10s cm-z, the diffusion constant of holes parallel to the dislocations will be about 100-200 cmT2 set-1, in adequate agreement with the experimental data. If polygonization of the dislocations is assumed, two additional adjustable parameters become available and a much larger anisotropy in diffusion constant can be accounted for. (b) P-type germanium Inp-type material the dislocations are, of course, still p-type. However, the negative charge density along the dislocation may be significantly higher than the acceptor density in the high-resistivity bulk material, so that there is still a space-charge region in which the electron potential energy is increased. The presence of this barrier will impede the capture of minority carriers (electrons) by the dislocations and, following the argument given by BELL and HOGARTH, provides a mechanism by which 7 may be anisotropic. The dislocations will clearly have no significant effect on the diffusion constant of the electrons, however, and no anisotropy of D can be expected. Acknowledgments-In preparing this contribution I have drawn on the results of various colleagues at R.R.E.. notably Mr. J. B. ARTHUR, Dr. J. W. GRANVIL.LE,and Dr. E. G. S. PAIGE.We are in turn indebted to Mr. P. J. HOYLAND, who supplied the dislocated crystals. The paper is published by permission of the H.B.M. Stationery Office, London.
REFERENCES 1. BELL R. L. and HOGARTHC. A., J. Electronics and Control 3, 455 (1957). 2. WERTHEIM G. K. and PEAFISONG. L., Phys. Rev. 107, 694 (1957). 3. HAYNES J. R. and SHOCKLEY W., Phys. Rev. 81, 835 (1951). 4. GIBSONA. F. and GRANVILLEJ. W., J. Electronics 2, 259 (1956). 5. SHOCKLEYW., Phys. Rev., 91. 228 11953). 6. MOORE A. R. aid WEBSTER- W. M., &oc. Inst. Radio Enars. 43. 427 11955). 7. GIBSON A. F. and PA~GE‘E. G: S., Phil. Mug. (to be published). 8. READ W. T., Phil. Mug. 45, 775 (1954).