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NOISE AND CURRENTVOLTAGE CHARACTERISTICS OF NEARBALLISTIC GaAs DEVICES By ROBERT ROY SCHMIDT A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1983 I ACKNOWLEDGMENTS The author wishes to express his sincere gratitude to Dr. C. M. Van Vliet and Dr. G. Bosman for their research guidance and aid in improving drafts of the work, and to Dr. A. van der Ziel, Dr. E. R. Chenette, and Dr. A. Sutherland for their helpful suggestions and kind interest. He wishes to thank Mark Hollis for fabricating the devices. Finally, the author appreciates the help of his fellow students in the Noise Research Laboratory, especially Bill Murray for drawing most of the figures contained herein. TABLE OF CONTENTS ACKNOWLEDGMENTS . . . ABSTRACT . . . . CHAPTER I INTRODUCTION . . . II THEORY OF VERY SMALL LAYERS IN GaAs . III EXPERIMENTAL PROCEDURES AND MEASUREMENT CIRCUITS . 3.1 CurrentVoltage Measurements . 3.2 The 0.24 Hz to 25 kHz Correlation System . 3.3 The 50 kHz to 32 MHz System . IV MEASUREMENT RESULTS . . 4.1 CurrentVoltage Characteristics .. 4.1a The n+nn+ Device . 4.1b The n+pn+ Device . 4.2 The n+pn+ Device Noise . . 4.3 The n+nn+ Device Noise . . V DISCUSSION OF EXPERIMENTAL RESULTS . . 5.1 The n+nn+ Device . . 5.1a CurrentVoltage Characteristic and Impedance . 5.1b Excess 1/f Noise . 5.1c HighFrequency Noise . 5.2 The n+pn+ Device . . 5.2a CurrentVoltage Characteristics and Impedance . 5.2b Noise . . VI CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER WORK . 6.1 The n+nn+ Device . . 6.2 The n+pn+ Device . . Page . ii 3 S. 12 . 12 12 . 14 . 40 S. 46 . 46 . 46 . 49 . 56 . 63 . 75 . 75 . 75 . 76 . 79 . 80 . 80 . 82 . 84 . 84 . 85 1 Page APPENDIX: COMPUTER PROGRAMS FOR THE HP 9825 ... 88 REFERENCES .. .. .. . .. ..... 93 BIOGRAPHICAL SKETCH . . . 96 iv 1 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NOISE AND CURRENTVOLTAGE CHARACTERISTICS OF NEARBALLISTIC GaAs DEVICES By Robert Roy Schmidt April 1983 Chairperson: C. M. Van Vliet Major Department: Electrical Engineering + + + + Conduction processes in novel submicron n n n and n p n mesa structures in GaAs are investigated. The widths of the ntype and p type layers are 0.4 pm and 0.47 im, respectively. These small devices are of interest due to the possibility of ballistic (collision free) transport which leads to picosecond switching times at femtojoule power levels, since the velocities of the carriers may be much greater than the collisionlimited drift velocity. Three noise measurement systems are described, covering the frequency range 0.2 Hz to 64 MHz. The first system, for frequencies up to 25 kHz, features a spectrum analyzer with a dualchannel fast Fourier transform algorithm. By exploiting the correlation feature, the system,when combined with very lownoise preamplifiers, has a background noise level for a lowimpedance source of less than 0.3 ohms noise resistance. The other two systems for radio frequencies use a tuned stepup transformer to bring the noise of the very quiet ntypF device to a level above that of the RF preamplifier. One is a conven tional singlechannel system. The other uses RF mixers before the dualchannel FFT analyzer to extend the range of the correlation system to high frequencies, depending on the preamplifier frequency response. Currentvoltage measurements, DC, pulsed DC, and AC impedance, are presented versus parameters bias level, temperature, and frequency, as are noise measurements. The ntype device is linear and relatively temperature independent exhibiting full thermal noise and, at low frequencies, very low levels of excess 1/f noise five orders of magni tude less than for the bulk material. This suggests that collisions are mainly absent from this device. The ptype device, conversely, is nonlinear with large levels of excess low frequency noise. There is a linear, noisy, and temperaturedependent lowbias region followed by a transition to a less noisy temperatureindependent highbias regime with large conductance. This may represent a transition from ambi polarly governed to nearballistic transport. 1 CHAPTER I INTRODUCTION There is great interest in very thin GaAs layers of lengths less than 1 m. This is of the order of the mean free path lengths of the dominant collision mechanisms for electrons. Under favorable conditions, carriers may cross the layers undergoing few or no col lisions, which leads to very high velocities, much greater than the collisionlimited drift velocity. The resulting socalled ballistic or nearballistic transport leads to very fastresponse devices for pico second switching at very low power or other new applications. These new devices have been made possible by advances in fabrication tech niques such as electron beam lithography and, in particular, molecular beam epitaxy. Such small devices enable us to investigate physical mechanisms on a small scale for which the traditional models no longer apply. In addition, practical knowledge of the performance limitations of these new devices is obtained. Early theories of these devices treated the ballistic motion of the carriers as the dominant mechanism affecting its characteristics. Recent measurements and later theories have shown that boundary condi tions, in particular the velocity dispersion of the carriers, have much greater influence. The organization of the chapters follows. In Chapter II some theories of small n n n and n p n layers in GaAs are surveyed and __ 2 predictions are noted. Measurement techniques and procedures are presented in Chapter III. In Darticular, a crosscorrelation noise measurement system with a noise resistance less than 0.3 ohms suitable + + for lowimpedance lownoise devices such as the n n n diode is dis cussed. In Chapter IV, experimental results of measurements on two + + devices are described. One is an n n n structure of length 0.4 pm. The other is an n p n device of length 0.47 um. Currentvoltage charac teristics, DC, pulsed DC, and AC to 25 kHz are presented, as are measurements from 0.2 Hz to 32 MHz. The measurements are repeated at 77 K to investigate the effect of temperature. In some cases, T is reduced to 12 K. Chapter V contains a discussion of the results. Finally, conclusions and recommendations for further work are presented in Chapter VI. An HP (Hewlett Packard) 9825A computer program used to automate the noise measurements is recorded in the Appendix. 1 CHAPTER II THEORY OF VERY SMALL LAYERS IN GaAs In most semiconductor devices, the drift current is limited by the rate of carrier collisions. However, for sufficiently thin devices few or no collisions occur permitting ballistic or near ballistic transport. The resulting carrier velocities can be much larger than in the collisiondominated case so that such parameters as switching speed can be greatly increased. Advances in fabrication techniques such as molecular beam epitaxy and electron lithography have resulted in submicron gallium arsenide devices with dimensions the same order of magnitude as the mean free paths of the dominant collision mechanisms. Thus nearballistic transport in these devices becomes probable. For polar optical phonon emission at room temperature, Eastman et al. [1] report mean free path lengths in GaAs of about 0.1 pm for electron energies near 0.05 eV to 0.2 um up to 0.5 eV. Phonon absorp tion is about four times less probable. They also report that the change in the direction of motion due to a collision is small (510 degrees). Intervalley scattering becomes significant for higher elec tron energies than 0.5 eV. Shur and Eastman [2] calculated the mean free path in the highpurity GaAs at 77 K to be 1.3 pm. Barker, Ferry, and Grubin [3] have suggested some complicating factors even for devices of length equal to the mean free path. First, in large area 1 devices, some carriers may move at angles to the length direction, thereby increasing the distance traveled and the number of collisions. Also, some of these carriers may collide with the device boundaries. Third, spacecharge effects may limit the current due to Coulomb scattering. Finally, short devices may be dominated by contact effects such as carrier reflection. The theory for pure ballistic transport was revived from the days of vacuum tubes by Shur and Eastman [2] based on a simple model. They assumed a onedimensional ntype device of length, L, neglecting diffusion and electron scattering. Then the current density is J = qnv (2.1) where q is the electronic charge, n is the free electron density, and the velocity, v, is found from 1 2 qVx m v (2.2) where Vx is the voltage at x due to the applied potential, V, and m is the effective mass. The initial velocity and the field at the injecting contact are assumed to be zero. Poisson's equation dV2 x 9 (n n) (2.3) dx2 EOr 0 is then solved yielding Child's law J / Er V3/2 (2.4) / m L for large bias voltage and large consequent injected space charge. For small applied voltages, the fixed charge due to ionized donors is greater than that due to the injected carriers and an electron beam drift region (J V1/2) results. The range of voltages for which their model is valid is kT AE < V < (2.5) q q where T is the lattice temperature and AE is the energy difference between the main conduction band minimum and the secondary valley. For very small voltages diffusion cannot be neglected. Intervalley scattering is the cause of the upper voltage limit. Some measurements [1,4] have been reported that appear to support this model, but their interpretation has been questioned as will be discussed. To take into account the effect of a few collisions, Shur [5] and Shur and Eastman [6] have extended the theory. Frictional "drag terms" to represent the collisions are added to the equations of balance of energy and momentum. In general, the effective mass, m , and relaxation times, Tm and TE' are functions of energy so that the balance equations are dm*(E)v m*(E)v dt qE T (2.6) dt T~M(E and dt qEv E (2.7) Shur assumes a constant energyindependent effective mass and a single momentum relaxation time, T, so that, neglecting (2.7), (2.6) is replaced by dv my m t = qE (2.8) Equations (2.1), (2.3), and (2.8) along with the boundary conditions for the ballistic case are then solved analytically. For T much longer than the transit time and sufficient applied voltage to justify neglecting no, Child's law is again obtained. Conversely T much less than the transit time yields (for V= iE where i is the mobility) the MottGurney law for collisiondominated spacecharge limited current 9 V J EEr (2.9) In the second paper, values of m*(E), v(E), Tm(E), and TE(E) are obtained from Monte Carlo calculations. Steady state is assumed and the boundary conditions are taken to be the initial field (and velocity) equal to zero. Again letting TE and Tm become large yields Child's law. For the general case, computer solutions are presented for various lengths and doping densities which show small deviations from their previous (ballistic) results. The interpretation that current proportional to V3/2 indicates ballistic transport has been seriously questioned by some authors. Rosenberg, Yoffa, and Nathan [7] have shown the crucial importance of the boundary conditions. For the ballistic case, they assumed a simple onedimensional model neglecting collisions. For each electron, they solve equations (2.1), (2.3), and the energyvelocity relationship S(v2 v) = q(Vi V) (2.10) where vi and V. are the initial velocity and potential, respectively. A displaced Maxwellian distribution of initial velocities characterized by a temperature and mean velocity is assumed. A set of nonlinear equa tions in V result which are solved numerically for specified electric field, E, initial carrier density, ni, and initial (temperature normal ized) mean velocity, v. They demonstrate that a wide variety of current versus potential curves can be generated for various choices of initial conditions [7, Figure 3]. The authors agree with others [3,8] that, conversely with Shur and Eastman, the currentvoltage characteristic may be used to infer the boundary conditions for a device that has already been shown to be ballistic by other means (such as noise behavior). Another significant effect in short devices is "spillover" of carriers from high to lowdoped regions. The result is that the effective length of the lowdoped region is shortened and hence the resistance is less than expected naively even for devices many Debye lengths long [7]. Universal curves have been calculated by van der Ziel et al. [9] from which the magnitude of the effect can be determined for a particular device. The authors calculated the nearequilibrium resistance for a 0.4 pm ntype device with doping densities Nd+ = 18 3 15 3 101 cm and Nd = 101 cm for liquid nitrogen and room temperatures. They solve Poisson's equation including electrons and ionized donors in the high and lowdoped regions, neglecting holes. They assume n(x) = Nd+ exo fIq()l (2.11) where Nd+ is the ionized donor density in the highly doped region and ,(x) is the potential. Reciprocal mobilities due to diffusion/drift and thermionic emission are added to give an effective mobility which agrees within 10/ of experimental results at room temperature. However, a similar calculation for an n pn+ device is off by more than an order of magnitude. They assumed that the electron spillover depletes the holes in the pregion such that they (holes) may be neglected. This is apparently not the case; the effects of the holes are important. The simple theory of ballistic [2] and nearballistic [5,6] transport does not take into account the energy or velocity distribu tions of the carriers. Cook and Frey [10] have suggested the inclusion of an electron temperature gradient term in the momentum balance equation for this purpose. This may be sufficient for first order effects such as the currentvoltage characteristic. Their treatment includes an average collective velocity dispersion term, in essence treating the ensemble as a single particle for such effects as the noise, a drawback they themselves point out in the simple theory. For the noise, more adequate analysis should include a Langevin or Monte Carlo approach based on the momentum and energy balance equations which include terms necessary to account for the significant physical mechanisms. Such a Monte Carlo calculation for a 0.25 pm n in diode at 77 K has been reported by Awano et al. [11]. Ionized impurity and intervalley and intravalley phonon scattering is included in their model but not nonparabolicity of the band edges. Particles with a distribution of velocities in equilibrium with the lattice at 77 K are injected at one contact. They find that a large proportion of the particles are transported ballistically. Groups of particles cor responding to one or two collisions and some backscattering from the collecting contact is observed. For potentials less than 0.5 volts, the currentvoltage characteristic is surprisingly close to Child's vacuum diode law. Holden and Debney [12] have presented a calculation based on ideas from Fry's theory for thermionic values [13]. They assume injec tion from both n regions of carriers with MaxwellBoltzmann velocity distributions. Scattering is neglected. Following Fry, the free carrier charge density is found by an integration over the injected carrier velocity distribution. A potential minimum due to injected spacecharge is assumed. They calculate currentvoltage characteristics for T = 77 K and lengths of 0.1, 0.2, and 0.5 um. The slopes at high bias are less than 3/2 (L = 0.1 and 0.2 pm give ,1.3, L = 0.5 pm gives <1.14). At low bias, the characteristics are somewhat more linear. The authors conclude that ballistic effects do not lead to a particular currentvoltage power law and suggest that another method must be found to determine ballistic effects. The noise behavior of thin structures gives important informa tion on the extent of ballistic effects. Currentvoltage character istics for these devices are dominated by events at the boundaries such as the carrier velocity distribution. In addition, practical knowledge of the performance limitations of the devices is obtained, as well as useful insight into the physical mechanisms causing the various types of noise. The noise is naturally divided into two frequency regions. At high frequencies the dominant mechanism, variously called thermal, velocity fluctuation, or diffusion noise is the electrical result of the Brownian motion of the carriers. While the theory is not yet complete, preliminary calculations have been made by van der Ziel and Bosman [14,15]. For the ballistic case (collisions neglected) at large bias, the device is modeled similarly to a dualcathode vacuum diode with opposing thermionic emission currents across the potential mini mum. Then correlated fluctuations in the current due to fluctuations in the minimum due to the space charge result in a lowered (64%) level of noise compared to the lowbias value of thermal noise of the AC conductance. For the collisionlimited case, at sufficiently large bias, the device is modeled as a spacechargelimited solidstate diode and the noise becomes twice the thermal noise of the AC conduc tance. Thus ballistic effects should reduce the noise and collisiondominated transport should increase it if sufficient bias levels can be achieved. At low frequencies, excess noise characterized by a spectral density with a 1/f frequency dependence dominates. According to most recent theories [16,17] such noise is thought to be caused by mobility fluctuations which represent fluctuations in scattering crosssections. The usual method to describe the noise is using Hooge's empirical formula [18] 2 aHl0 SI(f) fN (2.12) where I0 is the dc current, N is the number of carriers, f is the fre quency, and aH is Hooge's parameter. This is accurate for a uniform sample. However, for a non homogeneous sample or a mesa structure, van der Ziel and Van Vliet [19] have shown that the formula becomes m 2 r L maH 0 dx S (f) H dx S fAL2 nrx) 0 (2.13) where m is the number of layers, L is the length of one layer, A is the crosssectional area, and n(x) is the electron density. The value of aH was first thought to be constant (2 x 103). Later research has found that aH depends on material variation, the dominant scattering mechanism, and carrier heating [20] in Si. Thus the magnitude of aH should give a good measure of the rate of collisions or lack of them. CHAPTER III EXPERIMENTAL PROCEDURES AND MEASUREMENT CIRCUITS + + The procedures used in the measurement of both devices, n n n + + and n p n are similar, but the actual circuits depend on the device characteristics which are quite different. The ptype nearballistic diode (NBD) exhibited an impedance about 100 times that of the ntype device at small bias. It also generated excess noise at low frequencies orders of magnitude larger than the ntype NBD. Consequently, the noise of the nNBD was the most challenging measurement requiring an equiva lent noise resistance of the measurement system of about 0.2 ohms. 3.1 Current Voltage Measurements Three IV measurements were done on each diode: DC, pulsed DC, and AC from 2.4 Hz to 25 kHz. A block diagram of the nNBD circuit is shown in Figure 3.1. A threeterminal measurement is required since the resistance of the active region is comparable to that of the bonding wire to the top surface of the diode (the devices were mounted in TO5 cans). The resistance of the output wire, RL2, is negligible since it is in series with the large input impedance of the oscilloscope. Then V2 is the voltage across RX + RS which represents the series combina tion of substrate resistance and active region. The current through the device is Vl V2 I R R (3.1) RB + RLl nNBD DC OR PULSE POWER SUPPLY Figure 3.1 Block diagram of n +currentvoltage measurement Figure 3.1 Block diagram of n n n diode currentvoltage measurement I Pulsed measurements were done using an HP 214A pulse generator. The pulse width was 1 psec and the repetition rate was 100 Hz. The AC IV measurement employed the HP 3582A spectrum analyzer. The dualchannel feature was used to measure the transfer function as a function of frequency from 2 Hz to 25 kHz and as a function of DC bias. The circuit diagram of the AC resistance measurement is shown in Figure 3.2. Taking into account the parallel resistances, RX is found to be V 1 + RS + R V C R S RL S C RB R H= V R0R L (3.2) X VO II RC1 (3.2 1 1 + T~\ VC t RB The measurement for the ptype device is similar. Here, the lead resistances may be neglected since the impedance is large; the diode may be treated as a twoterminal device. 3.2 The 0.24 Hz to 25 kHz Correlation System For the electrical noise measurement of low impedance devices with resistances of the order of an ohm, the usual singlechannel method is not useful. Typical preamplifiers have noise resistances of about 10 to 20 ohms. The device noise is masked by the preamplifier noise. Therefore, the correlation method [21] is used as shown in Figure 3.3. A Hewlett Packard 3582A spectrum analyzer featuring a dual channel FastFourier transform is employed. Lownoise preamplifiers nNBD RC' R, Ll i RL2 B Rx V Vo I MR AC resistance measurement circuit for nNBD Figure 3.2  Vbatt I 1 CHANNEL A HP 3582A SPECTRUM ANALYZER CHANNEL B 2 p2= ISAB pASB SA' SB Correlation measurement setup Figure 3.3 precede each channel from a common input. By measuring the coherence (square of the correlation between the channels) the crossspectra can be calculated. The noise generated at the output of each channel is uncorrelated with the other and therefore averages out in the final reading. Only the noise generated at the input of each channel gives a contribution to the final reading. Hence a better device noise versus amplifier background noise ratio may be obtained. The device under test (DUT) noise is compared to a known cali bration signal which is applied through a series resistor much greater than the DUT resistance. The bias current is applied in a similar manner. An equivalent circuit of the measurement setup is shown in Figure 3.4. To characterize the noise of the system, we first neglect the effect of the device noise. The series resistors to the bias and calibration sources are R' and Rs. The calibration source, v al, transforms to a Norton equivalent, ic, where ci vocal (3.3) c Rm The source resistance is represented by RX. The preamplifiers are assumed identical (with uncorrelated noise sources, vl, i1, v2, and i2) with gain, G (assumed constant), and input resistance, Ri. The output of the amplifiers, u1 and u2, are then multiplied together and averaged. The measurement procedure is to compare the output for three con ditions. They are (1) shorted input (RX = 0) giving output reading M1, (2) open input (RX = W, RS J R >> Ri) giving M2, and (3) open input 18 IM rQ + "_ c. C4 r4a t, * U C) a, U, LO LL with calibration signal applied (i 3 0) which gives M,. The output c for each case is easily found to be M =0 (3.4) R? M2 G2 1(i* + i (3.5) 2 = l T4 and R M3 G2 (ili i2 + ) (3.6) In reality, M1 is not exactly zero. There is a residual back ground noise. This can be accounted for by introducing a correlated background noise source, vb, to the input voltage sources so that vI = vi + vb and v2 = v2 + vb. The primed portions are uncorrelated. Then the analysis yields Ml =G2 VbV (3.7) S __ ___ R2 =G2 v (i i* + i (8) M2 = bVb ( + i2i) (3.8) and M3 = 2 vb + (i i* + 2 + (3.9) 3 1 Writing the mean square averages as spectral densities, i.e., i i SiVc S = 2 f etc., and noting Sc = allows the current and S voltage noise to be calculated. M M S M3 M2 SIc K M S SI (3.10) M2 N11 Il + 12 Thus SV SII + SI2 ,S (3.11) KRP' S and R2S2 SVb i VM (3.12) 2 M2 ) The current noise spectra of the two channels are SI and SI2 and the background voltage noise spectrum is SVb' A significant source of background noise, after such causes as pickup of unwanted signal, ground loops, and power supply noise have been eliminated, is the finite averaging time of the spectrum analyzer. The sample coherence for shorted input versus number of averages is shown in Figure 3.5. It is a largeside biased estimate inversely pro portional to the number of averages which is limited to 256. The distribution of sample coherences is not easily described [22] for 1 10 100 10 NUMBER OF AVERAGES Figure 3.5 HP 3582A sample coherence of averages for shorted input versus number IT LJ z Z W LT 0 w small expected value. The residual nonzero value is interpreted as background noise. The system is used with two similar sets of preamplifiers denoted PAl and PA2; PA2 features five parallel input stages like that of PA1 but with larger bias current. They are shown in Figures 3.6 and 3.7; PA1 has larger voltage noise but extended low frequency response suitable for excess 1/f noise measurement. The PA2 has been designed for minimum voltage noise. The PAl features a capacitively coupled common emitter transistor first stage with shunt feedback. This reduces the input impedance but does not affect the noise. The primary motivation for this circuit is a practical one. The RC settling time of the base bias circuit is substantially reduced compared to the usual configuration. The tran sistor (GE 82) is chosen for large and small series base resistance. The second stage is a lownoise operational amplifier (BurrBrown OPA101BM) in noninverting configuration;12volt automotive batteries are used for the power supply. An equivalent circuit of the input stage is shown in Figure 3.8. It is the lowfrequency hybrid pi including source resistance, feedback, and noise sources. The source resistance is RS. Thermal noise of the series base resistance, rb, is vb. Shot noise of the base and collector currents are denoted ib and ic, respectively. To transform this circuit to a form similar to that in Figure 3.4, the closed loop gain, the input resistance, and the noise sources must be calculated. The closed loop gain is found by summing currents at 12 v 12K 487K IlOOhF SINPU GE 8, INPU I I +12V 9 or oPA01BM 0.5, F Figure 3.6. Model one (PAl) lownoise premplifier OUTPUT 91K the input v v. v. i + o i 0, (3.13) s RF 1 Rs and at the output V V. V i. + 1 0 (3.14) 1 R R F C where R = r R, i = gmv and the noise sources have been neglected vi for now. Letting r = rb + r so i = the output equation gives i in terms of v which, when substituted into the input equation allows i ri to be expressed as a function of v Then, for RF , the closed loop gain, A, is found to be v RF A R F (3.15) RF + RC + r. + 1 + S C 1 RF RSJ 1 + BRC The loop transmission is found by disconnecting the output current generator, gmv and applying a test current generator, iT, there. The output voltage, v is v = iT [RC (RF + RS r)] (3.16) The input voltage fed back is 27 r.RS (v v.) v. RS 0 (3.17) i r + R RF 1 S F so that ri RS (r + RS) RF i = vR (3.18) ri + RS) RF Then, noting that gmv, = gvi gives g v _Rc 1 RV rBR T RF + RS+ir ri + r(s C F R + r R S il( S) The input resistance is RS in parallel with Ri where, for RF >> RC and RF >> ri/, RC + R R. = r. (3.20) 1 + r R g 1+ Rcgm I The output resistance is R in parallel with RL where r.R R + r+ R R = r (3.21) 0 0 r. B+ 1 + RS Except for the noise, the circuit of Figure 3.8 now becomes that of Figure 3.9 where the voltage gain is lE  C) fR + R G = A R 1 (3.22) ( RSRi For the case that RF >> RC, ri, Ri >> rb, RS, and (riRF)/RS >> BRC, then BRC G Fr (3.23) The gain becomes open loop as the input is shorted by a small RS. To transform the noise sources, the effect of feedback is neglected. Open circuiting the inputs (RS = m) of Figures 3.8 and 3.9 and setting the output voltages equal yields i S rb Vb + 9C i =g rb m (3.24) a b r + r rb + r, Short circuiting (RS = 0) the inputs yields i a = Vb + + ibrb (3.25) gm Substituting numerical values for components with RS = 5.6 ohms and assuming 8 350 yields about 1200 ohms for the input resistance. The measured value for two PAls in parallel is 660 ohms as shown in Figure 3.10. The input resistance of two PA2s in parallel is shown in Figure 3.10. For the gain the calculated value is about 51 decibels. The result including the second stage is 90 decibels. The measured value is 89.5 decibels as shown in Figure 3.12. O 0 o 0 0 0 oN o o 0I O o o LO > o C\W ooz 0 oJ 0 LLj 0LOLL 0 m U O O 0 0 O o 0 0 0 0 o 0 c j 0 0 d (L>)30BNVISIS3 d IndNl 8 8 8 0 0 0 0 0 UL rno (2) 30NViSJI S3] N C) z O O 0* 0 W0L 0 O O J O 0 0 O II 0 t, w= r 0  Q) (8 0  0 =r O  O  0 0L o co too P (SP) N1V9 The noise resistance of PA1 and PA2 were measured to be about 35 and r2.5 ohms, respectively. The five in parallel input stage of the PA2 gives that factor of reduction in the noise resistance. Another factor of three reduction comes from increasing the input bias currents by that amount since, to first order approximations, the voltage noise sources are inversely proportional to bias [23]. Using the correlation setup gives about a factor of ten reduc tion in the voltage noise. This is seen in the ploLs of voltage noise resistance and current noise conductance for the two setups shown in Figures 3.13 through 3.16. The noise resistance, Rn, is defined by SVb = 4kTRn (3.26) Similarly, the noise conductance is defined by S1 + SI2 = 4kTGn (3.27) The current noise of the PA2 is much larger than that of the PA1 as would be expected from larger bias currents and five in parallel. For the sensitive measurements, the current noise is shorted by the small resistance of the DUT. This is the full current noise which remains in the correlation calculation. Of the voltage noise, only the background level remains. The device noise measurement procedure is very similar to that for the amplifier. Three conditions are recorded. They are (1) DUT on, (2) DUT off, and (3) calibration signal applied with DUT off. Any change in DUT resistance between the on and off state must be 0 0 0 0 " 0 D0 00 O 0 O 0 O O D 0 0 0 T 1 J CO 0D C j (SIHO) 33NVISIS3d 3SION S 4 N 0 > o LUL 40 D Sr 01 r .4 o 0 4 0 U N z 0 : O= 00 0 o L  00 CY O ( u o I ou t 10 0 o o 'o 'o  (N B"I lS) 0NV1.I,.Nf NOO "SION L 0 O S0O O 0 0 ) O 0 O 0 0 O0 O % 0% (SINHO) 3DNVLSISld 3SICN 39V71iOA ,0 1 ro I 02 > O  I a LiJ LL 0 I 0 0 .. O O O o i >  LLJ 0 u a> V) S ;5 o 1 r  O (n) ONVI.onON0O o ,  3SION IN3fnO , U ., 1 ,! taken into account (replacing the device by a resistance equal to its "on" resistance for the latter two measurements). The equivalent circuit of the input can be drawn as in Figure 3.17. The DUT is represented by the parallel combination of Rx, the AC resistance of the device, StLf, the thermal current noise generator, and V^'f the excess th x current noise generator. The calibration signal is ^cal f. The ampli fier is characterized by Ra, /Sif and /VLf A Af For the three conditions, the outputs are M = G2[(S + S + S)R2 + SI] (3.28) 1 x th A A M2 = G2[(Sth + Si)R2 + SA] (3.29) and M3 G[(Sth + SA + Sal)R2 S] (3.30) where G is the (constant) gain and R is the parallel resistance at the input. Then, as before, M1 M S K 1 2 x (3.31) M3 M2 Scal so that the device noise current can be written S = S +S = KS + (3.32) DUT x th cal R x L.LJ Z zI a) w , o S. 4, '4' 0) 4, v C, c, r, CL ar C a) 5 + + dV Note that for the nonlinear n p n diode, Rx d and depends on the bias current applied. An HP 9825A controller can easily automate this measurement and be used to average many data sets for increased accuracy. A program which does this for eight logarithmically spaced points per decade is presented in the Appendix. 3.3 The 50 kHz to 32 MHz System At radio frequencies, the same threemeasurement procedure as before is used for the noise. The circuit is more conventional, as shown in Figure 3.8. A stepup transformer consisting of four trans mission line transformers in cascade with capacitancetuned input and output is employed for the ntype device. See the Radio Amateur's Handbook for details. A Micronetics KSD20LEE solidstate noise source is used for the calibration signal. Frequency selection is done by an HP 8557A spectrum analyzer. The IF output signal from the spectrum analyzer is passed through a 21 MHz bandpass filter and detected with an HP 431C power meter. The preamplifier, designed and built by Christopher Whiteside of the noise research group, is shown in Figure 3.19. It employs two stages of lownoise FETs (2N4393) in cascade configuration to obtain 30 dB gain with a noise voltage resistance of about 70 ohms. This is less than the steppedup noise of the DUT. The HP 3582A spectrum analyzer can be used to make noise measurements in the radio frequency range. This is accomplished by mixing down the RF signal just before the analyzer as shown in Figure 3.20. Only a single channel is shown, although the correlation method F'1 0 O, 0 > * If I J. z n* 43 C C ULU U Z4  0tD LS tN 0 (n\ < U C~1) LO: can be used just as easily. The bias and calibration signal are applied as for the conventional RF system. The circuit for the ntype device, using a tuned stepup transformer, is shown. The 60 dB radio frequency preamplifier is actually two of the 30 dB preamplifiers shown in Figure 3.19 in cascade. The large gain is needed to overcome the poor noise performance of the mixer (MiniCircuits ZAD1). Using a standard RF signal generator with output amplitude capability of +7 dBm, the mixer has a frequency response of 100 kHz to 500 MHz. The same signal generator can be used to supply both mixers (correlation setup). Then two 60 dB preamplifiers are used and, if series resistance or unwanted pickup of the stepup transformer is not negligible, two of these also. Feedthrough of local oscillator har monics may overload the spectrum analyzer at some frequencies if a bandpass filter similar to that shown in Figure 3.21 is not employed. This simple filter had good (90 dB) attenuation to 100 MHz. The band width of a single FFT bin of the spectrum analyzer was extremely small compared to the frequencies of interest, causing significant inaccuracy. To reduce that, a program to average 80% of the bins to synthesize a bandwidth of up to 20 kHz was written. It is shown in the Appendix. 1 4 I O CHAPTER IV MEASUREMENTS RESULTS The nearballistic diode (NBD) is a sandwiched mesa structure of five lightly doped p or n layers, alternating with heavily doped n layers. A nottoscale sketch of the pNBD is shown in Figure 4.1. 18 3 + The doping densities of the various regions are 10 cm for the n 15 3 regions, approximately 2 x 10 cm for the n regions, and approxi 14 3 mately 6 x 101 cm for the p regions. The diameter of the mesas is 100 pm. The devices were manufactured by molecular beam epitaxy at the Cornell University Submicron Research Facility by Mr. M. Hollis. The mesas have very lownoise ohmic AuGe contacts. Currentvoltage measurements of three typesDC, pulsed DC, and ACare reported. Also presented are noise spectra versus parameters bias, frequency, and temperature. 4.1 CurrentVoltage Characteristics + + 4.1a The n n n Device + + The DC currentvoltage characteristics of the 0.4 m n n n device at room temperature and 77 K are shown in Figure 4.2 In contrast to the device reported in Eastman et al. [1], there is no vV region. The characteristic is linear for both temperatures up to very large bias. Pulse techniques are used for currents greater than 100 mA. The resistance of the active region is 0.75 ohms at room temperature I00/Lm DIAMETER 18 3 NJ 10 cm 15 3 Np r 10 cm I V2 GND Figure 4.1 Ptype nearballistic diode mesa structure 48 IOA z IA H (r o 0 300 K [] 77 K 100 mA 10mA ImA 0.1 mA ' 0.OlmA O.ImV ImV lOmV VOLTAGE Figure 4.2 DC IV characteristic of n n n 0.4 nm device ICOmV I and 0.68 ohms at 77 K. This is confirmed by the AC impedance measurement at 300 K for IDC = 75 mA shown in Figure 4.3. Low frequency equivalent circuits of the device at room temperature and 77 K are shown in Figures 4.4 and 4.5. The series resistances of the gold wires of the TO5 can are significant and must be taken into account. 4.1b The n p n Device + + Conversely, the n p n structure is quite nonlinear. The DC currentvoltage characteristics of the 0.47 im n+pn+ diode at 300 K and 77 K are shown in Figure 4.6. The device is linear for both temperatures up to about 100 mV. The characteristics then move through transition to a temperature independent high bias regime in which the slope falls off. The maximum slopes in the transition regions are 3 for the 300 K case and 4.5 at 77 K. The lowbias resistances at 300 K and 77 K are 90 ohms and 320 ohms, respectively. The measurements of AC resistance for several bias currents at T = 300 K shown in Figure 4.7 and for T = 77 K shown in Figure 4.8, display no frequency dependence. To compare the large and small signal resistances more easily, the exponent, B, can be examined where I = V Differentiating both sides and solving for g gives DC S DC (4.1) AC , 0 0 0 0 O 0 O o o 0 0 S' O Q o 4 0 0  0 + 0  O o o L.) c o 0r2 o 0  o00 0 4 0 0 a) C) 0 =U 0 vr o 1j S"RLI RL2 0.75 < R, RSI 0.37 0.08 RS2 Figure 4.4 Equivalent circuit of n n n structure at 300 K showing parasitic elements R LI RL2 0.08 0.10 0.68 R RSI 0.23 0.06 R2 RS2 ++ Figure 4.5 Equivalent circuit of n n n structure at 77 K showing parasitic elements 0 300 K E3 77K ImV lOmV lOOmV IV lOV VOLTAGE Figure 4.6 DC I characteristic of n 0.47 m device Figure 4.6 DC IV characteristic of n p n 0.47 ,im device IA  1OmA H SImA LU 0.1mA O.1mV E E < Ln < O C E7> O S7[> 0 [ > O Q o Q 0 0 0 E 0 [ 0 D 0 a o O o E 0 0 O E 0 So 7 0 E 0 S0 a o 0o o(  LD Lu IC Lu 0 o 0 0  0 7 0> 0 E71> 0 1> 0O ED> E< E E li 00 ro'E CBF 0 FDizK DD7 0 ED7 CD[7 Q oD 7 o ED 7 0 LID7 c cD 7 0 ED[7 0 (u)i) na 0o 0v O 0 0 o N C) z Cy LU 0 o r o I 0 r 0 I 1 I I E w E This is plotted in Figure 4.9 for small bias up to 34 mA. At higher bias, pulsed DC but not AC measurements were done. The resistance falls dramatically at high bias. This is seen more clearly in Figure 4.10 which presents the DC and pulsed DC conductance versus bias voltage for several temperatures down to 12 K. The largebias conductance in the temperature independent regime rises to a limiting value of about 0.33 Siemens. In the lowbias regime, the conductance quickly falls from its 300 K value by about a factor of three at 150 K to 200 K and remains nearly constant for further decreases in tempera ture. + + 4.2 The n p n Device Noise The n p n 0.47 ;m structure exhibited large levels of excess lowfrequency noise. The room temperature noise current spectrum (less than 25 kHz) for several bias currents is shown in Figure 4.11. The frequency dependence for all bias currents is about .75. In Figure 4.12 is shown the extension to 32 MHz of the spectra for bias currents of 100 pA and 1 mA. The 100 1A spectrum is approaching the thermal level at the highfrequency end. The turnover frequency to thermal noise occurs in the GHz range for larger bias. Figures 4.13 and 4.14 display the complete spectra versus frequency for bias currents of 34 mA and 68 mA, respectively. Both show slopes of 0.77 over many decades. No turnover frequency to a different slope is found. These spectra represent measurements in the lowbias regime and transition region. Measurement in the highbias regime requires pulsed techniques. iCD O O ro 0 1 I 1  qo d to() CiJ C Z LU :D 0 O 58 > = L) cc Ln   9> > L M 4) o F 0 a u *r L0 o V, I n c o P +1 ~C 00 0 0 0 0 O 0o 0 0 0 O o Q 0 0 .., ui "G 0 0o, ^ a^ So> A2 I 0r ! 0 0 N/ /" U \ 0 [ LJ LL. k,) 0 rn 0. S34 mA S13.6 mA 0 3.4 mA 0 I mA 0 400oA V o 'I SlooLA p7a 20 10 f (H:) Figure 4.11 Lowfrequency pNBD room temperature current noise spectra 14 10  15 16 10  10'6 < WO C), D 77 D Or7 00  7 77v 77 7 ImA D IOOA E 17 '7 77n v77 v77 v I \ Figure 4.12 UIK lOOK IM IOM lOOM FREQUENCY (Hz) Current noise of 0.47 pm n pn+ NBD at T = 300 K for I = 100 IA and 1 mA 10! IO17 N zJ` 7 1020 10 i ,, I I r0 0 2 0 0 0 N 1. o 0 0 (ZH/VIs 10 10 Sz 0 CO CO N o J = Q SO > o . o ac G 0 L S LL z ^r + o r 0 U 0 0 A C o C .?' I The room temperature current noise at 100 Hz is plotted versus bias current in Figure 4.15. There is an 12 dependence at lower currents up to a few mA. At higher bias the noise increases less fast. The deviation from 12 behavior appears to coincide with the transition region. The lowfrequency current noise spectra at 77 K for several bias currents are shown in Figure 4.16. The levels of excess low frequency noise are again large. The slopes are somewhat steeper than for the room temperature case, being approximately 0.85. Figure 4.17 displays the current noise at 100 Hz versus bias. Again, 12 dependence in the lowbias regime with a falloff in the transition region is observed. The temperature dependence of the noise in the lowbias regime was investigated further. The lowfrequency small bias (100 uA) noise spectra were measured for several temperatures (300 K, 250 K, 200 K, 150 K, 77 K, and 12 K) down to 12 K as shown in Figure 4.18. The magnitude increases with decreasing temperature down to about 200 K. Thereafter, further lowering of the temperature does not affect the magnitude very much, but the slope becomes more closely 1/f. The other temperatures are not plotted since they only obscure the figure. At 12 K, the slope is fully 1/f as shown in Figure 4.19. + + 4.3 The n n n Device Noise The magnitudes of the noise levels of the 0.4 um n+n n struc ture, both thermal and excess low frequency, are very small. The current noise spectra for several bias currents at room temperature is 64 14 10 15 10  10 16 10 17 i.j 10  19 2I 10 10 I 10 o0o I (mA) Figure 4.15 Current noise at 100 Hz versus bias current for pNBD at T = 300 K 0 ( Go ED 7 17 7 7 7 7 7 Y 10 0L A VIOOfiA , 400oA E ImA O 3.4mA ] 13.6mA 034mA 7  "V '77 7 100 0O 0K2 FREQ UENCY (Hz) Figure 4.16 Current noise of pNBD at T = 77 K I O13 N I c H CO 18 IOK lOOK 1013 1 x 71 x / u 4 N xg 1016 < 1017 1OU A lOOA ImA 10mA IO0mA I A CURRENT Figure 4.17 Current noise versus current of the pNBD for f = 100 Hz and T = 77 K I I I i 100 IK FREQUENCY (Hz) Figure 4.18 ++ Current noise of n+p n+ diode for I = 100 A and three temperatures lq 300K <0250K 0 12K 015 10 N IO  c0 1019 o1020 IOK lOOK 1015 16 10 N Ir cn 1 0 19 ,20 10 100 IK IOK lOOK FREQUENCY(Hz) Figure 4.19 Current noise of n p n diode at T = 12 K and IDC = 100 IA shown in Figure 4.20. Thermal levels and excess lowfrequency noise for some levels of bias can be seen. The frequency dependence of the lowfrequency noise is 1/f. The spectra from 1 to 25 kHz with the 1/f levels subtracted is shown in Figure 4.21. The lower three bias currents result in approximately thermal noise. There is still excess noise for 75 mA, however. Therefore, a spot noise measurement at 500 kHz for that bias was done. Less than J deviation from the thermal level was found. Values at 10 Hz obtained from the straight line approximations to the 1/f noise are plotted versus bias current in Figure 4.22. The expected behavior for 1/f noise, SI 12 is well satisfied. Spectra for T = 77 K and the same bias currents are plotted in Figure 4.23. Excess lowfrequency noise is found for all current levels. The noise in the thermal region is difficult to determine accurately because the expected value for full thermal noise (%0.17 2) is below the background noise of even the correlation setup. Also, for some bias currents, the noise is not yet flat at these frequencies. The 1/f noise at 10 Hz is plotted versus bias in Figure 4.24. Again, the I2 dependence of the magnitude is found. 017 10 , 75 mA 5C mA II mA 10 L  2C 19 K) rn r 10 I 21 10 10 00 IK OK JO0K f (H:) Figure 4.20 Current noise spectra of n 0.4 m NBD at T 300 K Figure 4.20 Current noise spectra of n n n 0.4 pm NBD at T = 300 K O 54 mA 2.5 x 1020 20 S27 mA S. (,VG.) I KHz 25 Hz 10 o mA 2.1 x 20, r  c, , lo/V C) ^ ^ ^ ^ ^ , V THEFi.!AL ,'CISE CF 075f1 i0 10 ';5_r f (H:) Figure 4.21 Thermal (like) noise of n n n device 19 zz 0O SLOPE 'L 2 !00 i (MA) Figure 4.22 Excess 1/f noise of n+nn+ device versus current 7 75mA E] 50mA O 27mA 0 IlmA "7 7 7 25 2.5 25 250 2.5K 25K FREQUENCY (Hz) Figure 4.23 Current noise of 0.4 nm nNBD at T = 77 K o16 10 N H CJO 1021 0 N I C\J 10 107 10 0OC DC CURRENT(mA) Figure 4.24 SI of nNBD at 10 Hz versus bias current for T = 77 K CHAPTER V DISCUSSION OF EXPERIMENTAL RESULTS 5.1 The n nn+ Device 5.la CurrentVoltage Characteristic and Impedance The IV characteristic of the ntype device is seen from Figure 4.2 to be linear for bias voltages up to about 1 volt and currents up to about 1 amp for both room temperature and 77 K. An attempt to apply pulses at higher bias resulted in melting the gold bonding wire at the top of the mesa. Higher bias was desired since a slight nonlinearity appears at currents greater than 1 amp. Neverthe less, 1 amp corresponds to a current density of 12,800 A/cm2. No V1/2 3/2 or V32 dependence is found which suggests that theories with these results are not adequate. Similarly, the collisiondominated MottGurney theory which predicts V2 current dependence at high currents does not apply. Indeed, sublinear current dependence at high bias seems to be indicated by the sparse data. The more realistic theory of Holden and Debney [12] gives a highbias current dependence of V14 for a 0.5 im device where collisions are neglected. At lower bias, their result appears to be somewhat sublinear, similar to the V /2 region of Shurand Eastman's [2] theory. The theory of van der Ziel et al. [9] gives linear behavior for small bias. They include spillover from the highly doped regions and calculate separately the mobilities due to diffusiondrift and thermionic emission. Setting the calculated and measured values of resistance at room temperature equal and solving for the diameter of the mesa gives a 96 pm diameter which is very close to the reported value of 100 jm. At 77 K, the measured value decreases 10%, the cal culated value decreases 23%. The impedance is purely resistive at these frequencies. 5.1b Excess 1/f Noise In the 1/f noise region, we would like to apply Hooge's empirical formula [18] which is Equation (2.12). For mesa structures or nonhomogenous samples [19], Equation (2.13) replaces it and is correct whether or not the transport is ballistic. In a ballistic or nearballistic device, many carriers do not undergo any collisions at all. This is in contrast with a typical semiconductor device in which every carrier collides many times. Hooge's formula was developed for the second case, requiring that N, the number of carriers in Equation (2.12) can be determined. In the nearballistic case, it is desired to exclude those carriers which are transported ballistically, including only those that contribute to the noise. This can be very difficult. Therefore, an alternative expression to describe "noisiness," avoiding this problem, will also be used. Noisiness is described as (S A(f)f)/I2 which is still dimension less unless the spectral slope is not 1. Substituting values gives, at 300 K, S i(f)f S = 1.6 x 1015 (5.1) and at 77 K, SI(f)f 14 2 1.4 x 104 (5.2) I If the slope is not 1, then just (S A(f)f)/I2 can be reported at a specified frequency. If n(x) = n is a reasonable approximation despite the complex nature of n(x), then Equation (2.13) becomes S A(f)fALmn Hb = 2 (5.3) I0 where the subscript, b, denotes validity for the ballistic case. If the transport can be characterized by a constant mobility, i, the measured resistance is L R m j n (5.4) q4A n5Tx) 0 Then, for the collisiondominated case, Hooge's constant becomes S A(f)f(mL)2 Hc = 2 (5.5) IoquR 2 2 Assuming that 300K = 0.74 mV 14 mec A = 7.9 x 10 m2 aHc = 7.2 x 108 (5.6) and aHb = 5 x 108 (5.7) At 77 K the values are cHc = 3.7 x 108 (5.8) aHb = 4.5 x 107 (5.9) These values are five orders of magnitude smaller than the value reported by Hooge et al. [16] for ntype bulk GaAs of 6 x 103. Thus the number of collisions for this device is very small. The remaining collisions involve polar optical phonon emission typically with a very small deflection angle [1], 6. Handel's [24] quantum theory of 1/f noise .2 1 indicates that the magnitude of the relative 1/f noise goes as sin2 so that the residual noise is very low. Comparing the ballistic and collisionlimited case temperature dependence of aH suggests that the device is not purely ballistic at either temperature since the low temperature ccHb is nine times larger than the value at room temperature. Further, the diffusiondrift resistance calculated by van der Ziel et al. [9] is larger than the thermionic emission resistance for both temperatures. However, the values are within a factor of two of each other. The very low value of aH suggests nearballistic transport. 5.1c HighFrequency Noise The device exhibits nearly 100% thermal noise at all currents to 75 mA if sufficiently high frequencies are attained. This current 2 corresponds to a current density of 960 A/cm2. There is no detailed theory for the noise as yet developed for all applied bias, just the preliminary calculations of van der Ziel and Bosman [14,15]. For the collisiondominated spacechargelimited diode at low bias, the noise is due to diffusion noise sources which transform via Einstein's relation to 4kT/R At high bias for which the MottGurney law For a bal (I V2) applies, the noise becomes 8kT/R where R For a bal x x di. listic device the noise is due to shot noise. At sufficiently high bias where soace charge effects dominate, correlations between current components due to fluctuations in the potential minimum caused by the space charge lead to subthermal noise. In order to differentiate between the two models, high bias must be achieved. Our measurements have found no deviations from thermal noise. One possible reason for this is that the current density of 960 A/cm2 is insufficient. Another is that the device is operating in between the two regimes as suggested by the 1/f measurements so that extreme bias may be required to see which effect dominates. 5.2 The n p n Device 5.2a CurrentVoltage Characteristic and Impedance + + The n p n device shows substantial nonlinearity. There are two regimes with a transition region in between as shown in Figure 4.10. At low bias, the device is linear with a DC conductance at room temperature 100 times smaller than the ntype device. It decreases with decreasing temperature to a limiting value a factor of 3 below its room temperature value. It reaches this value near 150 K and remains constant thereafter down to 12 K. The high bias regime is temperature independent and is also linear with a large conductance 4 times less than the ntype device. Due to the thinness of pregions and the large doping density of the n regions, the spillover of electrons into the pregions is not very different than for the ntype diode case [9]. Then current flow should be by nearly ballistic electron emission through the poten tial minimum. The resulting characteristic should be linear with a large conductance and nearly temperature independent. This appears to be a good model for the high bias regime. At low bias, the model fails, however. One possible reason for this is that enough holes re main in the pregion to control the transport ambipolarly. The details of such an effect are unclear, but a qualitative description by Dr. C. M. Van Vliet follows. The motion is not strictly ambipolar, since there is space charge, as indicated by the presence of the poten tial minimum [9], which even for V = 0 can be computed from Poisson's equation. Therefore, near the potential minimum the excess electron charge is small. Roughly speaking, only ambipolar pairs with energies within kT of the potential minimum are able to cross the minimum. If the injected carrier density An is less than p ata point (labeled x') approximately kT greater in energy than the minimum on the injecting side, then the current will be ambipolar (instead of ballistic). Clearly, with decreasing T, p(x') decreases. Thus, with decreasing T the ambipolar current decreases, and the transition to ballistic behaviorwhich is independent of Tsets in at lower bias. The I/V versus V characteristic is therefore as shown in Figure 4.10. In any case, assuming that holes control the mobility for low bias, and electrons at high bias gives a factor of 21 change in the con ductance at room temperature since typical values are pn 8,500 cm2 n Vsec 2 and p k 400 cm /Vsec [25]. There still remains to be explained the factor of 4 difference between the ndevice conductance and the high bias pdevice conductance. The potential barrier is larger in the ptype device [26]. In the p region, Poisson's equation is 2 q(n + N p) a (5.10) dx 0 where '(x) is the potential, n(x) the electron density, p(x) the hole density, Na the acceptor density, and ee0 the dielectric constant. For the ndevice in the n region d2 q(n Nd) x Nd (5.11) dx2 E0 If p(x) is assumed negligible due to electron spillover, then I 22 dp p (5.12) so that the ndevice barrier is smaller. The above model of ambipolar collisiondominated flow at low bias and nearballistic electron emission at high bias gives a qualitative explanation of the experi mental data. 5.2b Noise + + The n p n device showed much larger levels of noise than the ntype device. The frequency dependence of the spectra for all measured currents is about (1/f)75 for all measured bias currents at room temperature. Then Hooge's parameter is not well defined since it is not dimensionless (unless the slope is 1) and depends on frequency. The noisiness at 10 Hz and 100 pA is SAI(f)f Ix 9 2 = 6 x 109 I (5.13) This is about 4 x 106 times slopes do not change in the magnitude of the noise does seen at low bias, however. with the low bias transport the ntype device value. The spectral transition from low to high bias. The fall off at high bias from the 12 dependence This suggests that the noise is associated mechanism and that the high bias mechanism 83 is much less noisy. That is in good agreement with the conjecture of ambipolarly governed flow at low bias and nearballistic flow at high bias. The 0.75 slope is not common although van de Roer [27] has also found spectra going slower than 1/f in 6 im p n p punch through diodes. The spectra become more closely 1/f as temperature decreases. At 77 K the slope is 0.85 which is common for intermediate temperatures down to 12 K. The magnitude of the noise is = 3.3 x 108 (5.14) at 10 Hz and 100 pA. magnitude S A(f)f 2 I at 10 Hz and 100 yA. slope is unclear. Finally, at 12 K, the slope is fully 1 with (5.15) The explanation of this temperaturedependent S (f)f 2 I  10 CHAPTER VI CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER WORK 6.1 The n+n n Device + + The n n n device is nearly ballistic. Carrier transport is by both thermionic emission and by collisionbased diffusiondrift. Neither process can be neglected in a physical model. For the IV characteristic, it is desired to calculate n(x), v(x), and J(V) for any applied bias, expanding on the calculations of van der Ziel et al. [9] for small V. Alternately, the model of Holden and Debney [12] should be further developed to investigate the effect of adding col lisions to their model. Also the temperaturedependence and the effects of replacing the MaxwellBoltzmann velocity distribution by a FermiDirac distribution should be investigated. Since nonlinearity in the IV characteristic appears at high bias, very fast (1l nsec) pulse neasurements at yet higher bias would be very interesting. The mechanism causing this nonlinearity should be identified. For the noise, the measured value of aH found was extremely low. This is the best confirmation yet that lattice phonon scattering causes 1/f noise. It would be valuable to repeat the measurement on other mesas. Perhaps in such small devices Handel's [24] fundamental theory can be tested. In the thermal noise region, the results were inconclusive, neither subthermal nor greater than thermal noise was found. Perhaps the bias level was not great enough. In that case pulsed noise measurements could be attempted. This may be difficult since the noise is already very low. A detailed and complete noise theory for these devices is desirable, including collision effects and velocity dis persion. A Langevin equation based on the momentum and energy balance equations of Shur and Eastman [5,6] but including the above effects should give accurate results. + + 6.2 The n p n Device The ptype device was very intriguing, beginning with the IV characteristic. At present it is believed that at low bias the carrier transport is ambipolar and collisionlimited, centering about the potential minimum. At high bias, the injected electrons overrun the holes and nearballistic electron flow results similar to the n type device. The conductance is less because the potential barrier is greater. This model deserves further detailed investigation. The noise is very large and shows slopes becoming progressively less 1/f as temperature is increased. This interesting characteristic suggests that subtle and complex mechanisms may be taking place that are not well understood yet. To develop a theory that explains these effects would be a great step forward in our knowledge of very small devices. The 1/f noise extends to the GHz range to frequencies greater than our measurement system so that thermal noise measurements have still to be done. In addition, only these lowbias measurements have been done; pulsed noise measurements in the highbias regime are underway; they may yield valuable information. Investigation of this device has led to another novel device + + [28]. A p n p device with nonlinear characteristics is predicted to have negative differential conductance. At low bias, electron controlled ambipolar flow should give a large conductance. At larger bias, hole injection takes over with collisionlimited flow resulting in a low conductance. The transition region, therefore, should show the negative differential conductance. APPENDIX COMPUTER PROGRAMS FOR THE HP 9825 APPENDIX COMPUTER PROGRAMS FOR THE HP 9825 A.1 CorrelationSystem ThreeMeasurement Averaae This program calculates equations for the current noise spectral density of a device under test (DUT) using the threemeasurement technique. Actually, the magnitude of the calibration signal is also recorded. The program calculates data for eight logarithmically equal spaced points covering one decade in frequency. It is used with the HP 3582A spectrum analyzer which features a dualchannel Fast Fourier transform algorithm for frequencies from 0.02 Hz to 25 kHz. Many data sets can be averaged to increase the accuracy of the measurement which is described in Chapter III. 0: "REPEATED3 MEASUREMENT CORRELATION SYSTEM": 1: 0AABCD 2: dim A[24],B[24],C[24],F[8],M[8],S8][8] 8],Q$[1],LS[20] 3: dim MS[8,5];"MP12"M$[1];"MP18"M$[2] 4: "MP25"MS[3];"MP35"M$[4];"MP50"M$[5] 5: "MP70"MS[6];"MP95"*M$[7];"MP125"M$[8] 6: 12S[l];18S[2];25S[3];35+S[4];5Q0S[5];70S[6];95S[7];125S[8] 7: ldf 1,A,B,C,D,A[*],B[*],C[*],F[*],M[*] 8: fxd 0;dsp A,B,C,D,F[8];beep;stp 9: fit 3 10: "n"W$;ent "DUT ON (y/n)",QS;if Q$="y";gto "DUTON" 11: "n"Q$;ent "DUT OFF (y/n)",Q$;if Q$="y";gto "DUTOFF" 12: "n"QS;ent "CAL ON (y/n)",QS;if Q$="y ;gto "CALON" 13: "n"QS;ent "CAL MAG (y/n)",O$;if Q$="y";gto "CALMAG" 14: "n"QS;ent "CALC NOISE (y/n)",Q$;if Q$="y";gto "NOISE" 15: "n"*Q$;ent "CLEAR DATA FILE (y/n)",Q$;if QS="y";gto "CLEAR" 16: "DONE": 17: rcf 1,A,B,C,D,A[*],B[*],C[*],F[*],M[*] 18: fxd O;dsp A,B,C,D;beep 89 19: Icl 7i1;end 21: "DUTON": 21: for I=l to 8;gsb "GET" 22: if A=0;XA[I];Y*B[I];ZC[I];next I 23: if A0O;jmp 4 24: (X+A*A[I])/(A+1)+A[I] 25: (Y+A*B[I])/(A+1)B[I] 26: (Z+A*C[I])/(A+1)*C[I];next I 27: A+1A;gto "DONE" 28: "DUTOFF": 29: for I=1 to 8g:sb "GET" 30: I+8J 31: if B=0;XA[J];YB[J];ZC[J];next I 32: if B=0:jmp 3 33: (X+B*A[J])/(B+1)*A[J];(Y+B*B[J])/(B+1)*B[J] 34: (Z+B*C[J])/(B+1)C[J];next I 35: B+1B;gto "DONE" 36: "CALON": 37: for I=1 to 8;gsb "GET" 38: I+16~1 39: if C=0;XA[J];YB[J];ZC[J];next I 40: if C=0;jmp 3 41: (X+C*A[J])/(C+1)A[J];(Y+C*B[J])/(C+1)*B[J] 42: (Z+C*C[J])/(C+1)C[J];next I 43: C+1C;gto "DONE" 44: "CALMAG": 45: wrt 711,"AAIMN1MB1" 46: for I=1 to 8 47: wrt 711, "MP",S[I],"LMK";red 711,U,V 48: if A+B+C+D=0;VF[I] 49: if V#F[I];beep;dsp "FREQUENCY MISMATCH";stp 50: tnt(U/10)U 51: if D=0;UM[I];next I 52: uf D#0;(U+D*M[I])/(D+1)4M[I];next I 53: wrt 711,"AA0";D+1D;gto "DONE" 54: "NOISE": 55: ent "TEtFP",T;ent "DUT RESISTANCE",R 56: ent "CAL RESISTANCE",S;ent "DC CURRENT",U 57: gsb "LABEL" 58: for I=1 to 8;spc 59: v/(A[I]*B[I]*C[I])V;Y/(A[I+8]*B[I+8]*C[I+8])*W 60: v(A[I+16]*B[I+16]*C[I+16])X 61: (VW)/(XW)Y;M[I]/S/SZ 62: Y*Z+4*1.38e23*T/RN 63: prt "SI=",N;fxd 1;prt "FREQ=",F[I];flt 3 64: next I 65: spc ; spc ;gto "DONE" 66: "CLEAR": 67: 0A+BtCD 68: for I=l to 8;0F[I]1M[I];next I 69: for I=1 to 24;0*A[I]B[I]C[I];next I 70: gto "DONE" 71: "GET": 72: gsb "DATA" 73: if A+B+C+D=0;VF[I] 74: if V#F[I];beep;dsp "FREQUENCY MISMATCH";stp 75: asb "COH" 76: tnt(X/10)X;tnt(Y/10)Y;ret 77: "DATA": 78: wrt 711,"AAlMNlMB1" 79: wrt 711,MS[I] 80: wrt 711,"LMK";red 711,X,V 81: wrt 711,"AA0AB1LMK";red 711,Y,V 82: wrt 711,"ABO";ret 83: "COH": 84: wrt 711,"LFM,",76000+dto(4*S[I]),",4";red 711 85: for J=1 to 4;rdb(731)U;rdb(731)Z 86: ior(rot(U,8),Z)X[J];next J 87: wrt 711,"LFM,",77000+dtoS[I],",1";red 711 88: rdb(731)X[2];rdb(731)X[4] 89: wrt 711,"LFM,",75000+dtoS[I],",1";red 711 90: ior(rot(rdb(731),8),rdb(731))X[5] 91: wrt 711,"LFM,",75200+dtoS[I],",1";red 711 92: ior(rot(rdb(731),8),rdb(731))X[6] 93: wrt 711,"LFM,",77200+dtoS[I],",1";red 711 94: rdb(731)X[7];rdb(731)X[8] 95: X[1]*2t(X[2]15)X[1] 96: X[3]*2+(X[4]15)X[3] 97: X[5]*2+(X[7]15)X[5] 98: X[6]*2+(X[8]15)+X[6] 99:(X[5]+2+X[6]+2)/X[1]/X[3]+Z 100: ret 101: "LABEL": 102: ent "Label",L$;spc ;prt L$ 103: prt "DCI=",U;prt "TEMP=",T 104: prt "DUT RES=",R;prt "CAL RES=",S 105: prt "# SETS=",A 106: ret *19711 A.2 WideBand Filter Synthesizer This program is written for the HP 9825 computer in conjunction with the HP 3582A spectrum analyzer. The highfrequency 4/5 of the FFT bins are averaged to synthesize a wideband filter to use with the radio frequency FFT system which utilizes a mixer just before the spectrum analyzer. Without the use of this program, the accuracy of a measurement is poor since the bandwidth is then very small com pared with the frequency of interest. A bandwidth of 20 kHz can be generated by setting the frequency span on the 3582A to 25 kHz, although the program displays the filter output in normalized dBV//Vz. 0: "Last 4/5'ths of display averager: 1: dim B[10],Y[256] 2: wrt 711,"LFM,77454,5" 3: red 711 4: for I=1 to 10;rdb*711)~[I] 5: next I 6: wrt 711,"LSP";red 711,S 7: if B[3]>0;sfg 1 8: B[2]H 9: if H>127:H128H4 10: if H>63;H64H 11: if H>31;H32H 12: if H>15;H16H 13: if H>7;H8H 14: if H=0;250B 15: if H>1;68.87+B 16: if H>3;166.6667B 17: B[4]H 18: if H>127;H128H 19: if H>63;H56.H 20: if H>31;H324 21: if H>15;H16*H 22: if H>7;H84H 23: if H>3;H4+H 24: if H>2;sfg 2 25: 128N;if flgl;256N 26: if not flgl;B/2B 27: if flg2;256N 28: wrt 711,"LDS" 29: red 711 30: for I=1 to N;red 731,Y[I] 31: next I 32: Icl 711 33: if flg2;128+N 34: 0A;fxd 4 35: gsb "calc" 92 36: if flg2;XYY;0+X;128A;gsb "calc" 37: if flg2;prt 20*log(Y/.8N),201og(X/.8N);end 38: prt 201og(X/.8N);end 39: "calc": 40: for I=A+N/5 to A+N;X+tnt(Y[I]/20)//(S/B)X;next I 41: ret *30119 REFERENCES 1. 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