IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-26, NO. 6 , JUNE 1979

Experimental and Theoretical Studies of /-V Characteristics of Zinc-Doped Silicon p-n

Junctions Using the Exact DC Circuit Model

PHILIP CHING HO CHAN AND CHIH-TANG SAH, FELLOW, IEEE

937

Abstract-The exact distributed steady-state equivalent circuit model is used to calculate the forward current-voltage characteristics of zinc- doped p-n junction diodes. The experimental values of the Shockley- Read-Hall (SRH) coefficients at zinc centers in silicon and measured recombination center (zinc) density were used in the model. The theo- retical forward I- V characteristics are compared with experimental I- V over a wide range of temperatures, showing excellent ageement.

I. INTRODUCTION

T HE GENERAL equivalent circuit model for semiconduc- tor devices was formulated by Sah [ I ] -[3] . This distrib-

uted circuit model corresponds exactly to the set of governing differential equations. Using the equivalent circuit model, the solution of the coupled nonlinear partial differential equations is converted to a linear network problem. Green and Shewchun [4] showed that by using Newton-Raphson iteration technique for solving nonlinear equations, the small-signal equivalent cir- cuit of Sah can be used to obtain dc and transient solutions. The exact steady-state model was subsequently applied to a study of gold-doped [ 5 ] and zinc-doped [6] p-n junction di- odes under forward bias. The transient model has been ap- plied to study carrier transport along the surface in metal- oxide-semiconductor N O S ) capacitors [7].

In this paper, the forward current-voltage (I- V ) characteris- tics of zinc-doped p-n diodes calculated from the exact steady- state equivalent circuit are compared with experiment over a wide range of temperature.

11. EXPERIMENTAL SHOCKLEY-READ-HALL (SRH)

Zinc exhibits two acceptor levels in silicon. Electrical prop- erties of zinc centers can be characterized using the multiple- level SRH statistics of Sah and Shockley [8] , The energy level diagram and the definitions of the eight rate coefficients are given in Fig. 7 and Section 111-A of the preceding paper [6] .

COEFFICIENTS FOR ZINC IN SILICON

Manuscript received November 7, 1978; revised January 23, 1979. This work was supported by the Air Force Office of Scientific Research under Grants AFOSR-76-2911 and AFOSR-78-37 14, and by the Rome Air Development Center under Contract RADC-F-19628-77-C-0138 with the U S . Department of the Air Force. P. C. H. Chan was also sup- ported by IBM Postdoctoral Fellowship. This work is based in part on a Ph.D. dissertation submitted to the Department of Electrical Engi- neering, University of Illinois, Urbana.

The authors a e with the Department of Electrical Engineering, Uni- versity of Illinois, Urbana, IL 61801.

Since the experiments are performed in the dark, the optical transitions can be neglected, Also, the samples have high zinc recombination center density so that the Auger recombina- tions can be neglected. Thus in the theory, we assumed

ep = e p . t

Herman and Sah [9] have measured the thermal hole emis- sion coefficients of the neutral and singly ionized zinc centers. At low field the measured emission coefficients were fitted to the following equations:

epl = 2.6 X 10" (T/300)-' exp (-0.641/kT) s-l ( 1 )

epo = 7.41 X 10" (T/300)-' exp (-0.326/kT) s-'. ( 2 )

Thermal capture coefficients were also measured by a num- ber of authors [ 101 -[ 181 . The reported results for the four cap- ture coefficients were summarized in [ 171 and [ 181 .

In this work, the capture coefficients measured by Herman and Sah [18] will be employed. To obtain temperature de- pendence of cp2, Herman-Sah (H-S) data were extrapolated to low field. The extrapolated data are fitted to the equation

cp2 = 5.72 X (T/300)-3.93 cm3/s. (3)

Temperature dependence for the other three capture coeffi- cients was not given by Herman and Sah [18]. However, they were reported to be temperature independent over a wide range of temperature [lo] , [ 1 1 ] , [ 151 . cpl is obtained by ex- trapolating H-S values obtained at 83 K to low field. Field de- pendence for cnl and ca0 were not given by Herman and Sah. However, the value they gave agrees fairly well with the values measured by other authors [ 181 . The values for the three cap- ture coefficients, all independent of temperature, are

cpl = 9 X cm3/s (4) cnl = 5 X IO-'' cm3/s (5)

cno = 7 X 1 0 - ~ cm3/s. ( 6 )

The two thermal emission coefficients for electrons are still unknown. They are calculated from the detail balance relations

e,, =n?cn1cpzlep1 (7)

en, = n?cnocpl /epo. (8)

The temperature dependence of ni, the intrinsic carrier den-

0018-9383/79/0600-0937$00.75 0 1979 IEEE

IEEE 'I RANSACTIONS ON ELECTRON DEVICES, VOL. ED-26, NO. 6, JUNE 1979

TABLE I CHARACTERISTICS OF DEVICES

Device T y p e I m p u r i t y P r o f i l e Zn D i f f u s i o n N Z n ( ~ r n - ~ ) TemDerature

PZ1-Z6-2 P+/N 3 ~ 1 0 ~ ~ E x p ( - X ' / . 3 0 4 ~ 1 0 - ~ ) -5 . 7 x d 4 880C

B24-28-7 N+/P 3 ~ 1 O ~ ~ E X p ( - X ~ / O . 2 3 ~ 1 0 - ~ ) -. 8~10'~ l O 0 O C 8X1O14

J u n c t i o n d e p t h and d e v i c e l e n g t h are 5 and 2 0 3 . 2 m i c r o n s f o r b o t h d e v i c e s .

N+ GAT E

P

Fig. 1. Cross section of the devices used in the measurement.

sity , is calculated from the recent data on temperature depen- dence of the silicon bandgap.

111. DEVICE FABRICATION AND MEASUREMENT I n order to compare the theory with experiment, two sets of

p-rl junction diodes were used. They were fabricated by Her- man [ 191 . The p+-n diode was fabricated by boron diffusion inta n-Si with phosphorus concentration of 5.7 X lot4 The n+-p diode was fabricated by phosphorus diffusion into p-Si with boron concentration of 8 X l O I 5 ~ m - ~ . The diffu- sion times and temperatures were chosen so that in both de- viclzs the junction is about 5 pm from the surface. Zinc diffu- sio.ns were carried out in sealed quartz ampoules evacuated to 10"'' torr. The zinc diffusion time for the phosphorus-doped devicte P21-Z6-2 is 12 h at 880°C. For the boron-doped device B24Z8-7 the time is 12 h at 1000°C. The zinc concentrations were measured by transient capacitance technique [20] . The cross section of the devices is given in Fig. 1 . The characteris- tics, (of the devices are summarized in Table I.

The apparatus for the forward I-V measurement is described as follows. The bias voltage was generated from an HP6106A programmable power supply controlled by a voltage ramp gsn- erator designed in our laboratory. The current was measured us- ing a m HP425A dc pico-ammeter. The output of the HP422A and the bias voltage were traced on an x-y recorder. The currcnt-voltage characteristics for the devices were traced at every decade starting from the 100-pA scale up to 1 mA. In 0rde.r to measure current up to 10 mA, a 10-52 precision re&+ tor was placed in series with the diode. 'The voltage drop across the mistor was measured on the 0.1-V scale of the HP425A.. Temperature was kept to within 0.2 K throughout the mt:a.- surernent using a temperature controller designed in our latm-

0 P'N Control device (P21-C-5)

P+N Zlnc doped device(P21-26-2) A Difference + 2 W = X - X , (microns)

0 2 4 6 8 IO 12 14 16 18 20 22 24

Fig. 2. Zinc profile determined from the difference between the zinc- doped device and the control device (no zinc diffusion).

ratory. The MOS gate is biased to cut off surface channel leak- age. For p+-n device the gate is grounded. For the n+-p device the gate is biased at -20 V.

IV. DISCUSSION OF THEORETICAL AND EXPERIMENTAL RESULTS

The approximate spatial dependence of zinc concentration was determined from the steady-state dark junction capacitance versus reverse-bias voltage by Herman [19] . The junction was assumed to be abrupt and one side lowly doped compared with the other side. The result is shown in Fig. 2. The upper curve is obtained from the control device, which was fabricated on the same silicon wafer as p+-n diode P21-26-2, but with no zinc diffusion. At room temperature, most of the zinc is in the doubly negative charge state. The approximate spatial de- pendence of zinc is obtained by taking the difference of the curves and dividing by 2 due to the double negative charge, Zz.

The experiment and theoreticall-Vfor p+-n diode P21-26-2 at 257.5 K are compared in Fig. 3. The theory is obtained by numerical solution of the steady-state circuit model [5] , [6] . Detailed formulation and numerical procedure are given in the preceding paper [6] . In the constant NZn calculation, we have assumed N z , = 1014 cmT3. In the variable Nzn calcula-

CHAN AND SAH: I-V CHARACTERISTICS OF ZINC-DOPED SILICON p-n JUNCTIONS 939

I O 2 , I , , , , , , 1

K) I Device P21-26-2P?N

10-1 - 10-2 - 1

i 4

10-6 -

Theory Nir not constant Theory N,, constant 10-9 - 1 6 l O

- Experiment 0 0 2 0.4 0 6 0 8 1.0

' I I ' l i ' l l

01 0.3 0 5 0 7 '0.9 / . I VOLTS

VOLTS Fig. 6. Experimental and theoretical I-V for device B24-Z8-7 Fig. 3. Experimental and theoretical I-V for device P21-26-2 p+-n at 291.7 K.

257.5 K.

Device P21-26-2PVN

-- measured N,, --- extrapolated N,,

possible NTT

IO l4

loi3 0 5 10 15 20 25 30

X (microns)

Fig. 4. N z n profile for device P21-26-2 p+-n used in the calculation,

l o ' G l ' ' ' ' ' ' ' ' 0 . i 0.3 0 5 0 7 0 9 1 . 1

VOLTS

Fig. 5 . Experimental and theoretical I- V for device P21-26-2.

w l

N,, (assumed)

at

0 2 4 6 8 1 0 1 2

X (microns)

Fig. 7. N z n profile for device B24-28-7 n+-p used in the calculation.

io Devlce B24-28-7N+/P

3

'dloO 0.2 0 4 0.6 0 8 I O

VOLTS

Fig. 8. Experimental and theoretical I-V for device B24-28-7.

good at all three temperatures except at high current. The tion, the zinc profile in Fig. 2 was used. Since the data in Fig. high-current discrepancies are attributed to the back-contact 2 do not provide N Z n for x less than 7 p m , Nzn is extrapo- resistance, which is not included in the theoretical model. lated to x = 0 as shown in Fig. 4. The actual N z , may decrease This contact resistance will be estimated later in this section. in the p+ region due to the rejection of negatively charged zinc For the n+-p diode B24-28-7, zinc profile was not available. atoms by the negatively charged boron. The experimental and Calculation was made assuming constant Nzn. The agreement theoretical I-V using the extrapolated zinc profile of Fig. 4 at between theory and experiment was poor as indicated in Fig. 6 three temperatures are compared in Fig. 5 . The agreement is (solid dots) at 291.7 K. The calculated I-V is lower than the

IO ’

10-2 t J /- I

I 10-3

10-9 -

10-10 I I I I I 1 I I I

0.1 0.2 0.3 0.4 0.5 06 0.7 0.8 0.9 I O I

VOLTS

Fig . 9. The experimental and theoretical I-V characteristics for device P21-26-2 p+-n with the contact resistance subtracted out.

10-10 I I I , I I I

0 01 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 I O

V O L T S

Fig. 10. The experimental and theoretical I-V characteristics for dt:vic:e B24-28-7 n+-p with the contact resistance subtracted out.

measured I- V. A subsequent set of calculations was perforrned assuming zinc accumulation in the n+ diffused region. Such ii11

accumulation can be expected from the electrostatic interac- tion of the negatively charged zinc ions with the posithrely charged phosphorus ions. Good agreement between theory and experiment was obtained in Fig. 6 (triangles) if we assun1Ie the zinc concentration in the n+ diffused region to be 10 per- cent of the donor concentration. The assumed zinc profile is shown in Fig. 7. The experimental and theoretical I- V C U I ’ V ~ S

at three temperatures for the n+-p device B24-28-7 are given i n Fig. 8. Again, the agreement is good except at high current

Th.e tlll:ory-exl)elirrler~t discrepancy at high current can be shown as due - t o B temperature- and current-independent con- tact resis::ance as follows. The back-contact resistance can be estimated frmxm. the expermental data using the equation

i’ = Is exp [ Z F (V -. mj ] where R is the cclntact .resistance. The voltage drop across the (contact resistance ZIP can be estimated from the difference be- tween tl-le theoretical ard the measured I-V curves. The esti- mated contact res.istances were found to be about 11 S2 for the :n+-p device 1’21. -26-2 and 18 s2 for the p+-n device B24-28-7 at all .temperaturm measured. The theoretical forward I-V and :the experimental forward I- I/ with the contact resistances sub- .tracted according to (9)1 are shown in Figs. 9 and 10 for n+-p device P2 1-i!6-2 alnd p+-n. device B24-28-7, respectively. The agreements between theory and experiment are excellent.

V. CONCLUSION

The forward cwrent-voltage characteristics of zinc-doped p-n junctiorls were calculated using the exact distributed stealdy-state (equivalent circuit model. Experimental values for the SRH coefficients and lrecombination center densities in the bulk: were used in the theoretical calculation. The calculated forward I-V characteristics agree well with experiment at low- ,and intermediate-current level over a wide range of tempera- turels with. the adjustment of only one parameter-the recom- bination center density in the diffused region which is difficult i f not imposs,ible tlo measure. The discrepancy between theory ,and experi.ment at high current is shown to be due to the finite (contact resistance of the devices which is not included in the .theoretical model.

REFERENCES C. T. Sahl, “The equivalent circuit modelin solid-state electronics- Part 11: The single energy level defect centers,” Roc. ZEEE, vol. 55, p. 654,1967.

Part 11: ‘The m.ultiple #energy level impurity centers,” Proc. IEEE, vol. 55, p . 672, 1967.

Part 111: Conduction and displacement currents,” Solid-state Electron., vol. 113, p. 1547,1970. M. A. Green and J. Shewchun, “Application of the small-signal transmission line equivalent circuit model to the a.c., d.c. and transient analysis of semiconductor devices,” Solid-State Elec- tron., vol. 17, p. 941, :l974. 13. E. Maes and C. T. Sah, “Application of the equivalent circuit model for semiconductors to the study of Au-doped p-n junc- tions under forward bias,” IEEE Trans. Electron Devices, vol.

P. C. H. Chan and C. T. Sah, “Exact equivalent circuit model for steady-state characterization of semiconductor devices with multiple-energy-level recombination centers,” this issue, pp.

A. P. Ho and C. T. Sah, to be published. C. T. Sah and W. Shockley, “Electron-hole recombination statis- tics in semiconductors through flaws with many charge condi- tions,”Phys. Rev., vol.. 109, p. 1103, 1958. J. M. Herman, 111, andl C. T. Sah, “Thermal ionization rates and

Phys. Status Solidi(a), vol. 14, p. 405,1972. energies of holes at the double acceptor zinc centers in silicon,”

K. D. Glinchuk, A. D. Denisova, and N. M. Litovchenko, “Re- combination of charge carriers on zinc atoms in p-type silicon,”

- , “The equivalent circuit model in solid-state electronics-

- , “The equivalent circuit model in solid-state electronics-

ED-24, p. 1131, 1976.

924-936.

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-26, NO. 6, JUNE 1919 94 1

Sov. Phys.-Solid State, vol. 5, p. 1412,1964. [ I I ] K. D. Glinchuk and N. M. Litovchenko, “Recombination of cur-

rent carriers at zinc atoms in n-type silicon,” Sov. Phys.-Solid State, vol. 5, p. 2197,1964.

[ 121 B. V. Kornilov, “Optically induced charge exchange in zinc im- purity levels in silicon,” Sov. Phys.-Solid State, vol. 6, p. 2982, 1965.

[13] -, “Carrier recombination at zinc atoms in n-silicon,” Sou. Phys.-Solid State, vol. 7, p. 1446, 1965.

[ 141 -, “Determination of effective capture cross section for holes by singly negatively charged zinc atoms in p-silicon by the double injection method,” Sov. Phys.-Solid State, vol. 7, p. 2794, 1966.

[15] -, “Recombination of carriers at zinc atoms in p-type silicon,” Sov. Phys.-Solid State, vol. 8, p. 157, 1966.

[ 161 B. V. Kornilov and S. E. Gorskii, “Effective cross section for the capture of a hole by a double charged negative zinc atom in sili- con,” Sov. Phys.-Semicond., vol. 2 , p. 216,1968.

[I71 A. F. Sklensky and R. H. Bube, “Photoelectronic properties of zinc impurity in silicon,”Phys. Rev. B , vol. 6,p. 1328,1972.

[ 181 J. M. Herman, 111, and C. T. Sah, “Thermal capture of electrons and holes at zinc centers in silicon,” Solid-state Electron., vol. 16, p. 1133,1973.

[I91 J. M. Herman, 111, “High field emission and capture of electrons and holes at zinc centers in silicon,” Ph.D. dissertation, Univer- sity of Illinois at Urbana-Champaign, 1972.

[20] C. T. Sah, “Detection of recombination centers in solar cells from junction capacitance transients,” IEEE Trans. Electron Devices, V O ~ . ED-24, p. 410,1977.

Large-SignaI Performance of Microwave Transit-Time Devices in Mixed

Tunneling and Avalanche Breakdown

Abstract-A large-signal model for Read-type diode structures with narrow generation-region widths where mixed tunneling and avalanche exist is given. The generation region is modeled by use of a modified Read equation along with effective ionization rates. The injected cur- rent pulse, which is formed in the generation region, is calculated in isolation from the drift region in order that the effects of tunneling current can be clearly shown. The drift region is modeled by use of difference-equation versions of the device equations and is suitably interfaced to the generation region. The large-signal model of the total device is used to calculate the device admittance and efficiency. Large- signal results for GaAs and Si devices are given and the results are dis- cussed and compared.

R I. INTRODUCTION

EVERSE BREAKDOWN in a diode may occur due to an avalanche mechanism, a tunnel mechanism, or a mixed

avalanche-tunnel mechanism [ 11 -[4] . The breakdown is avalanche dominated for low electric fields and tunnel domi-

Manuscript received November 21, 1978; revised January 11, 1979. This research was sponsored by the Air Force Office of Scientific Re- search, Air Force Systems Command, USAF, under Grant AFOSR-

M. E. Elta was with the Electron Physics Laboratory, Department of Electrical and Computer Engineering, The University of Michigan, Ann Arbor, MI 48109. He is now with the Massachusetts Institute of Technology, Lincoln Laboratory, Lexington, MA 02173.

G. I. Haddad is with the Electron Physics Laboratory, Department of Electrical and Computer Engineering, The University of Michigan, Ann Arbor, MI 48109.

76-2939A.

nated for high electric fields. Microwave power may be gener- ated by a transit-time device in reverse breakdown from either mechanism. Three distinct modes of operation are identified for different widths of the generation region. These include the normal IMPATT mode, the MITATT (mixed tunneling and avalanche transit-time) mode where both tunneling and ava- lanche breakdown exist, and the TUNNETT (tunnel transit- time) mode where pure tunneling is present. The common IMPATT mode is inherently noisy but has a relatively large RF power output. The MITATT mode exhibits a noise perfor- mance-output power tradeoff. The TUNNETT mode would be useful as a low-noise amplifier, medium-power oscillator, self-oscillating mixer, and detector particularly at millimeter wavelengths. There are also indications that the use of tunnel- ing current will allow oscillation of GaAs devices above 100 GHz which will be useful for applications in millimeter-wave monolithic integrated circuits.

The possibility of TUNNETT-mode oscillation was con- sidered by Read [ 5 ] and others [6] -[9] but detailed calcula- tions were not attempted. Previous simple dc models of mixed tunnel-avalanche breakdown [4], [9] were not consis- tent with the experimental results of Lukaszek et al. [lo]. A more detailed dc model was developed by Elta and Haddad [ I I ] which included “dead-space’’ attenuated avalanche ioni- zation rates which correlates well with experimental results. Simple small-signal models o f transit-time devices in mixed

0018-9383/79/0600-0941$00.75 0 1979 IEEE