UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Bulk photo-effects in inhomogeneous semiconductors Cox, Clarence Donald 1959

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1959_A1 C65 B8.pdf [ 5.24MB ]
Metadata
JSON: 831-1.0085476.json
JSON-LD: 831-1.0085476-ld.json
RDF/XML (Pretty): 831-1.0085476-rdf.xml
RDF/JSON: 831-1.0085476-rdf.json
Turtle: 831-1.0085476-turtle.txt
N-Triples: 831-1.0085476-rdf-ntriples.txt
Original Record: 831-1.0085476-source.json
Full Text
831-1.0085476-fulltext.txt
Citation
831-1.0085476.ris

Full Text

Faculty of Graduate Studies PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of C. DONALD COX B. Sc., University of New Brunswick M. Sc., McMaster University IN ROOM 301 PHYSICS BUILDING MONDAY, JUNE 15th, 1959 at 3:00 P. M. COMMITTEE IN CHARGE DEAN G. M. SHRUM: Chairman R. E . BURGESS M. F. McGREGOR G. M. GRIFFITHS E . V. BOHN J. B. BROWN F. NOAKES A. M. CROOKER J. B. GUNN External Examiner: H. Y. FAN Purdue University BULK PHOTO-EFFECTS IN INHOMOGENEOUS SEMICONDUCTORS A B S T R A C T An observable e. m. f. exists in a semiconductor when a non-equilibrium carrier concentration is present in a region of electrostatic potential gradient. These two conditions may arise in a variety of ways, and the e. m. f. 's associated with various combinations of conditions are listed. One of these e. m. f. "s arises from a photo-generated non-equilibrium carrier concentration in regions of electrostatic potential gradient due to an inhomogeneous impurity distribution. The thesis is chiefly concerned with the extention of the theory of this bulk photo e.m.f. and its comparison with experiment. Previous work on the subject is reviewed and the theory is developed to cover all conditions of illumination in the region of an arbitrary impurity density gradient. An expression for the bulk photo e. m. f. is derived by two approaches, integration of the electrostatic potential gradient, and integration of the carrier quasi Fermi levels, over the illuminated region. The latter derivation is shown to be more general in its application, and is used to obtain an expression for the photo e. m. f. both in an illuminated p-n junction and in an illumin-ated bulk inhomogeneity. Using the general result, expressions for the e. m. f. are written for the extreme cases of weak and strong illumina-tion in extrinsic andnearly intrinsic semiconductors. The relation between bulk photo e. m.f. and photoconductive resistance decrease is examined. Measurements were made of the bulk photo e. m. f. as a function of light intensity. The proportionality of the effect at low levels of illumination was verified. Observations of the photo e. m. f. patterns showed a maximum of e. m. f. at positions of maximum con-ductivity gradient. Since the bulk photo e. m. f. is a function of conductivity gradient, light probe measurements give a sensitive tech-nique for the detection of inhomogeneous impurity distributions. With weak illumination, the measured ratio of photo e. m. f. to photocon-ductive resistance decrease was a constant independent of light inten^ sity. These observations verified the theory and suggested a use for this ratio in quantitative measurements of conductivity gradient. The photo e. m. f. at strong illumination was shown to be dependent on the impur-ity distribution outside the region of incident light. The conditions under which the ratio of photo e. m. f. to photoconductive resistance decrease is constant at strong illumination are shown to be in agreement with the theoretical treatment. Measurements of bulk photo e. m. f. as a function of temperature show a qualitative agreement with theory at low temperatures (extrinsic range). At high temperatures (intrinsic range) the results show a close agreement with the theoretically pre-dicted behaviour. G R A D U A T E S T U D I E S F i e l d of S t u d y : P h y s i c s Graduate Courses outside the Physics Department: Network Theory, A . D . Moore Electrical Engineering S ervomech anisms, E . V . Bonn Electrical Engineering Theory of Alloys V. Griffiths, Metallurgy P U B L I C A T I O N S M . W . Johns, C D . Cox, C . C . MacMullen, The Spins of Excited  States of Cdl l4 . Physical Review 86, 632 (1952) M . W . Johns, C D . Cox, R .J . Donnelly, C . C . MacMullen, Radioactive Decay of In^^. Physical Review 87, 1134 (1952) M . W . Johns, H . Waterman, D. MacAskill, C D . Cox, A. Description of a Large Double-Focussing Beta  Spectrometer and Its Application to a Study of the  Decay of In^^, Canadian Journal of Physics, 31, 225 (1953). BULK PHOTO-EFFECTS IN INHOMOGENEOUS SEMICONDUCTORS by CLARENCE DONALD COX B.Sc, University of New Brunswick, 1950 M.Sc, McMaster University, 1952 < A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1959 i i A B S T R A C T An observable e.m.f. exists in a semiconductor when a non-equilibrium carrier concentration is present in a region of electrostatic potential gradient. These two conditions may arise in a variety of ways, and the e.m.f.'s associated with various combinations of conditions are listed . One of these e.m.f.'s arises from a photo-generated non-equilibrium carrier concentration in regions of electro-static potential gradient due to an inhomogeneous impurity distribution. The thesis is chiefly concerned with the extension of the theory of this bulk photo •.m.f. and i t s comparison with experiment. Previous work on the subject is reviewed and the theory is developed to cover a l l conditions of illumination in the region of an arbitrary impurity density gradient. An expression for the bulk photo e.m.f. i s derived by two approaches, integration of the electrostatic potential gradient, and integration of the carrier quasi Fermi levels, over the illuminated region. The latter derivation is shown to be more general in i t s application, and is used to obtain an expression for the photo e.m.f. both in an illuminated p-n junction and in an illuminated bulk inhomogeneity. Using the general result, expressions for the e.m.f. are written for the extreme cases of weak and strong illumination in extrinsic and nearly intrinsic semiconductors. The relation between bulk photo e.m.f. and photoconductive resistance decrease is examined. i i i . Measurements were made of the bulk photo e.m.f. as a function of light intensity. The proportionality of the effect at low levels of illumination was verified. Observations of the photo e.m.f. patterns showed a maximum of e.m.f. at positions of maximum conductivity gradient. Since the bulk photo e.m.f. i s a function of conductivity gradient, light probe measurements give a sensitive technique for the detection of inhomogeneous impurity distributions. With weak illumination, the measured ratio of photo e.m.f. to photoconductive resistance decrease was a constant independent of light intensity. These observations verified the theory and suggested a use for this ratio in quantitative measurements of conductivity gradient. The photo e.m.f. at strong illumination was shown to be dependent on the impurity distribution outside the region of incident light. The conditions under which the ratio of photo e.m.f. to photoconductive resistance decrease is constant at strong illumination are shown to be in agreement with the theoretical treatment. Measurements of bulk photo e.m.f. as a function of temperature show a qualitative agreement with theory at low temperatures (extrinsic, range). At high temperatures (intrinsic range) the results show a close agreement with the theoretically predicted behaviour. In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. 0 iv T A B L E O F C O N T E N T S Chapter Page 1 INTRODUCTION 1 1.1 Intrinsic and Extrinsic Semiconductors 1 1.2 Semiconductor Impurities 4 1.3 Generation of an E.M.F. in Semi-conductors 6 2 THEORY OF BULK PHOTO E.M.F.'S 9 2.1 General 9 2.2 Imref Representations of Non-equilibrium Conditions 12 2.3 Analysis of the Bulk e.m.f. 23 2.4 Photo e.m.f. at an Illuminated Junction 34 2.5 Bulk Photo e.m.f. in a Near Intrinsic Semiconductor 41 2.6 Ratio of Bulk Photo e.m.f. to Photo Conductive Decrease of Resistance 43 3 PREPARATORY EXPERIMENTS 48 3.1 Filament Preparation 48 3.2 Metal-Semiconductor Contacts 50 3.3 Filament Conductivities 53 3.4 Apparatus 56 3.5 Identification of the Photo e.m.f. as a Bulk Effect 58 3.6 Measurement of Carrier Lifetime 61 3.7 Photo Conductive Effects 66 V Chapter Page 4 MEASUREMENTS OF THE BULK PHOTO E.M.F. 68 4.1 Linearity of the Photo e.m.f. with Weak Illumination Intensity 68 4.2 Dependence of the Bulk Photo e.m.f. on Impurity Density Gradient 71 4.3 Measurements of the Ratio of Bulk Photo e.m.f. to Photo Resistance 73 4.4 Saturation Effects 78 4.5 Temperature Dependence of the Bulk Photo e.m.f. 80 APPENDIX I Imref Gradients with Large Added Carrier Concentrations 86 APPENDIX II Dependence of Electrostatic Potential on Added Carrier Concentrations 88 APPENDIX III Evaluation of the Integrals 91 APPENDIX IV Imref Derivation of the Photo e.m.f. 95 APPENDIX V Added Carrier Densities with Uniform Illumination 98 APPENDIX VI Added Carrier Concentrations at Illuminated Junctions 101 BIBLIOGRAPHY 103 v i L I S T O F F I G U R E S Figure T i t l e Facing Page 1 Homogeneous n-Type Semiconductor 17 2 Homogeneous Intrinsic Semiconductor 18 3 p-n Junction (open circuit) 19 4 H^L-L) Junction (open circuit) 20 5 Bulk Inhomogeneity (open circ u i t ) 21 6 Assumed Carrier Distribution in an Illuminated Bulk Inhomogeneity 24 7 Carrier Distributions in an Illuminated Homogeneous Semiconductor 32 8 Dependence of Filament plus Contact Resistance on Current Density 51 9 Conductivity Temperature Dependence of the Germanium Filaments 54 10 Spectral Distribution of Radiation from a Zirconium Concentrated-arc Lamp 56 11 Scanning Mechanism and Filament Holder 57 12 Dependence of Bulk Photo e.m.f. Patterns on Filament Surface Conditions 58 13 Light Source and Balance Circuit for Lifetime Measurements 63 14 Photo Conductive After-pulse at a Metal-Semiconductor Contact 67 15 Dependence of Bulk Photo e.m.f. on Illumination Intensity 70 16 Comparison of Local Resistivity and Bulk Photo e.m.f. 71 v i i Figure T i t l e Facing Pa^e 17 Dependence of Photo Resistance and Bulk Photo e.m.f. on Illumination Intensity 74 18 Saturation Effects 78 (a) Strong Illumination Dependence of Light Probe Dimensions (b) Strong Illumination Characteristics of Photo Voltage and Photo Resistance 19 Temperature Dependence in the Extrinsic Range of Conductivity 83 (a) Photo Resistance (b) Bulk Photo e.m.f. 20 Temperature Dependence of the Bulk Photo e.m.f. in the Intrinsic Range 84 21 Experimental Verification of the Theory of the Bulk Photo e.m.f. in the Intrinsic Range 85 v i i i A C K N O W L E D G M E N T S I wish to thank Professor R.E. Burgess for his guidance throughout the course of this work, and for his valuable comments and constructive criticism during the preparation of this thesis. The assistance of the International Nickel Company of Canada through a Graduate Research Fellowship is gratefully acknowledged. I wish to thank the Defence Research Board for summer assistantships and research f a c i l i t i e s provided under grant number 9512-22. I am indebted to Mr. F.A. Payne for drafting the figures. C H A P T E R 1 - I N T R O D U C T I O N 1«1 Intrinsic and Extrinsic Semiconductors An electron moving in a crystal i s subject to a periodic potential arising from the atoms of the l a t t i c e . Quantum mechanics shows that an electron moving in such a periodic potential has certain ranges or bands of allowed energy states. If a band is incompletely f i l l e d the crystal w i l l have a metallic character; on the other hand, i f a certain number of bands are completely f u l l the electrons are not available for conduction, and the material w i l l be an insulator at absolute zero. At temperatures other than zero, some electrons in the f i l l e d band (valence band) w i l l be excited into the next highest empty band (conduction band). If the forbidden energy gap is several electron volts wide, the material w i l l remain an insulator for a l l practical purposes. For a forbidden gap of the order of one electron volt or less, however, a considerable number of thermally excited electrons appear in the conduc-tion band at room temperature and the material is cl a s s i f i e d as a semiconductor. When an electron becomes available for conduction by excitation across the forbidden gap, a vacancy occurs in the valence band. This vacancy or hole can also contribute to conduction, and behaves like a particle with positive mass and charge. An intrinsic semiconductor has an equal number of electrons and holes, i.e., the carriers have a l l been 2 produced by excitation of electrons from the valence to the conduction band, and the el e c t r i c a l conductivity i s an intrinsic characteristic of the substance. However, most semiconductors owe their conductivity to impurities, and these may be classif i e d e l e c t r i c a l l y as donors or acceptors. A donor impurity donates an electron to the conduction band while an acceptor traps an electron from the valence band, freeing a hole. In either case, an ionized impurity i s l e f t behind. When impurities of both types are present, the donors (Npj give up electrons to the acceptors (N^) producing an effective impurity concentration (N D - }. If the density (n) of electrons or (p) of holes originating from impurities is large compared with the intrinsic concen-tration (njj, one speaks of extrinsic or impurity semi-conductors. The intrinsic concentration i s a rapidly increas-ing function of temperature and at a sufficiently high temperature, a l l semiconductors w i l l show intr i n s i c characteristics. For the same reason, since no crystal can be entirely free from impurities, sufficiently low tempera-tures will, produce some extrinsic behaviour. Since electron and hole densities are partly intrinsic and partly extrinsic in origin i t i s often convenient to express them in terms of impurity density and intrinsic concentration. Assuming ele c t r i c a l neutrality and complete impurity ionization, we may equate positive and negative charges in the crystal. 3 h + i\lA = p + No The generation and recombination of electrons and holes i s a reversible pseudo-chemical reaction, and application of the mass action law gives These equations may be solved simultaneously to give carrier concentration in terms of impurity density and intrinsic concentration. Complete impurity ionization is a valid assumption in germanium except at very low temperatures, providing the impurities are predominantly the trivalent and pentavalent acceptor and donor elements which have very small activation energies (less than kT). At a l l points in the crystal, any separation of carrier types sets up a counteracting f i e l d tending to restore el e c t r i c a l balance, and electrical neutrality may be assumed except in regions of abrupt impurity density gradient (Shockley 1949), and even here significant depar-tures from neutrality exist only in narrow regions. np = constant = n* 4 1.2 Semiconductor Impurities Impurities may be chemical or physical in origin, i.e., foreign elements or crystal imperfections. In the group IV semiconductors for example, the group III and V elements act as acceptors and donors respectively, and enter the lattice substitutionary (Pearson 1949). This is explained in terms of the valence electron structure of these elements (Shookley 1950). On the other hand, copper and nickel take up i n t e r s t i t i a l positions and behave as acceptors, and there i s some evidence that copper substitu-t i o n a l ^ acts as a donor. The impurity characteristics of these elements are not thoroughly understood. Crystal imperfections may be produced by plastic deformation or radiation, or may occur during the crystal growth. These imperfections can act as donor and acceptor centers. Experimental evidence suggests that lattice vacancies act as acceptors and i n t e r s t i t i a l atoms as donors. Dislocations produced by plastic deformation or other means act as acceptor centers. This characteristic is identified with the unpaired electrons on the edge of the extra atomic plane of a dislocation. These tend to trap free electrons and form "dangling bonds" (Read 1954). When these acceptor levels are ionized the dislocation acts like a line of negative charge and scattering of electrons by charged dislocations affects the mobility of carriers in the crystal (Read 1955). Lattice stress in the neighbourhood of a dislocation causes local distortion of the band edges (Chynoweth 1958). 6 1.3 Generation of e.m.f. in Semiconductors Concentrations of electrons and holes in thermal equilibrium can be calculated by integrating the product of the Fermi function and density of states over the range of available energies. If the Fermi level is located in the energy gap several kT from a band edge, the Fermi function can be approximated by a Boltzmann factor, giving the familiar non-degenerate form of the carrier concentra-tions. Carrier densities may be alternatively expressed as the product of intrinsic concentration and a Boltzmann factor in which the energy term involves the difference of Fermi level and electrostatic potential in the exponent (Shockley 1950). Semiconductor parameters such as effective mass, energy gap, and density of energy states are implicit in the intrinsic concentration provided the bands have a simple parabolic energy versus momentum dependence. If two regions of a semiconductor in thermal e q u i l i -brium have different carrier densities, there w i l l be a gradient of electrostatic potential or "inner electric f i e l d " between these regions. According to the principle of detailed balance, the d r i f t current associated with holes moving in this f i e l d is exactly balanced by an equal and opposite hole diffusion current, and the same current balanc-ing applies to electron currents. Thus the gradient of electrostatic potential in the crystal w i l l be the sum of any externally applied and inner electric fi e l d s . In the following pages, electrostatic potential i s related to 7 conditions in the c r y s t a l , and e.m.f. or voltage to externally observed or applied potential differences. If a semiconductor i n equilibrium has an e l e c t r o s t a t i c potential gradient, i t can e f f e c t a charge separation of injected c a r r i e r s which appears as an externally observable e.m.f. In general, e.m.f.'s w i l l e x i s t when a non-equilibrium c a r r i e r concentration i s present i n a region of e l e c t r o s t a t i c potential gradient. Non-equilibrium c a r r i e r concentrations are produced by o p t i c a l or thermal e x c i t a t i o n , p a r t i c l e bombardment, or i n j e c t i o n . E l e c t r o s t a t i c potential gradients ex i s t with impurity density gradients, energy gap variations, thermal gradient, or inhomogeneous magnetic f i e l d s through s p l i t t i n g of energy l e v e l s in the bands. Thus there are a considerable number of p o s s i b i l i t i e s of e.m.f. generation with various combinations of these two sets of conditions. Many of them, however, are n e g l i g i b l y small, or cannot be observed separately. For example, thermal gradients produce an e l e c t r o s t a t i c potential gradient and also a gradient in energy gap. The r e l a t i v e contributions of these to an e.m.f. may be calculated but the e f f e c t s cannot be separately observed. An e.m.f. from p a r t i c l e bombardment of a p-n junction has been observed, but for p a r t i c l e energies above a threshold value, a p a r t i c l e flux s u f f i c i e n t to produce measureable e.m.f.'s would seriously damage the c r y s t a l under observation. Since an e l e c t r o s t a t i c potential gradient i s always 8 associated with some spatial gradient of physical charac-t e r i s t i c s of the semiconductor, an e.m.f. arising from thermal gradient may be classified as a thermocouple effect. Conditions which give rise to electrostatic potential gradients also produce a change of carrier mobilities, and Tauc (1957J has shown that a change of mobility ratio over the region of non-equilibrium carrier concentration w i l l also cause an e.m.f. This effect, however, is negligible compared with the e.m.f. from the electrostatic potential gradient. Most of the photo e.m.f.'s have been observed and identified (Tauc 1957J» Of these we w i l l be chiefly concerned with the photo e.m.f. due to illumination of a region of impurity gradient. The bulk photo e.m.f. and the illuminated p-n junction may be considered as two extremes of this effect. In the case of the bulk photo e.m.f. the electrostatic potential gradient is gradual, and ele c t r i c a l neutrality may be assumed throughout the crystal, while in the p-n junction there is a space charge region associated with the abrupt impurity gradient, and the analyses of the photo e.m.f. in the two cases are quite different, even though they are fundamentally of the same origin. 9 C H A P T E R 2 - T H E O R Y O F B U L K P H O T O E.M.F.'S 2.1 General The equilibrium carrier concentrations in a non-degenerate semiconductor may be calculated by integrating the product of the density of energy states and the Boltzmann distribution over the range of available energies. Alterna-tively, we may define an electrostatic potential ^ Q which is related to the energy of an electron (or hole) at rest in the conduction (or valence) band. For reasons of symmetry we make V© the energy of a unit charge placed at the level at which the Fermi energy would occur in the intrinsic semiconductor in thermal equilibrium. Then the equilibrium carrier concentrations have the form y>(*) - & Subscript zero refers to equilibrium conditions. Non-equilibrium densities are expressed by replacing the common Fermi level 0 O by quasi-Fermi levels or imrefs 0n and 0 p. Since the non-equilibrium value of the electrostatic potential is dependent on the carrier distribution, i t is customary to consider i t equal to the equilibrium value at some unperturbed point, and to specify a l l other values in terms of this point. The positional dependence of the n,(x) - n,- e x p - j ^ q 10 non-equilibrium electrostatic potential i s not in general equal to that in the equilibrium case, and this is indicated by dropping the subscript zero. coinciding with the Fermi level which is uniform throughout the system. When the carrier concentrations are perturbed, for instance by illumination, the imrefs separate, and take up positions describing the new carrier densities. At unperturbed points outside the illuminated region, the imrefs return to the equilibrium Fermi level. If the region of non-equilibrium gives rise to a difference of Fermi level between two points a and b on either side of the disturbed region, this difference is equal to the integral of either iraref gradient between the two points, and may be identified with the voltage difference (or e.m.f.) observed between the points a and b (6hockley 1950). expressed in terms of the imref gradients. Current densities consist of a f i e l d induced or d r i f t component and a diffusion component involving the carrier density gradients. By substituting for the carrier density gradients using the above relationships, total current densities may be written In equilibrium the imrefs may be thought of as a Electron and hole current densities may also be J P - -q^J.pqrad 0 p J"n= - < J ^ n qrad fa where yUn and yU9 refer to and the Einstein relation has been assumed valid. D n hole d i f f u s i v i t i e s . electron and hole mobilities, A _ - - .hi and D n are the electron and 12 2.2 Imref Representations of Non-equilibrium Conditions A non-equilibrium condition in a semiconductor can be graphically represented by an imref and electro-static potential diagram. This w i l l specify carrier concentrations, current densities, existing e.m.f.'s, and in fact contains a complete physical description of the electrical properties of the material. In undertaking an analysis of a semiconductor problem, a qualitative imref diagram can be helpful as was f i r s t demonstrated by Shockley (1950) for the p-n junction. Such a diagram, without being quantitatively exact until the solution i s reached, can be consistent and provide information that is not always obvious. For instance, a qualitative imref representation can predict the polarity but not the value of an e.m.f. and the direction but not the magnitude of a current density. By appealing to fundamental principles, a set of conditions can be formulated to be used as a guide to consistent imref diagrams. These involve (i) the continuity equation, ( i i ) ratio of d r i f t to diffusion current densities, and ( i i i ) e l e c trical neutrality. (i) The continuity equations for electrons and holes are 13 where g and r are the rates per unit volume of generation and recombination of carrier pairs. If the time derivatives are zero and using one dimensional geometry these become One dimensional geometry w i l l be a valid assumption in the case of photo carrier excitation i f the specimen is thin enough that the carriers are generated uniformly throughout the bulk under the illuminated area. We may also make this assumption in a current carrying filament with low surface recombination. Regions where generation and recombination rates are different are easily identified, and the continuity equations enable us to relate these regions to positional change of the current densities and hence of the imrefs. Where the time derivatives are equal we may write J"n + J~P - T = constant This helps in estimating the relative gradients of the imrefs. In particular, i f the total current is everywhere zero, qrad4>p bn and where b is the ratio of electron to hole mobilities. 14 With one-dimensional geometry the gradients depend on a single coordinate only, and may be considered vectors lying in the x-axis. Thus the products or quotients of gradients are scalar quantities with positive values when the gradients are in the same direction and negative values when they are in opposite directions. ( i i ) The equations for total current densities consist of a d r i f t and a diffusion term. The ratios of these terms are drift J P " cj.rad f and / • i diff. Xh g r a d ( 0 h d r i f t T„ g r a d y If the current density is predominantly diffusional, the electrostatic potential gradient is approximately zero. When the d r i f t current density predominates the gradients of imrefs and electrostatic potential are equal. ( i i i ) We assume crystal neutrality everywhere except in a junction transition layer, and therefore the imrefs must be consistent with an equal disturbance of the densities of both signs of carriers. The product of non-equilibrium carrier concentrations establishes the relative magnitude of the electron and hole imrefs. h p - n? e x p 15 and hence where An and Ap are the added densities of electrons and holes. Shockley (1949) has used an approximate form of this equation to obtain an expression for the injected minority carrier density in a biassed p-n junction. He assumes the added carrier density negligible compared with the majority carrier density, and thus in n-type material kT Misawa (1955) gives the exact form for the injected minority carrier densities. Using the neutrality condition, for instance in n material so that p ( p + n . - p . ) = h? exp (ft - ft) or + kT (ft - ft) and the minority carrier density p i s the solution of the above quadratic equation. Fletcher (1957) expresses non-equilibrium carrier densities in terms of the junction electrostatic potential step. However, his identification of the applied bias voltage as the difference of the 16 equilibrium and non-equilibrium potential step is incorrect. A subsidiary condition is applicable when the generated or injected carrier densities are large compared with the equilibrium densities. In this case the imref gradients are inversely proportional to the ambipolar diffusion length. In extrinsic semiconductors where the added minority carrier densities are smaller than the equili-brium majority carrier density but large compared with the equilibrium minority density g r a d <j>n qrad ^ kT kT where LQ and L P are the electron and hole diffusion lengths (Appendix I). Facing Page 17 FIGURE 1 HOMOGENEOUS n-TYPE SEMICONDUCTOR LIGHT I I I * H 1 IMREFS AND ELECTROSTATIC POTENTIAL V ^^Xr >g g > r 1 I V n ! r > g ^ ^ g >r i n,p >fln-nr n CURRENT DENSITIES In n CARRIER CONCENTRATIONS 17 The imref diagrams (Figures 1 to 5) show how incident light modifies the imrefs and electrostatic potential, and can in some cases give rise to an e.m.f. These diagrams are qualitative and are primarily for purposes of i l l u s t r a t i o n . For this reason the junction transition regions have been greatly expanded to show the behaviour of the imrefs and electrostatic potential in these regions, and except for the TT-L-V junction (Figure 4) the relative carrier densities are unspecified. The notations TT and JJ represent very small impurity densities of the p and n type respectively. Thus TT and p materials are nearly intrinsic and the TT-L-v junction is a "weak" p-n junction. When a homogeneous semiconductor is illuminated (Figures 1 and 2) the imrefs separate and the electrostatic potential changes from i t s equilibrium value. The latter is caused by a difference in carrier mobilities. Electrons and holes generated in the illuminated volume diffuse away from the region of high concentration, and the electrons, having the greater mobility, w i l l tend to move ahead of the holes. Such a tendency sets up an electric f i e l d which slows the electrons and speeds up the holes in their motion away from the illuminated region. Facing Page 18 FIGURE 2 HOMOGENEOUS INTRINSIC SEMICONDUCTOR! LIGHT III 1 I IMREFS AND ELECTROSTATIC POTENTIAL CARRIER CONCENTRATIONS 18 This f i e l d is identified with a gradient of the electrostatic potential under Illumination (see Appendix II), and in a homogeneous semiconductor open circuited (T -o) [b-i) g rad cf>n ( b - l ) g r a d 0 p g r a d y = _kT 9 ^ An_ Thus for a given illumination, the gradient of electrostatic potential w i l l be most pronounced in an intrinsic semi-conductor (Figure 2) where (b-l) kT qrad An In n-type (Figure 1) g r a d y = - ( b ~ i ) 9 r a d i & In the illuminated region where generation exceeds recombination, the space derivative of hole current density is positive and that of electron current density negative in accordance with condition ( i ) . The opposite holds where recombination is greater than generation. In homogeneous illuminated semiconductors the current densities are symmetrical about the mid-point of illumination, provided that surface conditions are uniform and light intensity is constant over the illuminated regions. Facing Page 19 FIGURE 3 p-n JUNCTION (OPEN CIRCUIT) CARRIER CONCENTRATIONS 19 In the illuminated p-n junction (Figure 3) carriers generated in the transition region separate because of the inner electric f i e l d and a photo e.m.f. is observed. The added majority carrier density is usually small compared with the equilibrium density and diffusional effects cause a negligible perturbation of the electrostatic potential. This is illustrated in Figure 3, in the carrier concentra-tion and imref diagrams. Total current densities are zero everywhere, but the current densities are not necessarily symmetrical about the x - 0 plane, since diffusion lengths are normally not equal in the p and n materials. Facing Page 2o FIGURE 4 IT-'i-jJ JUNCTION (OPEN CIRCUIT) LIGHT + I \ \ \ J 1 Photo emf X. r / X — — — — 1 / i ^ — i x, 0 (nG=2 IMREFS AND ELECTROSTATIC POTENTIAL 1 1 lg>r j /! x V\\\ \ jg>r, CURRENT DENSITIES CARRIER CONCENTRATIONS 20 The weak p-n ( or TC-i - X> ) junction (Figure 4) shows a considerable perturbation of the electrostatic potential under illumination, and these perturbations are not equal on the two sides of the junction. Thus the resulting photo e?.m.f. is partly diffusional in origin. The differences of imrefs and electrostatic potential at the edge of the transition layer have been drawn to corres-pond to equilibrium hole density in the TT side equal to twice the electron density, and an equilibrium electron density in the P side equal to twice that of the holes. Although the TT and O materials are very nearly in t r i n s i c there is a marked asymmetry of imrefs on either side of the junction. The diffusion lengths w i l l be approximately equal, and consequently the current densities are symmetrical about x » 0. Facing Page 21 FIGURE 5 BULK INHOMOGENEITY (OPEN CIRCUIT) LIGHT IMREFS AND ELECTROSTATIC POTENTIAL A . ' g > r ! 7 v / V Jn\ i /|\ [ XJp • g > r f J R v CURRENT DENSITIES CARRIER CONCENTRATIONS 21 The bulk photo e.m.f. arises from illumination of a gradual impurity density gradient (Figure 5). The diagram is drawn for n material and the chief component of e.m.f. arises from the difference of electrostatic potential at the illumination edges. As light intensity increases, the electrostatic potential gradient in the illuminated region approaches zero. However, the d i f f u -sional perturbations of e.m.f. on either side of the illuminated region are unequal, and the maximum e.m.f. w i l l always be less than the difference between the values of equilibrium electrostatic potential at the illumination edges. Since we assume the impurity gradient i s gradual, the diffusion lengths on either side of the illuminated region are approximately equal and the current densities are symmetrical about x = 0. 22 2.3 Analysis of the Bulk e.m.f. A theoretical analysis of the e.m.f. arising from illumination of an impurity gradient in a semiconductor was f i r s t published by Tauc (1955). Briefly, his derivation of an expression for the bulk photo e.m.f. involves integrating the gradient of electrostatic potential between two unperturbed points on either side of the illuminated region. This yields an expression for the bulk photo e.m.f. provided that the equilibrium electrostatic potential is the same at the integration limits. We have followed Tauc in making six basic assump-tions and these are included below for c r i t i c a l examination. (a) Electrical neutrality is assumed for both illuminated and equilibrium conditions, except in regions of abrupt impurity density gradients such as occur in p-n junctions. It should be emphasized that this assumption represents only a close approximation to elec t r i c a l condi-tions in the crystal since a tendency towards charge separation i s the agency which sets up opposing inner fields, i.e., electrostatic potential gradients. However, except in semiconductor, junctions, space charge effects are if 23 negligible and the crystal may be assumed e l e c t r i c a l l y neutral. (Gunn 1958J. The postulate of neutrality implies ambipolar carrier mobility (Rittner 1954) which i s automatically taken into account in the analysis of the bulk photo e.m.f. and gives rise to factors involving the mobility ratio. (b) Impurity atoms are assumed totally ionized. This assumption simplifies the mathematical treatment but i t s validity depends both on the semiconductor and the impurity. In germanium the activation energies of most impurities are small enough that ionization i s essentially complete, and this assumption is valid in our work. (c ) Classical s t a t i s t i c s are assumed in both the dark and illuminated regions. The most intense illumination used in the experimental studies was not sufficient to approach degeneracy since the added carrier density was 16 —3 estimated never to have exceeded 10 cm . (d) Ratio of electron to hole mobility is assumed constant. This assumption holds for a l l illuminations used since hole-electron scattering would not be significant. (e) Uniform generation of photo carriers is assumed across the bulk of the semiconductor. Using this assumption, the theoretical analysis becomes one-dimensional and gradients depend on a single coordinate only. This is not the case experimentally, but from the data i t would appear that this assumption i s not particularly restrictive. (f) Added carrier concentration is assumed zero Facing Page 24 FIGURE 6 ASSUMED CARRIER DISTRIBUTION IN AN ILLUMINATED BULK INHOMOGENEITY f * P i I * a a' F——. 3 1 ™ 1 1 b' b i t 1 ,i l li Agrad An 1 i , 24 outside a region which is ordinarily identified with the illuminated region, and is assumed constant within the illuminated region. This assumption simplifies the analysis of the bulk photo e.m.f., and i t s amplications are discussed in detail at the end of this section. (g) An additional assumption i s implicit in Tauc's work but is not stated, i.e., carrier lifetime is assumed constant. This holds for weak illuminations, but for large added carrier density lifetime becomes a function of carrier concentration. This implies that the excess carrier density due to illumination w i l l not be a linear function of the illumination. The e.m.f. which exists through illumination of a region of impurity density gradient may be expressed either in terms of the disturbance of the carrier imrefs or electrostatic potential. This is illustrated in Figure 6 for the case of a bulk inhomogeneity. When a region of inhomogeneity is illuminated between the points a' and b', the bulk photo voltage e B between unperturbed points a and b i s u ,b e B = /grad£, dx = / g r a d ^ d x = / qrad(f - dx a q a These integrals are entirely general, and apply wherever an e.m.f. arises from a non-equilibrium carrier density in a region of electrostatic potential gradient. However for 25 physical reasons their evaluation i s not always possible. In the p-n junction for instance, el e c t r i c a l neutrality does not hold in the transition region where the electric f i e l d i s non-uniform, and thus the integral involving this quantity cannot be simply evaluated. In the follow-ing we w i l l derive an expression for the bulk photo e.m.f. using the integrals of the electrostatic potential gradient, and also using the integral of the imref gradient. The former is essentially Tauc's (1957) method, although the approach is somewhat different, and this w i l l be evaluated f i r s t . By differentiating the expressions for non-equilibrium carrier concentrations, the imref gradients may be written k T f a d p grad fa = grad y +- — g r a d fa = g r a d y - ^- ^ _ — -In the open circuit condition, total current is zero and g r a d fa = _ bn g r a d fa P Dividing the above two equations and substituting for the ratio of imref gradients kT b 9 r a d n - g r a d p qrad y/ - — - ;  ^ v % bn t p 26 In equilibrium qrad Y o - - —r^— and from these, assuming c j r a d i n = qradhp b cjrad(v/-V4) d x a (b-/)qrad An f b + l U ^ q r a d ne dx q The integrands exist only where an added carrier concentration combined with an internal f i e l d exists or where a gradient of added carrier concentration exists, and this must be taken into consideration when the integrals are evaluated. The integration limits are arb i t r a r i l y chosen as two unperturbed points a and b on either side of the illuminated region. The equilibrium carrier densities at these points are not, however, involved in the solution since An and grad Au are zero there. Thus the equilibrium carrier densities which appear in the solution w i l l be the densities at the edges of illumination. The term involving the gradient of added carrier concentration i s identified with e^, the diffusional component of the e.m.f., and the term involving the gradients of the equilibrium carrier densities with e c, the internal f i e l d component. The integrals when evaluated (see Appendix III) give • • ' ( b + i ) M t - j - ~7T A ~ 111 and 2 k T ft> + i U n t when D is positive kT (b + AAn, T a n —i 2bna + An L(b+i) a* e c - - 1 V~D 2bn0 + ( b + 0 A n L - V : ^ 2bne t(b+i)An L T V - D b' when D is negative where D - 4 D * b - (b + lt/Al\ where A f i L , a constant, is the added carrier concentration within the illuminated region. The above expressions are simplified (see Appendix III) in the case of 28 (i) weak illumination (Ar\L « p„ + bn. ) n. In n - t y p e (bn 0>>p e] e s = e d + e c = - i ^ A n L ^ ^ in p - t y p e (p B >> bn0) ( i i ) strong illumination (A\)L > > p ft + brC) e 8 = ed+-.ec = JtT J_ % b + l In n in n - t y p e o*' These expressions can also be derived by applying the appropriate approximations before the integral is evaluated, and the integration i s much simpler in this case. The complete integral however, involves no assump-tions as to the ratio r,'//po and provides a convenient check on the derivation of the e.m.f. through the imrefs. Derivation of an expression for the bulk photo e.m.f. using the imrefs involves the same general approach used above. 29 b _ b and the imref gradients are expressed in terms of carrier densities and their gradients (see Appendix IV) , . kT i g rad (rip) grad 0 n - - — -bn + p g r a d <pp « — - — — — r ^ °, P b n + p under open circuit conditions. These are expanded and expressed as the sum of terms involving gradient of added carrier density and gradient of equilibrium carrier densities. The integration between limits a and b gives (see Appendix IV) XL 1 n rw + An, n ob' n0*' n o b. + AhL 4-kT b-l n | + (b + . ) A n / p o ^ b n o f t ,  + Z p . b ' + b r U ' Jsl In •V n o k ' +-Ai\ T a n - i 2bn0 + An t (b + 0 D 30 where D - 4 - n - b - (b + i)*An* In the case of the bulk photo e„m.f., the f i r s t terms of the diffusional and inner f i e l d components of e.m0f. cancel giving an expression for the bulk photo e.m.f. identical with that derived from the integral of the electrostatic potential gradients. The same fin a l solution i s also obtained from an integration of the hole imref over the illuminated region. In the gradual bulk inhomogeneity the electro-static potential is a slowly varying function of position and the approximations br),,^ p0 in n-type material and p„ >> bn o in p-type material may be introduced in the integrand when the photo e.m0f. i s derived using the electrostatic potential gradient,. The imref gradient however, is a rapidly varying non-monotonic function with opposite and almost equal excursions near the edges of illumination and the above approximations are not valid when applied to the integrand. In both derivations the analysis is simplified by assumption ( f ) , added carrier density i s zero outside the illuminated region and constant within i t (Figure 6 ) 0 This implies the following approximation: gradient of added carrier density i s small compared with gradient of equilibrium carrier density in the illuminated region, and at the edges the equilibrium carrier densities do not 31 change over the region of significant added carrier density gradient. Using this approximation, the e.m.f. integral i s separated into two terms involving different variables, n 0 inside the illuminated region and An at the edges, and these terms are identified with the inner f i e l d and diffusional components of e.m.f. respectively. However, when the width of the illuminated region is of the order of a diffusion length, the actual added carrier density distribution w i l l be quite different from the assumed rectangular one, and n Q may change appreciably over the region of significant grad A n . The validity of assumption (f) and the form of the e.m.f. integrals w i l l be examined under these conditions. When a semiconductor i s uniformly illuminated between points -a and a, the added carrier densities are (see Appendix V) G t r • - 2 a .. a - x A n = I - e x p — exp L (X >a) An- ^ - e x p -a a e x p T7 A n = Gt„ - e xp ~ cosh j ( - a < x < a ) where G is the rate of photo generation of carrier pairs per unit volume between points a and -a, Tp and L p are hole lifetimes and diffusion lengths. Facing Page 32 FIGURE 7 CARRIER DISTRIBUTIONS IN AN ILLUMINATED HOMOGENEOUS SEMICONDUCTOR CARRIER CONCENTRATION GRADIENTS (normalized) 32 Added carrier density and i t s gradient are plotted in Figure 7 for a ~ L p and a = /oLp . Under assumption (f) the gradient of added carrier density i s a delta function at the illumination edges, whereas Figure 7 shows that in reality a significant gradient of added carrier density exists over the smaller of 2 L p or L p + a at each edge of illumination. The integral may s t i l l be evaluated by the methods shown in Appendix III, but the equilibrium densities in the integrand are not those at the illumination edges, but are the weighted average of the individual densities over the region of significant grad An • Hence assumption (f) introduces negligible errors i f L, grad (In n0) < / The above equations and Figure 7 show that a significant added carrier concentration extends over the illuminated region plus a diffusion length on either side, i.e., over 2(q. + Lp) . Since the bulk photo e.m.f. arises from added carrier density and i t s gradient, the effectively illuminated region extends a diffusion length beyond the region where light is incident. In the inner f i e l d term, the integral exists over the region of significant added carrier concentration, and the integral may be evaluated by the methods of Appendix 33 III i f At)L i s considered the average added carrier density over the region of significant An . From these considerations, the form of the bulk photo e„m 0f. expression derived under assumption (f) i s approximately valid even though the carrier distribution is quite different from that implied by assumption (f)„ The bulk photo e.m.f. i s a function of the equilibrium conductivi-ties at the edge of the region of significant A h , and the incident light may be considered an experimental probe which samples the local conductivity variations. Figure 7 illustrates that the minimum sampling diameter with a vanishingly small illuminated region i s two diffusion lengths. Consequently i f the bulk photo e.m.f. i s used to investigate crystal impurity gradients, i t w i l l show only the variations which occur over the order of two diffusion lengths. Finer variations in impurity may be observed by decreasing the effective lifetime and diffusion length, for instance by surface treatment, but this w i l l correspondingly decrease the net added carrier concentration and photo e.m.f. at a given light intensity; i t would also make the recombination mechanism preponderantly s i g n i f i -cant at the surface compared with the bulk, and the photocarrier distribution would be markedly inhomogeneous across the filament normal to the illuminated surface. 34 2.4 Photo e.m.f. at an Illuminated Junction The photo e.m0f.'s arising from illumination of a bulk inhomogeneity and of a p-n junction represent two extremes of the same effect. However, the behaviour of the gradients of imrefs and electrostatic potential in the p-n junction differs from that in the bulk inhomo-geneity. In the p-n junction the imref i s the slowly varying quantity while the electrostatic potential has i t s maximum change in the transition layer. Electrical neutrality does not hold in this region and although e = / qrad (y~Y0) dx is valid, the approximate expression kT q r a d ( y - y o ) = ^ (b+l^qrad An (b + l) An g rad n c p o + bn 0 +(b+i)An bn,2 + An(b+i)n0 + used previously to derive the bulk photo e.m.f. is not applicable in the illuminated junction since i t assumes A n - A p and grad An • grad Ap . The integral of the imref gradient can be evaluated to a close approximation however, since the change of imref is very nearly zero across the transition region (Fig. 3). b „ a' ^ b' _ b g r a d 6 dx and 35 b' where a and b are two unperturbed points in the p and n regions and a' and b' mark the edges of the transition layer. This approximation is used by Shockley (1949) in his analysis of the biassed p-n junction, and is also jus t i f i e d in the illuminated p-n junction. Assuming the impurity density has an abrupt change of type at a given plane, the width of the transi-tion layer W may be calculated and is characteristically _4 of the order of 10 centimeters wide for impurity density to potential step ratio of 1015cm"3V~1. If the imref kT slope in the transition region approached — — , the strong illumination maximum slope of the minority carrier imref, (see Appendix I), this would correspond to an imref change of no more than - ^  — „ The diffusion length L a w i l l ordinarily be of the order of 0.1 millimeters which is small compared to W, and thus -^TW wi l l be small % L * kT compared with a typical weak illumination e.m.f. of or more. In addition to this, the continuity equations show that the space derivatives of the carrier currents change sign at the edges of illumination, and the carrier currents are zero at one point inside the illuminated region (Figure 3). Consequently the imref slopes are smaller inside 36 the illuminated regions than at the edges, and i f the illumination is confined to the transition layer the slopes are zero at one point inside i t 0 These considera-tions show that the assumption of zero imref change across the transition region is well ju s t i f i e d , and the p-n junction photo e.ra.f„ is - b e r = / q r a d fa p dx -f-/ c j r a d fa>? d x a -yb-Assuming the p and n regions are homogeneous, the integral of the electron imref gradient gives (see Appendix IV), ___ _kT j n n o p -f-(AiV)p n o n + ( b - l ) _kT | /Pop+ b r w ( W l ) S , , (b + \)(Ar>L)/ i 1 r 7 p . n + b n o n Where subscript p and n refer to carrier densities in the p and n materials and A l \ is the added carrier concentra-tion at the edges of illumination, i.e., at the boundaries of the transition layer. Integration of the hole imref between these same limits must give an identical expression for the photo e.m.f., and this provides a convenient check on the derivation. Using the hole imref 37 = kl_ / N POP + (An t) <S Pop p., + (An,.), -f Since the two expressions have the same second term, the f i r s t terms must be equal, and this can be shown to follow from the assumption of zero imref change across the transition layer. When the imref change is zero, the differences of imrefs are equal at the transition boundaries, i.e., kT i n. Pr where n n, p n and n p, p p are the carrier concentrations at the boundaries of the transition layer. Thus or 9 p., n f kT | poP kT | n n.„ + ( A O , 38 and since •on Pc n, 'op 'on the f i r s t terms of the two expressions for the e 0m„f. are equal. different in the p and n regions the imrefs w i l l tend to separate less in the region of low lifetime. This produces a sharp gradient of imrefs across the transition layer which corresponds to a large flux of carriers of both signs into the low lifetime side, maintaining the separation of the imrefs essentially equal at both edges of the transi-tion layer. Thus the relative magnitudes of the carrier and are dependent on the equilibrium majority carrier densities in the material on either side of the junction (see Appendix VI). with no restrictions as to the type of material on either side of the junctions, and can be applied to any type of junction where zero imref change across the transition layer may be assumed. It should be noted that zero imref change has been used to imply equal differences of imrefs at the transition boundaries, which is a less restrictive condition, and the derivation w i l l be valid so long as the imref changes across the transition layer are equal or zero. With weak illumination the added carrier densities are negligible compared with the equilibrium majority carrier If the carrier lifetimes are substantially concentrations The above expressions for photo e.m.f. were derived 39 densities, and the p-n junction photo e.m.f. becomes n op 1 in Pan and the photo e.m.f. i s proportional to log light intensity when (-An,) l i e s between the majority and minority carrier densities. At strong illumination the photo e.m.f. satur-ates to kT Pon + bnot, + 7. r -ir The second term of the saturation e.m.f. involves the log of the ratios of conductivities of the material on either side of the junction, and w i l l ordinarily be small compared with the f i r s t term. The saturation e.m„f0 i s then approxi-mately equal to the equilibrium electrostatic potential step at the junction. The weak illumination photo e.m.f. in a n-n+ junction becomes KT (An L) I e r b on which is equivalent to the result obtained from illumination of a bulk inhomogeneity (see section 2.3J when h o n + > > Pi o n These expressions for the photo e.m.f. at illuminated junctions hold for an arbitrary concentration of photo generated carriers. The form of the e.m.f. expression i s the same as that derived by Cummerow (1954) for very weak 40 illumination (At1 L <^ ftop , p o n J where the e.m.f. is proportional to light intensity, and for weak illumination ^ n o p < A / 1 L < pop) where the e.m.f. is proportional to log light intensity. Cummerow's treatment however, does not show saturation e.m.f. at strong illumination. In a p-n junction, the second term of the e.m.f. expression w i l l be very small since the conductivities of the p and n materials w i l l ordinarily be of the same order of magnitude. In this case the only significant contribution to the e.m.f. arises from the f i r s t term, which is identical with an expression derived by Fan (1948). 41 2.5 Bulk Photo e.m.f. in a Near Intrinsic Semiconductor Assuming complete impurity ionization and electrical neutrality, the equilibrium carrier densities in a semi-conductor may be written in terms of intrinsic carrier concentration, n^, and the net positive impurity ion density, ND"*NA ( g e e section 1.1). n 0 = + n-No - N , We consider a semiconductor to be near intrinsic when n; » , and in this case the equilibrium carrier concentrations are approximately The analysis of the bulk photo e.m.f. in section 2.3 gives a general expression for the e.m.f. in terms of the added carrier concentration in the illuminated region and the equilibrium carrier concentrations at the edges. The e.m.f. in near intrinsic material is obtained by substituting the above.carrier densities into the general expression. If the semiconductor is illuminated between points a' and b' and 42 assumptions (a) to (g) used in the general treatment are valid, the e.m.f. between unperturbed points a and b i s 8 (b + iT H L n ; for weak illumination (AnL << fyj and for strong illumination ^ A M L > > D T ^ 43 2.6 Ratio of Bulk Photo e.m.f, to Photoconductive Decrease of Resistance Illumination of a semiconductor filament increases the number of carriers available for conduction and thus decreases the filament resistance. In the following this photo conductive resistance decrease A R w i l l be referred to as the photo resistance, and is equal to the filament resistance in the dark minus the illuminated resistance. If the illuminated region has an impurity gradient, the added carrier density w i l l also give rise to a bulk photo e.m.f. Both these effects originate from the same carrier distribu-tion and their ratio does not contain the added carrier density. This gives a convenient method of relating the bulk photo e.m.f. to the semiconductor impurity gradient without knowledge of the illumination intensity in terms of the number of effective photons absorbed per second. For weak illumination, the bulk photo e.m.f. in an n-type semiconductor may be written (see section 2.3) ru. n, lob ' " o a 1 where At)L is the added carrier concentration in the illuminated region, and nob< and Doa. are the equilibrium carrier densities at the illumination edges. When the filament is uniformly illuminated producing a constant added carrier density throughout the volume under 44 the illuminated area, AR = RG 'L where RQ is the equilibrium dark resistance, A is the crossectional area, and the integration is over the entire length, L, of the filament. A CJ , the conductivity increase due to illumination, is assumed small compared with O~0 , the equilibrium conductivity. Thus AR - - J - ^ + ^ A n . / ^ a' The integration limits are the edges of illumination since A n t is zero elsewhere and c r 2 becomes OJ L z , the average equilibrium conductivity over the illuminated region. The ratio of bulk photo e.m.f. to photo resistance becomes, e B 2 A kT cr0ck, - C E V AR = b+l S b'-a' The illuminated region acts as an experimental probe which samples the equilibrium conductivities at the illumina-tion edges a' and b' (see section 2„3j. If the illuminated region is sufficiently narrow that the equilibrium conductivity fluctuations may be regarded as monotonic within i t , then the ratio is approximately e B , _ _2A_ kT d o ; AR = b + l % dx In impurity semiconductors the conductivity i s very closely proportional to the impurity density, and the bulk photo e.m.f. may be expressed in terms of the conductivity gradient. However, conductivity has a minimum where and in near intrinsic material the ratio of bulk photo e.m.f. to photo resistance must be expressed in terms of the impurity density gradient. Using the approximations and results of section 2.5 e B _ 2 kT/\b _ d _ ^ D - ^ ) AR x M b + i) The ratio of bulk photo voltage to photo resis-tance was derived assuming a uniform added carrier density throughout the volume under the illuminated area, and this brings the filament crossectional area into the expression for photo resistance. Consequently, when the ratio is used to estimate the conductivity gradient, the condition of uniform added carrier density must be closely approxi-mated. With a non-uniform added carrier concentration, the effective crossectional area is unknown and a quantitative estimate of the conductivity gradient cannot be obtained. The photo resistance i s linear with light intensity for weak illumination A d << . As the light intensity i s increased the illuminated region approaches zero resistance and the increase in the photo resistance is predominantly due to carrier diffusion away from the 46 illumination edges. As a result the photo resistance i s proportional to log light intensity with strong illumina-tion since the photo resistance there depends chiefly on diffusional spread of carriers and i s insensitive to impurity gradients. The bulk photo e.m.f. i s linear with light intensity for weak illumination. However, at strong illumination, the e.m.f. component in the illuminated region may reach saturation, and i f the added carrier density is increased further, any e.m.f. change must arise from carriers that have diffused far enough to arrive in an unsaturated region. This has the effect of increasing the region of significant added carrier concentration beyond 2(cl-fLa^ (see section 2.3). If the region outside the area of illumination has a r e s i s t i v i t y gradient opposite to that in the illuminated region, the net effect i s a lowering of the observed "saturation" e.m.f. On the other hand, i f the impurity gradient increases monotonically along the filament, the strong illumination increase of the region of significant added carrier concentration w i l l give rise to e.m.f. components of the same polarity as the saturated component, and the bulk photo e.m.f. w i l l increase monotonically with strong illumination. In the f i r s t case, the behaviour of the strong illumination photo e.m.f. i s unlike the photo resistance behaviour and the ratio of these quantities is constant at weak illumination only. In the second case, the photo e.m.f. increase with strong illumination depends 47 entirely on diffusional carrier spreading, and the ratio of photo e.m.f. to photo resistance i s constant at a l l light intensities. 48 C H A P T E R III - P R E P A R A T O R Y E X P E R I M E N T S 3.1 Filament Preparation The germanium filaments used for photo e.m.f. observations were cut from monocrystalline ingots using a reciprocating wire saw. Typical filaments had a rectangular crossection of ~ i/nm. and a length of ^  1.5 cm . After the filaments were cut and cleaned, the ends were sandblasted and electroplated to obtain a metallic contact to the material (see section 3.2). The filament was then heated in an inert atmosphere, tin was melted on the plated areas and 0.010 inch diameter wire leads were attached. The filaments were mounted by securing the leads in phosphor bronze clips which were bolted on a plastic holder. The clips served as terminals by which the filament was connected to the measuring ci r c u i t s . If the photo e.m.f. is used to investigate satura-tion photo effects, a long carrier lifetime is desirable, while investigation of filament conductivity variations requires a short lifetime and carrier diffusion length (section 2.3). With a I mm crossection, the effective carrier lifetime in the filaments was strongly dependent on surface conditions through the surface recombination velocity. The filament effective decay constant (reciprocal of lifetime) is the sum of a bulk and a surface decay constant (Shockley 1950). 49 and where s i s the surface recombination velocity (cm./sec), 2B and 2C are the crossectional dimensions of the filament and D a is ambipolar dif f u s i v i t y . High surface recombination velocity and short effective lifetimes were obtained by sandblasting or polishing the filament surfaces and the surface recombination velocity was reduced by etching the filament in a 3% solution of hydrogen peroxide. Hydrogen peroxide has the advantage of being easier to handle than the fluoride etchants, and does not attack the metallic contacts. 50 3.2 Metal-Semiconductor Contacts Metal to semiconductor contacts in general show some degree of rectification depending on the method of contact fabrication. When an n-semiconductor filament is heated and a metal wire alloyed with an acceptor element melted onto i t , a p-n junction i s formed at the contact. If the wire is alloyed with a donor element, a low-high (L-H) n-n+ junction is formed. The p-n junction i s unsuitable as a contact since i t s reverse resistance i s ordinarily large compared with a typical filament resistance and because i t is photo sensitive and can cause injection effects. The L-H junction has negligible resistance for both directions of current flow and provides an ele c t r i c a l l y "smooth" metal to extrinsic semiconductor transition. The alloying technique however, i s not particularly successful in contacts too near intrinsic material. In this work we were not primarily concerned with the characteristics of metal to semiconductor contacts, which were considered acceptable i f their resistance change with a moderate current density increase was small compared with the resistance of filament plus contact. A one per cent overall resistance change with a current density of plus or minus one Q.mp/cm2 was chosen as a tolerable upper limit of non-ohmic behaviour, and an attempt was made to achieve this figure by using soldered or electroplated contacts. Facing Page 51 FIGURE 8 DEPENDENCE OF FILAMENT PLUS CONTACT RESISTANCE ON CURRENT DENSITY 3 + O a> o c 2 + IS) o O a> o c o £ 2 in a) a: 3 + ° 40 cm, n- type * 10 ft-cm, n- type • 3 & - c m , p-type J (amps/cm z) 51 Each solder or electroplate was given two t r i a l s , once on a sandblasted, and on an etched surface since these correspond to the two extremes of surface condition at the contact. The soldering t r i a l s were made using pure tin, pure lead, antimony-doped t i n , antimony-doped lead, and both pure and doped 50-50 solder. These contacts gave non-ohmic behaviour which ranged from five to twenty per cent resis-tance change with one amp./cm. current density increase. Plated contacts gave more satisfactory results. Nickel, copper and rhodium plating was tried, with various temperatures, plating current densities and G@ surface conditions. Optimum contact characteristics were obtained using a rhodium plate (Weisberg 1953) on a sandblasted surface 2 at room temperature and 20 ma/cm. plating current density. This gave a resistance change of one to two per cent at 1 2 amp./cm. with the n-type and near intrinsic materials (Fig. 8). With the 3 ohm-cm p-type, rhodium plate on a sand-blasted surface gave less than 0.2% resistance change up to 2 15 amp./cm. . Plating on etched surfaces gave markedly non-ohmic contact with n and near-intrinsic material, but in the p-type was not noticeably different from results obtained by plating over a sand blasted surface. These observations are in agreement with those of Borneman (1955). The conductivity of germanium in the intrinsic range is a rapidly increasing function of temperature, and with the near intrinsic filaments at room temperature, as much as five per cent filament resistance change could result from one degree Centigrade 52 temperature change. Hence the measurements of contact linearity were made with an ambient temperature constant to better than 0.1 degrees Centigrade and to avoid e l e c t r i -cal heating the filament resistance was measured on a wheatstone bridge using 60sec. pulses at a 30 pps repeti-tion rate as the bridge current source. The bridge balance point was obtained using an oscilloscope with a differential input amplifier. The departure from linearity of the resistance versus current density plot indicated the quality of the contact; i f the variation was symmetrical the indication was that the contacts were e l e c t r i c a l l y equivalent and would be reproducible. When asymmetry was observed, the implication was that the contacts under study would not be closely reproducible. Plated Cu3Ge (Fink 1948) was also given a t r i a l as a contact to germanium filaments. This compound has the electrical characteristics of a metal, and gives a bright silver coloured deposit that i s inert to concentrated n i t r i c or sulphuric acid. In this respect i t s chemical properties are similar to germanium. Considering the characteristics of this material i t seemed plausible that i t might provide a smooth electrical transition from metal to semiconductor. The'electrical characteristics of the plated contacts of Cu3Ge were acceptable but not better than the rhodium contacts. In the course of a few months the Cu^Ge plated on germanium lost i t s lustre and darkened, while a deposit on a metal backing remained bright and untarnished. This behaviour was a further reason for rejecting the compound as a contact material. 53 3 . 3 Filament Conductivities A semiconductor in the intrinsic range has an equal number of carriers of both signs, and the conductivity is The intrinsic concentration n^ and the mobilities are both temperature dependent. In germanium at temperatures from 100°K to 500°K the electron mobility i s proportional to T " 1 , 6 6 and the hole mobility to T ~ 2 , 3 3 (Morin 1 9 5 4 ) . The variation of these quantities with temperatures is slow compared with that of the intrinsic concentration -r 3/2 £3 Y)Lcc T e x p - ^ j r and thus to a close approximation cr = C exp 9 2kT A plot of log C T versus 1/.J. gives a straight line in the intrinsic range of conductivity, from which the energy gap at absolute zero, Eg, is determined. At lower temperatures, the intrinsic concentration becomes negligible compared with carriers arising from impurity centers, and i f the impurities may be assumed totally ionized (see section 1 . 1 ) the conductivities are CT= ^ / I N ^ - N A ) in n-type CT = <^^?[NA- NO) in p-type Facing Page 54 FIGURE 9 CONDUCTIVITY TEMPERATURE DEPENDENCE OF THE GERMANIUM FILAMENTS 54 Using published data on mobility these extrinsic conductivi-ties give an estimate of (Np - N^). The germanium filaments used in this work were cut from ingots of 3 ohm-cm p-type, 10 and 40 ohm-cm n-type, and from a near intrinsic ingot with residual n impurities. Filaments were also taken from a "graded" n ingot having a monotonic r e s i s t i v i t y gradient from approximately 3 ohm-cm to 30 ohm-cm over a 5 cm. length. The conductivity temperature dependence of these materials are plotted in Figure 9. The graded specimen is not included because of i t s extreme inhomogeneity. The filaments were glued in turn to a ther-mometer bulb and heated in an o i l bath to 150°C. Resistance and temperature readings were taken simultaneously as the bath cooled to room temperature. For low temperatures a copper-constantan thermocouple was attached to one terminal of the filament. The filament was immersed in petroleum ether cooled to liquid nitrogen temperature and resistance and temperature readings were taken simultaneously as the bath warmed to room temperature. With this relatively simple experimental set-up, the values of energy gap obtained were close to the accepted value of 0.785 ev. (Morin 1954) and this leads us to place considerable confidence in our values of (N D - N A). 55 Table I Material Resistivity at 20°C (ohm-cm) Apparent Eg (ev) < % " NA> (cm - 3) P 3.1 0.783 1.14 X 10 1 5 n 9.1 0.725 1.75 X 10 1 4 n 40.0 0.750 2.87 X 10 1 3 near intrinsic 56.0 13 approx. 10 Close agreement with the accepted value of germanium zero-temperature energy gap was obtained with the p specimen confirming the ohmic quality of the contacts to this material. Facing Page 56 FIGURE 10 SPECTRAL DISTRIBUTION OF RADIATION FROM A ZIRCONIUM CONCENTRATED-ARC LAMP 56 3.4 Apparatus When a germanium filament i s illuminated with a small light spot in a region of impurity density gradient, a bulk photo e.m.f. i s observed between the end contacts. If the spot is scanned over the length of the filament and the voltage plotted as a function of position, the e.m.f. pattern obtained is characteristic of the filament and i s related to i t s impurity distribution. Two light sources, a one hundred watt "Point-o-Lite", and a twenty-five watt zirconium arc lamp were used for developing the bulk photo e.m.f. The point-o-lite is essentially a black body radiator while the zirconium arc has a continuous spectrum and many very intense lines. The spectral distribution of the zxrconium arc as supplied by the manufacturer i s shown in Figure 10. The zirconium arc was a more intense source than the point-o-lite, and was used for most of the photo e.m.f. work. Both sources produced intense spots of approximately 0.1 inch diameter with the Point-o-Lite, and 0.03 inch diameter with the zirconium arc. These were demagnified using a two lens system to 0.1 millimeter diameter or larger at the germanium filament surface. The filament on Its plastic holder was bolted to a small carriage which moved vertically in a brass track, and the second lens of the optical system was mounted ri g i d l y in front of i t (Figure 11j. Adjusting screws were provided so that the carriage and track could be moved in a horizontal plane parallel or perpendicular to the direction of the Facing Page 57 FIGURE 11 SCANNING MECHANISM AND FILAMENT HOLDER 57 light. These were used to position and focus the light spot on the filament. The carriage was driven by a reversibl synchronous clock motor, and the e.m.f. was recorded on a 0-10 mv recording potentiometer as the light spot scanned the filament. Typical e.m.f. patterns obtained in this way are reproduced in Figure 12, section 3.5. For scanning at temperatures higher than room temperature the whole assembly was placed in an insulated box and the light beam was brought in through a lucite window The temperature inside the box was maintained constant within 1°C using an electronic thermal regulator. With this arrangement the change of e.m.f. pattern with temperature was observed up to 70°C. Facing Page 58 F I G U R E 12 DEPENDENCE OF BULK PHOTO E . M . F . PATTERNS ON F I L A M E N T SURFACE CONDITIONS r-3 P O L I S H E D S U R F A C E H E A V I L Y ETCHED SURFACE 58 3.5 Identification of the Photo e.m.f. as a Bulk Effect The magnitude and pattern of the positional variation of the photo e.m.f. observed with a given filament changed markedly with different surface conditions. This suggested the possibility that in some cases the observed photo e.m.f. originated from surface rather than bulk irregularities. To identify the origin of the effect, the photo e.m.f. was examined as a function of illuminating position with surface conditions varying from the. extremes of sand-blasted (high recombination velocity) to heavily etched (low recombination velocity). A light spot was scanned along a five centimeter long specimen and positional Variation of photo e.m.f. was recorded. Three photo e.m.f. plots were made for each surface condition using light spot diameters of approximately 0.40, 0.80 and 1.40 millimeters. Surface conditions investigated were, in order, polished, sandblasted, and etched. The filament was subjected to four successive etches of four minutes in three per cent hydrogen peroxide at 45°C. and was scanned after each etch. Photo e.m.f. plots are shown in Figure 12 for the two extremes of surface conditions. The e.m.f. pattern observed with a polished surface showed many small peaks superimposed on larger peaks. As the light spot diameter was increased these smaller peaks were no longer resolved and the pattern reduced to several relatively broad peaks. The sandblasted surface showed 59 essentially the same patterns but with-reduced amplitudes. A l l major e.m.f. peaks were in the same position. The f i r s t three etches produced a progressive de-emphasis of the minor peaks with the smallest light spot diameter and after the third etch, the light spot diameter had very l i t t l e effect on the photo e.m.f. pattern. The fourth etch produced no appreciable change in the e.m.f. pattern, indicating that the surface had reached the limit -ing condition obtainable with a hydrogen peroxide treatment. Throughout the entire range of surface conditions, the positions of the major peaks were unchanged. Surface etching increases the carrier lifetime and diffusion length, and consequently increases the effective light spot diameter, i.e. diameter of the region of s i g n i f i -cant added carrier concentration. The effective light spot diameter can never be smaller than twice the diffusion length (see section 2.3) and this explains the insensitivity of photo e.m.f. to light spot diameter with the heavily etched surfaces. The disappearance of the smaller peaks when the surface i s etched is also explained by the increase in the diameter of the region of significant added carrier concen-tration. As the sampling width increases only the grosser variations in impurity density are detected and the fine structure is averaged out. This explanation of the e.m.f. pattern change is borne out by the polished and sandblasted surfaces where carrier diffusion length i s much shorter, and the effective light spot diameter approaches the actual 60 light spot diameter. In these cases the e.m.f. plot with small light spot diameter is superficially dissimilar to the etched surface plot while the pattern with large diameter spot is very similar to i t (Figure 1 2 ) . On the basis of these observations we may definitely identify the observed photo e.m.f. as a bulk effect, and consider surface conditions as one parameter determining thepdimensions of the sampling light probe. This conclusion is supported by e.m.f. patterns obtained by scanning the four sides of a filament. These were essentially the same even with a polished surface. It was unlikely that surface irregularities were the same on a l l sides of the filament, and thus the similarity of the patterns indicate that the observed e.m.f. originated from variations in the bulk of the filament. 61 3.6 Measurement of Carrier Lifetimes In a semiconductor in equilibrium, the volume rate of thermal generation of electron-hole pairs is exactly balanced by an equal volume rate of recombination. If excess carrier pairs are momentarily generated by some means, their effective lifetime t is defined by j _ = 1 d p t p-fc dt When the excess concentration is small compared with the equilibrium concentration, V is the time required for the excess concentration to decay by a factor e Decay constants are defined as reciprocals of lifetimes. The experimental arrangement for observation of effective lifetime is as follows. The germanium filament in series with a variable resistance is connected across a constant voltage source. A pulse of light generates excess carrier pairs in the germanium and the resulting increase of conductance causes a voltage increase at the resistor terminals which decays back to the steady state voltage as the photo-generated carrier pairs recombine. The time dependence of the photo-conductive voltage pulse across the resistor is the same as that of the excess carrier concentra-tion provided that (i) no carriers are swept out of the filament ( i i ) filament surface conditions, bulk lifetime, and conductivity are uniform, ( i i i ) no excess carrier genera-tion occurs during the time of observation. When the illumination consists of a light spot, 62 condition (i) is satisfied i f the f i e l d conforms to the sweep-out condition where d is the distance from the light spot to the contact towards which the carriers are moving. If the whole filament is illuminated, carriers w i l l be swept out unless the f i e l d is small enough that carrier motion i s predominantly diffusional r k T i.e. r < —— where LA= "/^^ i s t n e ^ b i p o l a r diffusion length. With light spot illumination the f i e l d may cause appreciable d r i f t without violating the sweep-out condition. In this case condition ( i i ) , uniform surface conditions and bulk lifetime must apply, for otherwise the recombination rate changes as the carriers move along the filament. The effects of non-uniform conductivity are discussed in section 3.7. If surface and bulk conditions are not uniform, the f i e l d must be reduced to conform with the diffusional condition, and the photo-conductive pulse gives a measure of the carrier lifetime at the point of illumination. The measured effective decay constant may be considered the sum of a bulk and a surface decay constant (section 3.1) and by measuring decay constants under the extremes of high and low surface recombination velocity, a measure of the bulk lifetime can be obtained (Stevenson 1955). Fac uig Pago 6-j FIGURE 1J LIGHT SOURCE AND BALANCE C I R C U I 1 FOR LIFETIME MEASUREMENTS : >5M ; i o n 6 KV = gap< D.C. ! .OOl^f 0 Ge sample lenses 3> Shield-Shield ? i r • Amplifier Difference Amplifier o 'Scope Differentiating Exponential Amplifier Network Generator 63 From condition ( i i i ) the photo-conductive decay method of lifetime measurement requires a light source which can be switched off in a time short compared with carrier lifetime. Since lifetimes may be as short as a few microseconds, the light source should go from f u l l to zero intensity in 0.5yU sec. or less. A spark gap i s the most convenient light source capable of meeting this ,.„ requirement. Such a light source was obtained by charging an 0.001AF condenser through a high resistance and discharg-ing i t across a gap using the circuit shown in Figure 13. By adjusting gap width and placing a small resistance in series with i t , the light pulse duration measured with a photo multiplier and fast oscilloscope was reduced to 0.1/U sec. The gap consisted of an auto spark plug with approxi-mately one millimeter between the electrodes and was operated in a i r . The condenser, charging resistor, and gap were enclosed in a metal container to reduce el e c t r i c a l inter-ference and the spark was focussed on the filament using a two lens system (see Figure 13). Decay constants of the photo conductive pulses were measured using a null balance method. This method required generation of exponentially decaying pulses with adjustable and accurately known time constants. The generated pulses were balanced against the photo conductive decay pulses using an oscilloscope with an input difference amplifier. When the amplitudes of the two pulses were equal, the decay constant of the generated pulse was adjusted 6 4 to give a zero signal. An attempt was made to generate the desired pulses by reflecting part of the light from the spark gap to a photo multiplier whose load resistance was shunted by a variable capacitor. For this system to be satisfactory i t was necessary that a givenrnumber of photons should always produce the same excess carrier density, and that a constant fraction of each light pulse be incident at the photo multiplier. Because of variation of spark path across the gap, neither of these requirements was closely attainable and this method of pulse generation was abandoned. The technique f i n a l l y developed involved genera-tion of standard decay pulses directly from the photo conductive pulses (see Figure 13). The photo conductive pulses are amplified and fed to a lyU sec. RC differentiating network and the resulting spikes are applied to the grid of a triode biassed to cutoff. The cathode resistor was made much larger than the tube transconductance and was paralleled by a variable capacitor. As the spike was applied to the grid the condenser was charged. Approximately a microsecond later, the tube had returned to the cutoff condition, and the capacitor discharged through the cathode resistor supplying the necessary pulse of adjustable decay time. To ensure that the generated pulse heights were proportional to the photo conductive pulse heights, i t was necessary for the spike amplitude to be large compared with the difference between applied bias and cutoff bias. The lower limit of pulse time constant measureable with this system was approximately 3yW sec. Shorter time constants were estimated directly from the photo conductive pulse displayed on the oscilloscope. The performance of the c i r c u i t was tested by measuring an accurately known time constant of an exponen-t i a l l y decaying voltage pulse. The null point was sharp enough to allow a balance to within one part in four hundred of the variable capacitance, i.e., of time constant. This figure however, represents operation under optimum conditions) the pulses were of constant amplitude, exactly exponential, and free from noise. The photo conductive pulses did not have a uniform amplitude, and in most cases had an unavoidable noise background. In addition to this any pulses with amplitudes smaller than the difference between the triode applied and cutoff bias appeared without a corresponding balancing pulse, and tended to make the balance point less well defined. With these practical d i f f i c u l t i e s , the uncertainty in balance was within five per cent. 66 3.7 Photo-Conductive Effects In the lifetime measurements described in section 3.6, the f i e l d applied to the filament i s kept small enough that carrier motion is predominantly diffusional F < — When the f i e l d is increased beyond this value, the packet of carriers photo-generated by the spark image moves along the d filament decaying as i t goes, and with a f i e l d larger than -^-^ carriers w i l l be swept out of the filament before decaying (see section 3.6). If sweep-out fields are applied to the filament with the circ u i t shown in Figure 13, one expects the observed wave-form across the resistor in series with the filament to be a truncated exponential. The expected wave-form was observed with the 3 ohm-cm p-type filaments only. With filaments of the other materials the wave-form was exponential 'immediately after the light impulse but was followed by a subsequent increase in pulse height which f e l l rapidly to zero (Figure 14). With a constant f i e l d in the filament, the time between the original spark impulse and the peak of the "after-pulse'1 was proportional to the distance of the spark image from the contact towards which the carrier packet was moving, and approached zero as the spark image approached the contact. With the spark image at a fixed position on the filament, the time between the spark impulse and the after-pulse peak was inversely proportional to the sweeping f i e l d . Facing Page 67 FIGURE 14 PHOTO CONDUCTIVE AFTER-PULSE AT A METAL-SEMICONDUCTOR CONTACT metal contact ELECTRIC FIELD F > n - semiconductor light pulse (t = 0) contact metal EQUILIBRIUM CONDUCTIVITY n An(t =0) An(t = At) An(t = 2At) At 2At n, ADDED CARRIER DISTRIBUTION Volts across filament 'dark" V(t=0) 0 At 2At PHOTO CONDUCTIVE PULSE 6 7 In both cases i t was necessary for the f i e l d to be large enough to produce sweep-out. These observations indicate that the after-pulse is associated with a contact effect. A given density of excess carriers produces a larger r e s i s t i v i t y decrease in a low conductivity region than in a high conductivity region, and this fact is used to explain the pulse wave-forms. A pulse of light produces a packet of photo-generated carriers in the filament which dri f t s towards a contact decaying and spreading as i t goes (Figure 14). The local conductivity increase due to the carrier packet causes a decrease of filament resistance. As the packet decays the filament resistance returns to i t s unilluminated value. However, i f the crystal is inhomo-geneous, the carriers may move into a region of lower conductivity, resulting in a lowering of the filament resis-tance even though the total excess carrier density i s decreasing through recombination. In our case the region of lower conductivity is the metal-semiconductor contact (see section 3.2) and as the carrier packet moves out of the filament, the photo conductive after-pulse is observed. This effect was not present in the 3 ohm-cm p-type filaments, and this confirms the ohmic quality of contacts to this material. 68 C H A P T E R IV - M E A S U R E M E N T S O F B U L K P H O T O E.M.F. 4.1 Linearity of the Photo e.m.f. with Weak Illumination  Intensity Most of the bulk photo e.m.f. measurements were made using filaments from the 40 ohm-cm n material and the graded material (see section 3.3). In the 40 ohm-cm material the r e s i s t i v i t y fluctuates about a mean value throughout the length of the ingot while in the graded material the r e s i s t i v i t y increases monotonically along the ingot. Photo e.m.f. scans of the 3 ohm-cm p and 10 ohm-cm n filaments showed bulk e.m.f.'s of a fraction of a m i l l i v o l t , and although these were useful in assessing the homogeneity of the ingots, the voltages were not large enough for detailed investigation of photo e.m.f. behaviour. In measurements of the bulk photo e.m.f., the incident light i s considered "weak" i f the photo-generated carrier concentration is small compared with the equilibrium majority carrier concentration. When excess carrier density is comparable with the equilibrium concentration saturation e.m.f. effects are observed (see section 2.3). Thus for our purposes illumination intensity refers to the ratio of photo generated ;to equilibrium carrier densities and not to photons incident per unit area per second. For example, photo generated carrier density is dependent on the filament effective lifetime (see Appendix V) and a given illumination 69 which approaches saturation intensity when incident on an etched surface may be weak when the surface is sandblasted. Also, the equilibrium carrier density in the intrinsic range increases rapidly with temperature and an illumination which has saturation intensity at room temperature w i l l be weak at some higher temperature. The light incident on the filament was varied using a stepped wedge with equal density increments per step. Optical density i s defined as density = log ( 0 f-where I± is the incident intensity and I t the transmitted intensity. When a collimated beam of light is incident on the wedge the total transmitted intensity consists of an attenuated beam plus forward scattered light. The diffuse density is defined with I t equal to the total transmitted intensity and the specular density is defined with I t equal to the intensity of the transmitted beam. The manufacturer's data for the wedge specified optical neutrality for wave-length from 0.4 to 1.0 microns and included a calibration in diffuse densities. In our experiment set-up, the transmitted beam was focussed on the filament and thus i t was necessary to recalibrate the wedge for specular density. A germanium photo diode was used to make this calibration, f i r s t , because the short c i r c u i t photo current was known to be directly proportional to the incident light intensity (Cummerow 1954) and second, because i t s spectral sensitivity was identical with that of our filaments thus giving a Facing Page 70 FIGURE 15 DEPENDENCE OF BULK PHOTO E.M.F. ON ILLUMINATION INTENSITY Wedge segment no. ( Intensity increasing »•) 70 calibration which automatically took account of any departure of the wedge from optical neutrality at wavelengths greater than 1 micron. Using the photo diode the measured wedge specular density was 0.234 per step, or 2.34 db per step. The theory of the bulk photo e.m.f. predicts a linear dependence of photo voltage on weak illumination intensity (see section 2.3) and this was verified using the calibrated wedge. Log photo voltage was plotted against wedge step, and from the wedge calibration a linear relation between voltage and illumination corresponded to a straight line of 2.34 db per step, i.e., slope of one in log voltage versus log illumination intensity. Observations were made with both the 40 ohm-cm n material and the graded material at various e.m.f. peaks in the filaments. In a l l cases the slopes were between 0.94 and 1.01. Typical plots are shown in Figure 15. These results verified the linearity of photo e.m.f. with weak illumination intensity. Facing Page 71 FIGURE 16 COMPARISON OF LOCAL RESISTIVITY AND BULK PHOTO E.M.F. e B (mv) -4 Probe Voltage -3 0.4--2 --1 -1 0.2-"-2 ^ ^ ^ ^ proportional to o—' - 3 0.1--4 -71 4.2 Dependence of Bulk Photo E.M.F. on Impurity Density  Gradient The bulk photo e.m.f. arises from an impurity gradient in the semiconductor and the final form of the e.m.f. expression involves the equilibrium carrier densities at the edges of illumination for both weak and saturation intensities (see section 2.3). Except in the near intrinsic case the equilibrium carrier densities may be identified with the equilibrium conductivities (see section 2.6), and in extrinsic materials the photo e.m.f. with a given illumination should have a maximum value where the impurity density gradient, i.e., conductivity gradient, has a maximum. Thus for an n-type specimen with weak illumination intensity we may write (see section 2.6) The proportionality of bulk photo e.m.f. to the differences between equilibrium r e s i s t i v i t i e s at the illumina-tion edges was experimentally checked by comparing a probe measurement of local r e s i s t i v i t y with a photo e.m.f. pattern obtained over the same length of filament. A constant current was passed through a germanium filament and two metal probes with a fixed spacing of approximately 0.7 millimeters were moved along i t . The voltage between the probes was propor-tional to the local r e s i s t i v i t y , and the positional variation of this voltage is shown with the corresponding photo e.m.f. pattern in Figure 16. Points in the plot where the photo e.m.f. has a maximum or minimum correspond to 72 points where the re s i s t i v i t y gradient has a maximum or minimum, and the curves suggest that the photo e.m.f. is proportional to the local r e s i s t i v i t y gradient as predicted by the theory. Shortly after these observations were made the proportionality of bulk photo e.m.f. to res i s t i v i t y gradient was adequately confirmed by Frank (1956) and further investigation of this point was unnecessary. The photo e.m.f. gives a measure of impurity gradient and is thus better suited for investigation of crystal inhomogeneities than probe measurements of res i s t i v i t y . This is especially true in near int r i n s i c material where conductivity has a minimum (see section 2.6). Figure 16 illustrates the advantage of light probe measurement of impurity gradients; the fine structure in the photo e.m.f. pattern may be identified with r e s i s t i v i t y gradients which were not detectable by the probe measurements. 73 4.3 Measurements of the Ratio of Bulk Photo e.m.f. to  Photo Resistance From the analysis of section 2 . 6 the ratio of bulk photo voltage to photo resistance should be constant at weak illumination. With a uniform density of photo-generated carriers, the ratio is directly related to the impurity density gradient, i.e. e B ^ 2 A _kT d o ; A R S " b + ! H d x To verify this expression experimentally, an accurate probe measurement of local conductivity, and uniform generation of carriers in the bulk of the filament are necessary. Although a non-uniform photo-generation rate corresponds to an effective crossectional area less than A (section 2 . 6 ) , the ratio should remain constant at weak illumination even under this condition. For measurement of the ratio of bulk photo e.m.f. at weak illumination, the light spot was focussed on a position in a filament exhibiting a large e.m.f. peak. The stepped wedge was used to decrease the light intensity by known increments, and the photo e.m.f. was measured by a potentiometer. The illuminated and dark resistance measured on a wheatstone bridge gave the photo resistance. The added carrier distribution must be the same in the e.m.f. and resistance measurements, and for this reason an a.c. bridge source is preferable. The bridge balance point was deter-mined using a high gain oscilloscope with an input difference Facing Page 74 FIGURE 17 DEPENDENCE OF PHOTO RESISTANCE AND BULK PHOTO E.M.F. ON ILLUMINATION INTENSITY 74 amplifier. Plots of log photo resistance and log photo voltage were parallel with a slope of approximately one (Figure 17). These observations show that the ratio of photo e.m.f. to photo resistance was constant at weak illumination as predicted by theory. However, since the photo-generated carrier density was not uniform throughout the filament crossectional area, the results could not be used for a quantitative verification of the expression for the ratio of photo voltage to photo resistance. Photo resistance measurements were also made using a d.c. bridge source. In this case the f i e l d applied to the filament must be kept small enough that carrier motion is predominantly diffusional, i.e., The maximum allowable f i e l d from this condition was small enough that the photo e.m.f. substantially affected the bridge balance. This d i f f i c u l t y was removed by bucking out the photo voltage with an equal and opposite voltage applied across a known resistance in series with the filament. With this method, the photo resistance was found to be dependent on direction of current flow, and plots of log photo resis-tance against log light intensity were parallel to each other and to the log photo voltage plot (Figure 17). The change of photo resistance with current direc-tion was explained by the dependence of dark and illuminated resistance on current density. The dark resistance of the 75 filament used for the d.c. bridge measurements showed considerable asymmetry with current densities in opposite directions. With light incident on the photo e.m.f. peak, this asymmetry decreased with light intensity and at strong illumination the filament resistance was not affected by the direction of current up to the maximum allowable from the diffusional condition. Thus the observed dependence of photo-resistance on current direction originated in the asymmetry of the dark resistance with current density. When the filament was illuminated at a point where the photo voltage was zero, the illuminated resistance showed a depen-dence on current density similar to the dark resistance. The characteristics of the dark and illuminated resistances corresponded to a weakly rectifying junction at the point of illumination. This region exhibited a large peak photo voltage, i.e., had a large local impurity gradient, and the asymmetry in the dark resistance as a function of current density corresponded to the direction of rectification implied by the polarity of the photo e.m.f. The voltage drop across the weak junction was sufficient to perturb the added carrier distribution from i t s open ci r c u i t value even though the total f i e l d applied to the filament conformed to the diffusional condition. This could account for the change in photo resistance with current direction. At saturation illumination intensity, the electro-static potential step associated with the impurity gradient was flattened out, and the cause of rectifying behaviour removed. As a result the illuminated resistance of the filament showed less non-ohmic behaviour than the dark resistance. These observations showed that the elec t r i c a l contacts to some of the 40 ohm-cm n filaments were better than was indicated by the methods of section 3.2 in the sense that a part of the observed non-ohmic behaviour originated in the bulk of the filaments. At the higher light intensities the ratio of photo e.m.f.to photo resistance was measured using a method described by Tauc (1957). A constant current was passed through the filament and the light beam chopped using a rotating slotted disc. The light pulse incident on the filament produced a photo resistance pulse which appeared at the filament terminals as a voltage pulse equal to the product of current and photo resistance. When the light was incident on a region of impurity density gradient, a photo voltage pulse also appeared at the terminals. The resultant voltage across the filament was displayed on a high gain oscilloscope, and the direction and magnitude of the current was adjusted to reduce the observed pulse to zero. At this point where i f is the current through the filament. With photo voltages of the order of a mi l l i v o l t the oscilloscope gain was not sufficient to allow an accurate pulse balance, however, photo voltages of this magnitude are ordinarily in the linear range (Figure 17) where the ratio of photo 77 voltage to photo resistance is known to be constant. The accuracy of the compensation method described above improves with increasing photo voltage and was used chiefly to investigate the behaviour of the ratio in the saturation range of illumination intensities. Facing Page 78 FIGURE 18 SATURATION EFFECTS (a) STRONG ILLUMINATION DEPENDENCE OF LIGHT PROBE DIMENSIONS Wedge segment no. (b) STRONG ILLUMINATION CHARACTERISTICS OF PHOTO VOLTAGE AND PHOTO RESISTANCE 78 4 - 4 Saturation Effects When the illumination intensity is sufficient to produce saturation e.m.f. in the volume under the illuminated area, a further increase in intensity affects the photo e.m.f. only through carriers which have diffused far enough to arrive in an unsaturated region. These unsaturated components w i l l always be present in the observed e.m.f. unless the impurity density gradient occurs only in the region where light is incident. Since this is not normally the case, the observed "saturation" e.m.f. w i l l increase or decrease with light intensity depending on the impurity distribution in the immediate neighbourhood of the light spot (see section 2.6). If the light spot is considered a sampling probe, an increase of illumination intensity into the saturation range may be thought of as increasing the probe dimensions beyond 2(a + L p) (see section 2.3). In the 40 ohm-cm n material, the impurity density fluctuates about a mean over the length of a given filament (Figure 16). In this material the unsaturated components wi l l generally have a polarity opposite to the saturated component, and the photo e.m.f. w i l l decrease as the illumination intensity is increased into the saturation range. This is illustrated in Figure 18(a) which consists of records of successive scanning of a heavily etched 40 ohm-cm filament with illumination intensity decreasing in the saturation range. The corresponding decrease in probe dimensions permits a better resolution of the e.m.f. peak and a consequent higher e.m.f. at lower illumination intensities. In the graded material the impurity density increases monotonically along the filament and a l l e.m.f. components are of the same polarity. In this case the photo e.m.f. continues to increase up to the maximum illumination intensity. The dependence of photo e.m.f. and photo resistance on light intensity is shown in Figure 18(b). In both materials the photo resistance behaviour is the same, and because of the impurity distribu-tion in the graded material the ratio of photo voltage to photo resistance is constant up to the maximum illumination intensity. This i s discussed in section 80 4.5 Temperature Dependence of the Bulk Photo E.M.F. The temperature dependence of the bulk photo e.m.f. can be calculated from the results of section 2.3. For weak illumination in extrinsic material 2kT b in n - t y p e and in p - t y p e The impurity centers may be assumed totally ionized in germanium at temperatures greater than 50*K and thus the equilibrium concentrations are independent of temperature. The temperature dependence of b, the ratio of mobilities of electrons and holes i s calculated from published mobility data (Morin 1954) b - 4.9 X I07 T r 4 . 65 X I0'Z T o. 6fc The added carrier concentration A h L i s dependent on l i f e -time, and i f the carrier lifetime i s temperature dependent this w i l l influence the temperature dependence of the bulk photo e.m.f. Where lifetimes are constant in the extrinsic material T -r- o . 3 3 b e 8 <* OC T in n - t y p e e 8 cc bT oc T fn p - t y p e 81 In near intrinsic material (section 2.5) In this expression the temperature dependence of the factor involving the mobility ratio may be neglected, and where AtlL is temperature independent dependence were made over a temperature range of ~120°C to 100°C. For measurements at low temperatures the filament was placed in a dewar flask with an aperture in the si l v e r -ing, and the light was focussed on a position exhibiting a large photo e.m.f. peak. The temperature was lowered by pumping liquid nitrogen through a c o i l of copper tubing inside the dewar and temperature measurements were made by means of a thermocouple connected to one contact of the filament. To minimize temperature differences between the contacts, the filament was mounted horizontally. In the f i r s t measurements the dewar was f i l l e d with petroleum ether to provide thermal inertia and stabilize the temperature in the region of the specimen. This was unsatisfactory at the lower temperatures as the petroleum ether became increasingly cloudy below -30°C. and froze at approximately -80*C. Since no other suitable coolant was available, the petroleum ether was removed and the filament temperature was lowered by cooling the surrounding air in the flask. e 8 T Observations of the bulk photo e.m.f. temperature 82 When a sufficiently low temperature was reached, the pump was shut off and measurements of the photo e.m.f as a function of illumination intensity were made at 10* intervals as the filament warmed to room temperature. One dis-advantage of this arrangement was the rapid rate of warming at low temperatures which made i t d i f f i c u l t to get complete observations of the photo e.m.f. as a function of illumina-tion intensity before the temperature had changed appreciably. measuring the bulk photo e.m.f. at low temperatures had been developed, the filament was incorporated into a ci r c u i t which allowed concurrent measurement of both the bulk photo e.m.f. and photo resistance. Two equal constant currents were passed through the filament and a variable resistor which was made equal to the filament dark resistance. After the necessary experimental techniques for 1 / * ^ 2 i o recorder o— ,2 o o VW> 11— filament Facing Page 83 FIGURE 19 TEMPERATURE DEPENDENCE IN THE EXTRINSIC RANGE OF CONDUCTIVITY -120 -J 1 1 1 1 i i i i i i i • • I 14 12 10 8 6 4 2 Wedge segment no. (Intensity increasing >) 83 With the switch in position 1 as shown, one terminal of each of the resistor and the filament was connected together and the voltage between the other terminals (i) was zero with no illumination and ( i i ) with filament illuminated was equal to the bulk photo voltage plus the product of current and photo resistance. This voltage was measured on a recording 0-10 mv meter. With the switch in position 2, the current was shut off and the photo voltage recorded. By subtracting the two voltages a value of the photo resistance was obtained. This method of measuring photo resistance is subject to considerable error at weak illumina-tion but the accuracy improves at strong illumination where the difference between the two voltages i s greater. Since the photo resistance is known to be directly proportional to the photo e.m.f. at weak illumination (see section 4.3), inaccuracy at low light intensity i s not a serious disadvantage. At temperatures from -120°C. to -30CC. both the photo e.m.f. and photo resistance increased with temperature (Figure 19a and b) in a 40 ohm-cm filament. As the tempera-ture was increased further, both quantities decreased slowly until the temperature had reached approximately 10°C, and then decreased more rapidly as the filament entered the intrinsic range of conductivity. With the graded filament the behaviour of photo e.m.f. and photo resistance was similar to that described for the 40 ohm-cm filament. With both materials the increase of photo e.m.f. with temperature in the extrinsic range is too rapid to be Facing Page 84 FIGURE 20 TEMPERATURE DEPENDENCE OF THE BULK PHOTO E.M.F. IN THE INTRINSIC RANGE _J 1 I I l l I l I i I I 1 12 10 8 6 4 2 Wedge segment no. (Increasing intensity *-) 84 accounted for by the T/b term (see above) and is attributed to the lifetime dependence of the At)L factor. This is borne out by the photo voltage versus illumination charac-teri s t i c s for the 40 ohm-cm material which shows a progressive decrease of e.m.f. with light intensity at strong illumination from -120°C. to -4o°C. This corresponds to an increase in the effective light intensity, i.e. increasing lifetime (see sections 3.2 and 4.1). In successive low temperature measurements, the magnitude of the photo e.m.f. with a given light intensity was not closely reproducible, In a l l cases however, the photo e.m.f. and photo resistance were linear with weak illumination intensity and the curves showed the other characteristics of Figure 19. By drying the air in the dewar flask the variation between successive sets of low temperature measurements was reduced, but not entirely removed. Because of this behaviour, an analysis of At)u through lifetime measurements at low temperatures was not attempted. The same methods of measurement were used to investi-gate temperature dependence of the photo e.m.f. in the intrinsic range. The filament was heated in air in a dewar flask and at a sufficiently high temperature the heater was shut off and observations of photo e.m.f. as a function of illumination intensity were made as the filament cooled to room temperature (Figure 20). The photo e.m.f. showed progressively less saturation behaviour as the temperature was increased because of the rapid increase of equilibrium carrier concentration, and consequent decrease in the ratio Facing Page 85 FIGURE 21 EXPERIMENTAL VERIFICATION OF THE THEORY OF THE BULK PHOTO E.M.F. IN THE INTRINSIC RANGE > £ GD 0 . Temperature (°C) 8 0 i 7 0 6 0 _ 2 9 1 0 0 T . 3 0 ( l/°K) SO 4 0 1 0 . - 2 8 ID E o 1 0 cvu c . - 2 9 . 3 3 85 of photo-generated to equilibrium carrier densities (see section 4.1). At temperatures above room temperature, successive sets of photo e.m.f. observations were accurately repro-ducible, and i t was possible to obtain an experimental check on the weak illumination temperature dependence of the photo e.m.f. predicted by the theory. The measurements were made from 20*C. to 100aC. in the manner described above except that the light intensity was maintained constant and the photo voltages were measured on a potentiometer. The carrier lifetime decreased slowly with increasing tempera-ture, showing a total change of approximately 10% over the temperature range investigated. Thus the A l \ factor has negligible effect on the bulk photo e.m.f. temperature dependence and from the theory (see above) T ~ n? 2 The value of n± was taken from published data (Morin 1954) i\ = 3.1 x 10 T exp 0 . 7 3 ? k T This relationship was checked by plotting log e B and log ey/2T as a function of reciprocal temperature, and comparing the slope of the plot with log ^/r\ as a function of reciprocal temperature (Figure 21). The slope of the log e8/£j and log '/n2 plots are equal within experimental error, and this confirms this feature of the theoretical treatment of the bulk photo e.m.f. in the intrinsic range. 86 A P P E N D I X I Imref Gradients with Large Added Carrier Concentrations Equal added carrier densities n a and p a are present at the plane x = 0. If a l l carrier motion is diffusional, then p(x) - pe + p Q exp = h o -H n a exp where L a is the ambipolar diffusion length. _X_ Then - y(x)j = ^ In pa + p a e x p ( - f a ) In extrinsic material where carrier motion is predominantly dif fusional, grad (f>? (x) w i l l be large compared with grad and where p a e x p then Similarly i f then Cjrad (fip(x) kT n a exp q n a d <&,(x) _kT Ia extrinsic semiconductors, the added minority carrier density from injection or photo generation may be large compared with the equilibrium minority density but is usually negligible compared with the equilibrium majority carrier density. In this case the diffusion length i s that of the minority carriers (Rittner 1954) and the imref gradients are approximately where L n and Lp are electron and hole diffusion lengths. q r a d aWx) - — r i n n - t y p e 88 A P P E N D I X II Dependence of Electrostatic Potential on Added Carrier  Concentrations A non-homogeneous semiconductor is illuminated producing a uniform concentration of added carriers throughout the bulk under the illuminated area. The concentrations of the electrons and holes are n = n£ e x p i (y-fa) p » h, e x p ^ r ( 0 P - f ) and the concentration gradients are q r a d n q , , . N n = "SX ^ r a c / ( ^ - ^ ) No net current flows in the open ci r c u i t condition so that b n cjrad fa + p cj rad 0 P = o and we use this relation to substitute for grad (f>p in terms of grad fa p q r a d n g r a d (-yv - 0„) Thus n q r a d p bn . ; , , ^ ^ - - p - 9 m d & - qrad f 89 and n qradfp- hn) , . qrad U/ = — J - y — ; — - qrad S, p q r a d ( b n - p ) , . similarly q r a d y = ~~ 7 7 — q r a d 0 P ^ b g r a d (np) J If the semiconductor is homogeneous, (assumption (a), section 2.3) and g r a d n 6 = q r a d p0 = o g r a d n = q r a d p = qrad An where Ah is the concentration of added electrons and holes. The above equations are simplified in this case and become (b- l ) p q r o d < £ p h q r a d <ph ^ r a d ^ " - T { ^ ) = fTTp In the homogeneous semiconductor the electrostatic potential gradient can also be expressed in terms of the gradient of added carrier concentration. Since grad n = grad AH ^ j q rad An then q r a d 0„ = - — 1 — -f q r a d U> J H h J which leads to /, \ kT qrad An qrad y = / b - l U i -intrinsic material where M = p ^ y ( b + l ) ^ h 91 A P P E N D I X III Evaluation of the Integrals b e B grad - •%) dx a i l l b (b-j) grad An (b-1) An g r a d na dx p. + bn, +^b +1) An bn e 2+An(b+i^n 0f h-a The f i r s t term i s identified with the diffusional e.m.f., e d. From assumption (f) section 2.3, a significant gradient of added carrier density exists only at a 1 and b', the edges of illumination (Figure 6), and the f i r s t term of integrand is f i n i t e only at these points. The equilibrium carrier densities are slowly varying functions of position and may be considered constant at the two points where the integrand exists. Thus we write where 6 is very small and 92 The second term is identified with the inner f i e l d component of e.m.f., e c. In this case the integrand exists only in the region where an added carrier concentration exists, i.e., between the points a' and b'. The added carrier concentration is assumed to be constant, A n L , between these points (Figure 6) and the variable of integration is the equilibrium carrier density. Thus the integral becomes ^' kT A / ( b ^ d n 0 6r = A n L / — T . — \ -% j bn* + A n L ( b + ))ry+.n? which is readily evaluated giving 2kT AnL (b + l) -P - i Zbn e + An L (b+1) 1 an • D J when D i s positive e _ kT An,(b+t[ | 2bn.-r(b-f()An, 1 V~D~ 2 b n 0 t(b+i) An.+y^D when D is negative D = 4 n t 2 b - ( b + l ) "An t 2 These solutions are unwieldy, and i t is more convenient to consider the two extremes of weak and strong illumination i.e., added carrier density At\ much smaller or much larger 93 than the equilibrium majority carrier densities, (a) Weak Illumination When A l \ << De+ bn0 , the diffusional component of the e.m.f. is approximately i l l In the inner f i e l d component we assume Then since tan s = 9 i f e is small 9. n* + b n o W r w Ml (b+l)An t and e n = e j + e. 2 kT An, n in n-type material ( b r\>>p o Or 2bkT in p-type material ^ po > > b ftj) 94 (b) Strong Illumination When An L >> p o+bn o , the diffusional e.m.f. component becomes approximately e - i l l ( t i l l In p ° b ' * b r V  d % (b + i) p „ + b n „ . In the inner f i e l d component of e.m.f. the log form of the integral solution is used, and assuming ^b+l)*4f)Ll >>4fl* b e c = _ i L L | n If the substitution in the original integral expression i s made in terms of p Q, then I n P o b ' e c = TT The complete expression for strong illumination or saturation e.m.f. i s then in n-type material (bn£ >>>p„) o r e e d + e c - - p ~ In £ - b ' 8 d b +1 in p-type material (p a A P P E N D I X IV Imref Derivation of the Photo e.m.f. Using the gradients of the carrier concentrations and relating the imref gradients through the open cir c u i t current condition b n q r a d fa + p c j r a d fa = 0 we may write • , kT I q rad(np) bn + p and q rod 0 D = —— — —r • ^ Y p % P bn + p This is expanded to a more suitable form using assumption (a) section 2.3, An = Ap and grad An = grad Ap The expression then contains terms which may be identified with the diffusional and inner f i e l d components of the e.m.f. The integration w i l l be carried through using the electron imref. b e B = / g r a d fa dx b b kT / fn + p) qrad An kT / An qrad(n 0+p 0^ / J / Q )( % I n(bn + p) % / n(bn + p) 96 The f i r s t term is written b kT / (n0 + p„ +2An") dAn ~% (n0+An) ]^  + bn.-4-(b + l)An] a. dAn and expanded by partial fractions to b , b ^ _ JiL / r_dA Ln , JlL/b-A / a. a. . These integrals are evaluated using the methods outlined in Appendix III, giving kT In n O A . + An L n o b . n « « - n o t .+ An u Using the relation grad p. = - qradne the second term is written b k j / An (no 1 dn n0(n0+Ztn) bn0 + An(b + i) n 0 + n t ' and expanded to b' dne k T / d n 0 _ kT / dn e k J A n / b , , \ S / n* S / n o + A n « - 1 / b n 6 ^ 4 n L ( b4 i ) n 0 + nt2 a' 97 These are evaluated giving kT in n o b- n c a . + An , ^ • - i 2 b n 0 + A n L ( b + i) Ian _ z k T Ant(b4-Q where D = 4 n? b - d n * (D-H? The arc tan term above takes a log form when 0 is negative (see Appendix III). 98 A P P E N D I X V Added Carrier Densities with Uniform Illumination A homogeneous n-type semiconductor is uniformly illuminated between the points a and -a. If the time derivatives are zero and one dimensional geometry is assumed, the continuity equation for holes i s d p where g and r are the rates per unit volume of carrier genera-tion and recombination. Carrier diffusion w i l l be ambipolar under the above conditions; but i f the total hole density is small compared to the total electron density, the ambipolar diffusivity, Da, is approximately equal to Dp, the hole dif f u s i v i t y . Assuming a uniform rate, G, of carrier pairs are photo generated per unit volume, the generation and recombination rates are <g = ~ (x >a and x < -a) q = y- •+ Gr (-a < x < a ) — — ( tor a l l X T where T p is the hole lifetime and AlT\(x) = Ap(x) i s * n e added carrier pair density. 99 Tor X ><3 and X<~a the continuity equation becomes a Ari(x) _ An(x) dx* D P t p = ° Using the boundary conditions An * o x - ± oo An. = Ar\(a) x = ± a the solutions of this equation are Ar\(x) = An(a) exp (x >a) Lp Ah(x) =Ahfa) exp ^ (x <-a) where Lp = i/D?1p =• hole diffusion length. In the illuminated region the continuity equation becomes orAn.fr) _ An(x) • = Q dx1 DPTP Dp An(x) = Q<Cp + C , e x p ( - f ) + ^ e x P ( x ; N ) and nfr) = Qc, +C, exp(-{ The added carrier density Anfx^ and i t s derivative must be continuous at the edges of the illuminated region X = =t (X f and from these conditions c, = c2 - c a D d A n fa) = G t f 4- 2 C c o s h ^ - ) Substituting for Ar\(a) C * - G i : P e x p ( -100 and An(X) * 6 Cp (- a < x < GI) At>(x) - ~ , - e x p f i f i j e x p ^ ) . ^ - e , p , - — )| exp - j -L The r a t i o An(o) 2 shows that the added c a r r i e r concentrations at the edges of the illuminated region are always greater than one half the concentration at the mid point. 101 A P P E N D I X VI Added Carrier Concentrations at Illuminated Junctions When the differences of imrefs at the edges of a junction transition layer are equal, i.e., when then POP + (Ar)L)p n,P + (AnLV Assuming h 0 h >> p, to (A"i)o POP + and p 0 p >>n, P (An,), Pop this simplifies Then for weak illumination, when the added carrier densities are negligible compared with the majority carrier densities and for strong illumination, when the added carrier densities are large compared with the majority carrier densities Similar relations apply to junctions in which one side is intrinsic and to low-high (L - H) junctions where there is a large abrupt increase of impurity density. In the i -n junction (A h i 2 Pi (weak illumination) 102 and (strong illumination) ( A - (An] In the n - n +junction n0„ (weak illumination) (strong illumination) 103 B I B L I O G R A P H Y Borneman, E.H., Schwarz, R.F., Steckler, J.J., Rectification  Properties of Metal-Semiconductor Contacts, Journal of Applied Physics, 26, 1021 (1955). Chynoweth, A.G., and Pearson, G.L., Effect of Dislocations on  Breakdown in, Silicon p-n Junctions, Journal of Applied Physics, 29, llb3 (1958). Cummerow, R.L., Photovoltaic Effect in p-n Junctions,  Pnotovoitaic EffectPhysical Review, 95, 16 (1954). Fan, H.Y., Theory of Photovoltaic Effect of p-n Barrier in  Semiconductor] Physical Review, 75, 1631 (1949). Fink, C.G., Dokras, V.M., Electrodeposition of Germanium, Transactions of Electro-chemical Society, 9$~, 80 (1949). Fletcher, N.H., General Semiconductor Junction Relations, Journal of Electronics, 2, 609 (1957). Frank, H., Photo-electric Measurement of Inner Electric Fields  in Inhomogeneous Semiconductors, Czechoslovakian Journal of Physics, j>, 433 (1956). Gunn, J.B., On Carrier Accumulation, and the Properties of  Certain Semiconductor Junctions, Journal of Electronics and Control, 4, 17 (1958). Misawa, T., Emitter Efficiency of Junction Transistor, Journal of the Physical Society of Japan, 10, 362 (1955). Morin, F.J., Maita, J.P., Conductivity and Hall Effect in the  Intrinsic Range of Germanium, Physical Review, 94, 1525 (1954). Pearson, G.L., and Bardeen, J., Electrical Properties of Pure  Silicon Alloys Containing Boron and Phosphorus, Physical Review, 75, 865 (1949). Read, W.T., Scattering of Electrons by Charged Dislocations  in Semiconductors, Philosophical Magazine, 46, 111 (1955). Read, W.T., Statistics of the Occupation of Dislocation Acceptor  Centers, Philosophical Magazine, 45, 1119 (1954). 104 Read, W.T., Theory of Dislocations in Germanium, Philosophical Magazine, 45. 775 (1954). Rittner, E.S., Electron Processes in Semiconductors, Photoconductivity Conference Atlantic City, John Wiley and Sons Incorporated, 1954. Shockley, W., Electrons and Holes in Semiconductors, D. Van Nostrand Company, 1950. Shockley, W., The Theory of p-n Junctions in Semiconductors  and p-n Junction Transistors, Bell System Technical Journal, 28, 435 (1949). Stevenson, D.T., Keyes, R.J., Measurements of Carrier  Lifetimes in Germanium and Silicon, Journal of Applied Physics, 26, l"90 (1955). Tauc, J., Generation of an e.m.f. in Semiconductors with Non-equilibrium Current Carrier Concentrations, Review of Modern Physics, 29, 308 (1957). Tauc, J., The Theory of a Bulk Photovoltaic Phenomenon in  Semiconductors^ Czechoslovak!an Journal of Physics, 5, 178 (1955). Weisberg, A.H., Heavy Rhodium Electroplates Now Possible, Materials and Methods, 37, Number 3, 85 (1953). 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085476/manifest

Comment

Related Items