llf Noise in bipolar transistors

Transcription

1 J. Phys. D: Appl. Phys., l8 (1985) Printed in Great Britain llf Noise in bipolar transistors C T Green and B K Jones Department of Physics. University of Lancaster. Lancaster LA1 4YB. UK Received 23 March in final form 20 July 1984 Abstract. Measurements have been performe don the low-frequency. excess. in noise silicon. bipolar. npn transistorsat frequencies between lo"hzand25 khz. at temperatures between 77 K and 375 K and over a wide range of bias. The DC characteristics have also been studied in detail. The excess noise can be represengd completely by a single current noise Source between the base andemitterwith intensity;: = K''lb'hJ Af uherefbi5 the non-ideal ba\e current. 'J is an exponent near unity which varies only slightly with temperature. A is the base-emitter contact area and K' is a constant for each device which is independent of temperature and other bias quantities. Discussion is presented of the connection between the excess noise and the non-ideal, recombination current and of the relationship between these results and the various noise models. 1. Introduction l/f noise is a low-frequency noise, in excess of the white shot and thermal noise. It is a resistance, or emission, fluctuation which occurs in most electrical systems. The fractional fluctuation has variationz/r2 a =A f/f where Afis the measurement bandwidth and y is an exponent very near unity. In most electrical systems the magnitude of the fluctuation depends on the sample volume or surface area and can be linked with some form of electrically active defect. However, there is a large body of opinion which believes that the fluctuation is so widespread because it is due to some fundamental process such as a bulk, phonon-related, mobility fluctuation (e.g. Hooge et a1 1981) or a quantum-mechanical Bremsstrahlung effect (e.g. Handel 1975, 1980). Although p n junctions are widely used in bipolar diodes and transistors the study of their excess noise has not been detailed. very Before surface passivation was improved there was considerable evidence for a noise source at the oxide interface near the base-emitter junction. Previous experimental work has shown the general form of the bias dependence but there has been clear no indication of the processes involved in high quality, low-noise devices. There is general agreement that the noise increases with bias and can be represented, mainly, by a base-emitter current generator but there have been many variations on these general ideas. Current theories of excess noise specifically in bipolar devices will be described in more detail in the discussion section but they are based either on the Hooge mobility fluctuation equation or on fluctuations of the recombination of carriers in the space charge region or at the oxide surface or at dislocations. In the experiments reported here we have carried out a detailed investigation of the electrical noise in high quaiity, low-noise npn bipolar silicon transistors over a wide /85/ The Institute of Physics 77

2 78 C T Green and B K Jones range of experimental variables and compared the noise results with the device DC characteristics also measured over a wide range of bias conditions. Analysis of these noise and DC characteristics has enabled us to obtain an understanding of the details of the device operation and establish appropriate models and equivalent circuits. The comparison of the noise and the DC characteristics has been done in two parts. In this paper we report detailed measurements on few a devices over a wide range of variables and it is shown that there is a very simple parameter which can be used to describe the excess noise in a device. This result helps to determine the processes causing the noise and hence aids in a comparison of the various theories of excess noise. Except where stated otherwise, the device used in this work is the BC413 BP made by Ferranti Electronics Ltd. This a low-noise, high gain (hfe- 290), npn silicon bipolar transistor made by a double-diffused planar process with a dumb-bell geometry using a low-concentration phosphorus doped emitter. 2. Experiments 2.1. Static characteristic experiments The static characteristics, or DC I-V relationships, of the transistors were measured using an HP4140B picoammeter and voltage source controlled by an HP85 computer. The measurements were made between the current limits of the instrument, and lo-* A, so that there were no high-injection or collector series resistance effects at high currents. Typical results for the reverse leakage and breakdown of the base-emitter and base-collector junctions are shown in figure l I I v Figure 1. A typical example of bipolar diode reverse current flow. Lg I (in A) versus lg V (in V) for the emitter-base junction (+) with the collector open circuit and the collector-base junction (0) with the emitter open circuit. Device BC413 : A378. In these experiments the unused terminal is open circuit. In both junctions the reverse I-V relationship 'follows a power law at low voltages with I = and the currents are comparable at the same voltage, high At voltages the current increases due to junction breakdown. The collector junction can sustain higher voltages.

3 l/f noise in bipolar transistors 79 The base-emitter junction forward characteristics and the transistor action shown are in figure 2. For these measurements the collector voltage is kept at 5 V. The collectorbase reverse leakage current has the opposite sign to the base-emitter forward current so that for a small base-emitter voltage the base current is zero and a cusp is formed on the Ig 1 IB 1 - V,, curve. The characteristics were measured at temperatures between 77 K and 375 K with the device in an Oxford Instruments type DN704 cryostat. l V,,lV) Figure 2. A typical example of bipolar transistor forward current flows. Lg 1 I1 versus baseemitter voltage VBE for the base current (0) and collector current (----) is shown for VcE = 5 V. T = 296 K. Device BC413 : A Noise experiments The noise of the devices was measured using the device in the common emitter configuration as its own preamplifier. For most measurements a constant 5 V collector bias was used and this was set manually or maintained by a negative feedback circuit. The devices were screened for burst noise and none were used that showed bursts in the output voltage noise seen on an oscilloscope trace for frequencies in a band between 50 Hz and 10 khz. The devices were biased over a wide range of base currents and the temperature was controlled at values between 77 K and 375 K. A range of source resistors was used. The measurements were made in a screened room with a filtered mains supply. The noise signal was amplified by a low-noise preamplifier and its spectrum measured and calculated by a Hewlett Packard HP3582A spectrum analyser controlled by an HP85 computer. The spectrum between Hz and 25 khz was computed and averaged over many readings. The data were fitted to the sum of a white component and low a frequency component with a Ilf dependence. The white (frequency independent) noise intensity,

4 80 C Green T and B K Jones the exponent, y, and the intensity of the non-white component at 1 Hz were recorded. In all cases this fit gave a good representation of the observed spectrum. The output noise voltage was referred to device the input by dividing by the measured device voltage gain. This was derived by feeding a white noise voltage signal, much larger than the device noise, into the input and using the HP3582A spectrum analyser to calculate the transfer function between the input and output. The details of all the measurements have been given by Green (1983). 3. Results 3. l. DC analysis From the forward DC characteristics the low-current, low frequency operation of the transistor was derived. The lg ICVBE curve was accurately linear and the slope enabled the device junction temperature to be determined since IC = L-0 exp(evbe/kt). The IB-VBE data could be fitted accurately to the sum of two terms (Ashburn er al 1975) IB = Ib -k 1; = Ibo exp(evbe/mkt) -k Iio exp(evbe/kt) (1) where 1 m is called the ideality factor. This fit is shown in figure 3. The first term is attributed to the recombination or non-ideal base current which is the part of the base v,, (V) Figure 3. A typical example of the fit of the base current IB to the sum of two components as given in equation (l). The dashed curve and the dots are the measured of data figure 2. The two straight lines are the fitted base current components. The steeper, ideal current is fitted parallel to the 1, data and the non-ideal current line is fitted to the low-current le data. The curve fitting w,s done between the marked points. The measured values are: junction temperature 296 K, Ig I,, = -13.7,lg I ~ = o and m = 1.71.

5 llf noise in bipolar transistors 81 current which does not contribute to the transistor action since the carriers involved recombine before they reach the collector. This will be discussed in more detail later. The second term is the ideal base current due to the conventional diffusion process. In the transistors studied the two terms were of comparable size in the voltage range where the noise measurements were of most interest. A curve-fitting procedure was used which fitted Zbo, Zi;o and m to the data over most of the measured range of VBE. The device gain is FE = ZC/ZB and this was computed, and is shown in figure 4(a). At high ZC it decreases due to high injection and collector series resistance effects but these are not significant here. At low ZC it decreases because of the increase in importance of the non-ideal current term. At very low currents the effect of the cusp is seen as an apparent rise. The non-ideal nature of the device is often expressed by another ideality factor which we call here m' and is given by IC)/vBE]/d[ (lgzb)/vbe]. (]g m' E d[ (2) It is thus the local value of the ratio of the slopes of the forward characteristic curves. It has a value m at small voltages where the non-ideal current dominates and unity high at voltages where the ideal, diffusion, current dominates. At very low voltages it is influenced by the cusp in the ZB curve. This is shown in figure 4(b) lg (I, In A) 3. - (61 2 m' - 1, l, l,,,, ~ ~, VBC(V) Figure 4. (a) The static current gain FE presented as Ig ( h ~ against ) Ig IC, (b) The ideality factor m' d[(lg k)v~~]/d[(lg IB)/VBE] presented against VBE,

6 82 C Green T and B K Jones 3.2. White noise analysis The white noise intensity was studied as a function of bias and temperature. was It found that the results could be accurately described by the shot and thermal noise contributions of the device elements. This described by the equation and the noise equivalent circuit is shown for figure 5 (e.g. Unwin and Knott 1980). Here re is the emitter junction slope resistance, rbb is the base spreading resistance and Rs is the source resistance. The term 4kT re/2 may need modification if the emitter current is significantly non-ideal but this is not usual since the emitter current is dominated by the collector current. To fit the data the device constants were determined from the DC characteristics except rbb which was determined from the white noise component measured at an optimum value of ZB and with Rs = 1Q. Thus, from the DC and white noise analysis we could have confidence that we understand the device operation and its important parameters. Figure 5. Bipolar transistor noise equivalent l circuit. (After Unwin and Knott 1980.) 3.3. l/f Noise analysis The excess noise was found to have a l/fy dependence. The data were reduced to an equivalent voltage at the input and fitted over as wide a frequency range as possible. The values of y and the intensity at 1 Hz were computed. The noise equivalent circuit of a system can usually be represented by a current and a voltage generator at the input. The excess noise measured with large source resistance, Rs, will be dominated by any current source while that with Rs = 0 will de-emphasise the current source contribution. We will show later in this section that no voltage generator was needed to represent the data and we will now discuss the current generator. The source of the excess noise is not yet established so that various fitting procedures were attempted to determine a simple bias and temperature dependence for the excess noise. The base-emitter junction has been considered a likely source and a plot of the current noise intensity against ZB is shown in figure 6(a) for different temperatures. AS found by other workers, the fit is not a simple power law or a simple temperature

7 bipolar in llf noise transistors 83 dependence. However, if the same data are plotted, in figure 6(b), againstli then it is seen that a simple, temperature independent, power law is obtained. Thus - (Zi)bf = KZA2 A f/p with K temperature independent and ynear unity. 1- Temperature (K1, L -7-6 Temperature (K1 A lg (non-ideal base current In AI Figure 6. The excess noise intensity referred to the input (calculated at 1 Hz) against the total base current (a) and the non-ideal base current (b). Temperature is the parameter. Lines with a slope of 2 are drawn to lead the eye.

8 84 C T Green and B K Jones This is consistent with the results of Stoisiek and Wolf (1980a) and Higuchi and Ochi (1977) who produced similar data but did not continue their analysis far enough to derive this simple result. The square-law dependence fits in with the observations in all the other linear systems in which Vfnoise has been well studied and has been interpreted as some sort of fluctuation in the resistance, emission, scattering or recombination. The representation of the excess noise by only a current source at the input now has to be justified. Figure 7(a) shows the excess noise intensity calculated as a voltage at the input displayed as a function of Rs. It is seen that the voltage noise intensity is proportional to R; at large RS so that a current generator is indicated. At low Rs the intensity becomes constant and this might indicate a voltage noise source. However. although the external source resistance, Rs, can be.reduced to a negligible value there are resistances, internal to the device, through which any noise current must flow and hence will generate an apparent input voltage noise. In figure 7(b) the noise constant K is plotted against (Rs + rbb + re)* and it is seen that the current noise flowing through this total source resistance can account for the data and no voltage noise generator is necessary. Thus the noise is referred to the input as voltage a " (d)l,f= (ii)ly(rs + rbb + re)' (5) which is generated by the noise current flowing through the input circuit. The use of re rather than the normal bipolar input impedence rrr = hfere is because the non-ideal current, and hence the noise current, does not take part in the transistor action so that there is no negative feedback from the output current generator through re. Figure 7. (a) The excess noise voltage referred to the input as a function of the source resistance, Rs, with base current as parameter. (b) The same data reduced to the noise constant Kfrom equation (4) and displayed against (& + rbb + re). Lines with Slopes * 2 are shown. The strong connection between the non-ideal base current and the excess noise appears to be independent of the details of the non-ideal base current. In figure 8(a) is shown the K value measure of the noise as a function of temperature for one transistor with repeated measurements. This measure of the excess noise is thus independent of

9 llf noise in bipolar transistors 85 temperature. This is significant because the non-ideal base current varies with temperature and the ideality factor m describing the non-ideal base current is also shown to vary with temperature in figure 8(b) , J, " Device junction temperature (K1 Junctlon temperature (K) Figure 8. (a) The variation of the noise constant Kwith temperature. 3. results of repeated measurements on one sample; C, estimates of the mean. with deviation indicated by [ 1. Specimen BC413 : A249. (b) The variation of the ideality factor m with temperature for devices2n930:s2(o).bcj13:a249(0)andbclrjlc:sj(o) BFY 90 2N930-4, Figure 9. The variation of the noise constant K with emitter area for the four devices given in table 1. A slope of -1 is drawn to the lead the eye. The data were taken at T = 300 K. Z~=3x10-6AA,V~~=5V.R~=10552.

10 86 C T Green and B K Jones In this work we did not have the facility to fabricate devices with different geometries or sizes on the same slice but did attempt to investigate the geometry dependence by performing the same analysis on a variety of devices. The K value is shown against base-emitter junction area in figure 9 and the device details are given in table 1. It is seen that there is statistical evidence of a dependence on 1/A. This was also seen by Higuchi and Ochi (1977) in a controlled experiment. This behaviour is what one might expect if each fluctuating element were independent and they uniformly were distributed across the area. However, the quality of this experimental result is rather surprising since the devices are made for different purposes and are from different manufacturers using different processes. The relationship, if not fortuitous, suggests either some fundamental phenomenon or a consequence of a very common element in the manufacturing process or the materials used. Table 1. Physical characteristics of the devices used for figure 9. Emitter junction periphery area Base Emitter area type Device (lo- mm2) (10- mm ) (mm) Geometry BFY Interdigital 2N dot Ring BC Dumb-bell BUY emitters Finger The variation of the exponent of the frequency y is shown against Tin figure 10. It is seen to have a slow variation and this is not yet explained. Because of this change there will be some uncertainty in the value of the excess noise intensity used: here the value is that at 1 Hz. In a few devices the DC characteristics and noise were studied when the collector and emitter contacts were reversed. Normal transistor operation was observed but with very low gain. The ideality factor was found to be near unity and thexcess noise reduced. However, these results did not give conclusive evidence that the noise factor K was less in this configuration Llevlce ~unctlon temperature (K) Figure 10. The variation of the excess noise exponent y with temperature for device BC413 : A31 measured at IE = 1.1 x lo- A.

11 l/f noise in bipolar transistors Discussion The results show that the appropriate way of assessing the l/f, excess, noise in bipolar transistors is through a measure of the non-ideal or recombination current in the baseemitter junction. This method of analysis is found to be appropriate for several planar silicon devices and may be generally applicable to diodes and other bipolar devices. The noise can be represented by a single current source - (ii)uf = K Zh2 A f/af :. The constant K is independent of T, m,zh/zb and yis nearly unity. This representation has great attraction since it is consistent with most other l/f fluctuations. It represents a fluctuation in some scattering or recombination process distributed over some area or volume. The square-law dependence on current is the same as that found in other systems and the dimensions of the constant are integral except for the slight deviation of yfrom unity. We find a value of K = 60 X cm2 which compares with 1 x 10 cm2 found by Higuchi and Ochi (1977). These results can indicate the relative suitability of the several models that have been proposed Base resistancepuctuations One suggestion has been that th excess noise is caused by a fluctuating base resistance. rbb (Stoisiek and Wolf 1980a). Thisfollowsfrom the observation that manyl/ffluctuations are resistance fluctuations. This resistance the geometric or spreading resistance due to the semiconductor base connection. If the noise were caused by such a mechanism one would expect the noise to depend on the total base current Zi and not just the non-ideal component Zb2. We also have statistical evidence which suggests that the excess noise does not depend on fluctuations in the base resistance rbb (Green et a1 1985). To investigate this model further a study was made of the dependence of the transistor parameters on the magnitude of the collector voltage V,. The collector voltage has only a small effect on the transistor parameters; the Early effect. As the collector-base voltage is increase, the depletion volume increases partly at the expense of the base volumn so that the base width decreases. Hence gain the increases since the carriers can be swept through the base region more easily and also the base resistance, Ybb, increases. Measurements were made on the gain, rbb, and the excess noise variation with collector voltage. The results were not very accurate since the effect is small. However. we can say that any changes in the noise constant are small and perhaps not significant but they are in any case not faster than linear in &b. Bulk theories (Hoppenbrouwers and Hooge 1970) or surface theories (Black et a1 1982) suggest that an r3 or r2 dependence should be observed. We therefore conclude that the noise is not caused by a base resistance fluctuation Mobilitypuctuations The mobility fluctuation expression for l/f noise has been interpreted for bipolar diodes by Kleinpenning (1980) and in turn for bipolar transistors by Van der Ziel(l982). In an npn transistor there are two contributions due to the mobility fluctuations. These are in the electron current ZEB passing through the base from emitter to collector and in the

12 88 C T Green and B K Jones hole current ZBE injected into the emitter from the base and recombining in the emitter. These terms are where rn and tp are the diffusion lifetimes and WB and WE are the base and emitter widths. The former contribution is likely to dominate but these expressions bear little resemblance to the results found here experimentally. The details of the recombination processes are not considered. Stoisiek and Wolf (1980a) use a version of this model to analyse their results, but large assumptions had to be made in the complex analysis Base currentflow The connection of the excess noise with the non-ideal current is of help to determine the source of the noise. However, the processes involved in the generation of the non-ideal current are themselves not well established. The current components within a transistor are shown diagrammatically in figure 11. Here I,, I,, IC represent the total terminal currents and ZEB is the ideal, diffusion electron current between the emitter and collector. The base current is made up of IBRC, the generation or leakage current derived from the base-collector junction depletion region; ZBE the current due to hole flow from the base Figure 11. A schematic representation of the current flow in an npn bipolar transistor. ZE, ZB and IC are the total terminal currents. ZEB is the ideal, diffusion electron current between emitter and collector. ZBRC is the generation, or leakage current in the base-collector junction depletion region. ZBE is the base current due to hole flow from the base recombining the neutral emitter. ZBR is the electron-hole flow recombining in the base-emitter depletion region and ZBB is the electron current from the emitter recombining in the neutral All base. the recombination processes can also take place in the surface, shown as dashed curves, as well as in the bulk. 0, electrons; 0, holes.

13 transistors bipolar in llf noise 89 recombining in the neutral emitter; ZBR the electron-hole flow recombining in the base-emitter depletion region andzbb the electron current from the emitter recombining in the neutral base. Each of these recombination processes can be located either in the bulk or in the appropriate surface region. The non-ideal current is normally associated with the component ZBR and especially the part ZBRS where the recombination is at a surface. We will consider models based on ZBRS and ZBR before considering the possibility that the other components may be involved Bulk recombination centres The conventional description of the non-ideal current is based on the recombination of carriers through Shockley-Reed-Hall (SRH) centres in the junction depletion region. This model is due to Sah et a1 (1957) who showed that the dominant traps are those situated at the Fermi level which is near mid-gap and that for this process the m value is 2. This model has been successively refined to account for the observed m values both greater than and less than 2 and also for the fact that m often does not depend on VBE. Buckingham and Faulkner (1969) introduced a trap density distribution which was non-uniform in two dimensions. Nussbaum (1973), Ashburn et a1 (1975) and Dhariwal and Srivastava (1977) have extended the calculations to consider different trap energies and distributions and also the effect of removing the depletion approximation. Lefferts (1981) has done a detailed analysis. A simple generation-recombination process cannot account for the vf noise spectrum. In many vf noise models the spectrum is formed from the sum of spectra derived from a distribution of time constants. These time constants have to be long to correspond to the low audio-frequency spectrum of the noise and have to be derived from some natural process which gives a suitable weighting to the time constants to give the spectrum. For junctions with a low trap density the relevant time constant in the SRH process is the carrier transit time which is very short (Lauritzen 1968). To explain the phenomenon observed a fluctuation in the efficiency of the SRH process is needed. If such a fluctuation could be introduced which is independent of the recombination process itself, that is the trap distribution in energy and space, then it might account for the lack of sensitivity of the noise on the value of m, the temperature and VBE Surface recombination Recombination at the surface near the base-emitter junction would be an attractive explanation for the link between the noise and the non-ideal current since vf noise is strong in MOS devices where the carriers interact strongly with an oxide interface. For these devices there exists a well-established and credible model due to McWhorter 'which has been adapted for bipolar devices (e.g. Hsu 1970, Blasquez 1976, Stoisiek and Wolf 1980a). In this model, traps are assumed to be uniformly distributed throughout the oxide and these fill and empty by a temperature independent tunnelling process. The fluctuating occupancy produces fluctuations in the carrier density, surface potential, depletion region volume and surface recombination velocity. Black er a1 (1983) have suggested that glass properties of the oxide produce these fluctuations. Surface recombination and a link with l/f noise has been established in some devices (Blasquez and Roux-Nogatchewsky 1980). Kwok and Lee (1980) report that the surface recombination current has a very low activation energy, about 0.28 ev, and an ideality factor m greater than 2, whereas we find here values between 0.44 ev and 0.53 ev and m much less than

14 C T Green and B K Jones 1.8 at room temperature. Lefferts (1981) has produced a model of the non-ideal current based on recombination at surface states. The non-uniform distribution of surface states with energy is a possible cause for both the m factor and the vfnoise. We do not rule out surface recombination effects, although there is evidence of an area, rather than perimeter. dependence (Higuchi and Ochi 1977) Metallic precipitates Lefferts (1981) has analysed the SRH recombination process in bipolar transistors in some detail and has concluded that it cannot account for the observation that m is independent of VBE. He produced another model based on recombination at metal particle precipitates embedded in the semiconductor junction. The particles act as recombination centres as they charge and discharge by carrier transport over the Schottky barriers at their boundaries. This model produces value a of m which changes little with VBE although the temperature variation has not been considered. Since Schottky diodes show vf noise one might expect it to be apparent in the recombination current itself. The noise mechanism in Schottky diodes (Grant eta1 1978) is due to the fluctuation in the occupancy of the traps within the potential barrier in the depletion region round the metal. Since a single electron entering or leaving a particle will change its potential one might also expect slow fluctuations in the recombination velocity at these surfaces also. The major difference in behaviour between junctions incorporating SRH centres or metal particle precipitates is in the reverse leakage current and breakdown. Junctions containing SRH centres within the space-charge region produce a generation current proportional to the number of traps and hence the depletion volume for a uniform spatial distribution of traps. Thus one would expect the reverse current show to a V#; variation for an abrupt junction and a V@ variation for a linearly graded junction. At high voltages the breakdown is very sharp. For junctions containing the metal precipitates Lefferts calculates the leakage current to be soft right up to the breakdown region with a V variation and 1 < n < 7. The devices used by Lefferts show a variation of leakage current and breakdown corresponding to the presence of metal particles. However, for our devices shown in figure 1 we find a V,3 and V$i3 variation in the reverse leakage current as expected for a nearly uniform distribution of traps in a slightly graded junction, and the breakdown is fairly sharp. Gettering is known to reduce the density of these particles and also the l/f noise (Stoisiek and Wolf 1980b, Lefferts 1981) so in some devices this may be a contributing mechanism. Our devices had not been intentionally gettered. We conclude that our devices do not contain a high density of metal particles and that the l/f noise does not derive from this source. The details of recombination at, or near, the metal-semiconductor emitter contact interface have not received much attention. The A1-Si junction may itself be a Schottky barrier and these are known to be noisy since a small amount of trapping can produce a fluctuation in the barrier height and hence a large current fluctuation. Recently considerable progress has been made in the understanding of the structure of the contact (Pramanik and Saxena 1983a, b). The solubility between A1 and Si is such that Si dissolves in A1 at high temperatures but recrystallises out as p-material on cooling. Aluminium spikes are also formed pointing into silicon. the The aluminium reduces the silicon oxides to make good a metal-semiconductor contact but A1203 is produced. Thus the expected interface is complex with a rough transition region with semiconductor and

15 llf noise in bipolar transistors 91 perhaps metal particles and also oxide layers and oxide particles. This type of structure allows fluctuating recombination processes similar to those already mentioned for the oxide interface and the Schottky barrier processes. A fluctuation in the contact recombination rate would produce a fluctuation in the recombination current since the carrier density gradient would also fluctuate. 5. Conclusions For several low-noise, planar silicon bipolar transistors we have established a simple relationship which gives a measure of the l/f noise and links it with the recombinatibn processes in the base-emitter junction which result in the non-ideal base current. Acknowledgments We wish to thank D H Bidle and P M Lee for their help in performing the experiments and M J Turner for advice and encouragement. Financial support for the research and a studentship (CTG) was provided by SERC and Ferranti Electronics Ltd. References Ashburn P, Morgan D V and Howes M J 1975 Solid State Electron. l Black R D, Restle P J and Weissman M B 1983 Phys. Rev. B Black R D, Weissman M B and Restle P J 1982 J. Appl. Phys Blasquez G 1976 Phys. Stat. Solidi a Blasquez G and Roux-Nogatchewsky M 1980 Rev. Phys. Appl. IS Buckingham M J and Faulkner E A 1969 Radio Electron. Engng Dhariwal S R and Srivastava G P 1977 Solid State Electron Grant A J, White A M and Day B 1978 Noise in Physical Systems ed. D Wolf Springer Series in Electrophysics vol. 2 (Berlin: Springer) pp Green C T 1983 PhD Thesis University of Lancaster Green C T, Jiagbogu 0 A and Jones B K 1985 J. Phys. D: Appl. Phys. at press Handel P H 1975 Phys. Rev. Lett Phys. Rev. A Higuchi H and Ochi S 1977 Proc. Symp. on llf Noise, Tokyo pp Hooge F N, Kleinpenning T G M and Vandamme L K J 1981 Rep. Prog. Phys Hoppenbrouwers A M H and Hooge F N 1970 Philips Res. Rep Hsu S T 1970 Solid State Electron Kleinpenning T G M 1980 Physica 98B Kwok K W and Lee K-F 1980 Solid State Electron Lauritzen P IEEE Trans. Electron Deu. ED-l Lefferts R B 1981 PhD Thesis Stanford University Nussbaum A 1973 Phys. Stat. Solidi a Pramanik D and Saxena A N 1983a Solid State Tech b Solid State Tech Sah C-T, Noyce RN and Shockley W 1957 Proc. IRE Stoisiek M and Wolf D 1980a IEEE Trans. Electron Deu. ED b Solid State Electron Unwin R T and Knott K F 1980 IEE Proc Van der Ziel A 1982 Solid State Electron