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Mixed conduction in Cr-doped GaAs
J: Phys. Chem. Solids, 1975, Vol. 36, pp. 1311-1315.
Pergamon Press.
Printed in Great Britain
MIXED CONDUCTION IN_ Cr-DOPED GaAst D. C. LOOK Physics Department, University of Dayton, Dayton, OH 45469,U.S.A. (Received 18January 1975;accepted 18April 1975) Abstract-Hall-effect and magnetoresistance measurements have been carried out in GaAs:Cr as functions of magnetic field strength (B = O-18kG) and temperature (T = 125420°K). Independent solutions for the mobilities, CL. and ppr and the carrier concentrations, n and p, are obtained from the basic mixed-conductivity equations. These quantities, as well as the intrinsic carrierconcentration,B, are then calculatedas a function of temperaturefor one sample,and subsequentanalysisyieldsthe followingvalues in the range T = 360-420”K:an acceptor (presumably Cr) energy E, = 0.69 2 0.02 eV (from the valence band); the bandgap energy E, = Ego+ crT,with Ego= 1.48 f 0.02 eV, (I = 3.2 X lo-’ eVpK; p” = 2700+ 100cm*/V set, decreasing slightly with temperature; pLp= 350+ 50 cm’/V set; and an acceptor-to-donor concentration ratio, &IN, =8. The electron mobility appears to be limited by neutral impurity scattering, with N, = 2 x lOI cmb3. Several other samples tiere also investigated but as a function of temperature only (at E = 0). At room temperature both positive @type) and negative (n-type) Hall coefficients were observed. 1. INTRODUCTION
It is well-known that GaAs doped with chromium is semi-insulating and thus is useful as a substrate for devices produced by diffusion, implantation, or epitaxial growth. Because of this its properties have been rather extensively explored by several techniques, including Hall-effect, photoconductivity, and optical absorption measurements [ l-51. Although the Hall coefficient (R) has always, in the past, been found to be negative, it has nevertheless been apparent that holes also contribute appreciably to the conductivity, precluding the usual simple calculation of the electron concentration (n) and mobility (h) from the zero-magnetic-field values of Hall coefficient (Ro) and conductivity (~0). Inoue and Ohyama[4] have studied R and u as a function of temperature (but not magnetic field) and have fitted their data by using the mixed-conductivity model with assumed values of CL,,, cup,and Ego, the bandgap at T = 0. They find a deep acceptor level at 066 eV from the valence band, and an acceptor-to-donor concentration ratio of about 2.1. In another recent study, at room temperature only, Philadelpheus and Euthymiou[S] combined Ro and u. with the magnetic-field dependence of u to obtain CL.,CL~, n and p ; however, their calculations were only approximate because the field dependence of R was too small to be measured. In our study we have investigated six GaAs : Cr samples from various manufacturers. Three of these were p-type (i.e. positive Hall coefficient) at room temperature, a phenomenon not reported before, to our knowledge. Furthermore, each of these three changed to n-type upon heating, thus indicating mixed conduction. The roomtemperature data are summarized in Table 1, including the “activation energy” of the resistivity (p memEdkr).Some tWork performed at Aerospace Research Laboratories, Wright-Patterson Air Force Base, under Contract No. F33615-71C-1877. Sin fact, within a few degrees of the Hall-zero (i.e. the temperature at which the Hall coefficient changes sign) R can even change sign as a function of B; however, this was not the case for any of these samples at room temperature.
of the Hall coefficients listed in the table were measured at B = 18kG and thus the values of Ro/pomay be somewhat different than the true Ro/po.SAlso listed in Table 1 is the number of the temperature run, each run extending from 300-420”K. It is observed that heating the sample to 420°K in an He atmosphere can, in some cases, appreciably affect the subsequent room-temperature values of p. and Ro. In one case, sample C, p. decreased by a factor 20 to a value low enough to possibly make it unfit for use as a substrate. For some of the samples the electrical properties annealed at room temperature, such that after several days they were about the same as those before heating. Another observation from Table 1 is the wide variation in values of p. and E,, not unrepresentative of those reported in the literature. Thus, it appears that the electrical properties of this material are not well understood and we have undertaken a more complete investigation by measuring a,,, in sample F, from 125-420”K, R. from 195-420°K, and the field dependences of R and u from 300420°K. To determine CL.,h,,, n and p above 300°K we have written independent solutions for these quantities starting from the basic mixed-conductivity equations. Then their temperature dependences make possible a determination of the acceptor activation energy and the concentrations of donors and acceptors. 2. EXPERIMENTAL
1311 IPCS Vol. 36 No. 12-A
CONSIDERATIONS
A conventional five-contact configuration[6] was employed for the Hall measurements, with each of the three voltage leads connected to an electrometer with triaxial cable. To reduce effective cable capacitances, the unity-gain output of each electrometer was connected to the middle shield of its respective cable [7]. Furthermore, the current through the sample was passed to ground through another electrometer operated in the “fastfeedback” mode. This scheme dramatically reduces the time constant for attainment of equilibrium, such that it is only about 20sec at 10”R. The sample was surrounded by a He exchange gas of about 50 mm pressure in a Pyrex dewar system which was capable of rapid, stable
1312
D. c.
LOOK
Table 1. Hall-effectand resistivitydata for several GaAs:Cr samples.Eachrun extends from 300420°K
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operation from 5.5-6OO’K.The maximum magnetic field available was 18 kG. The samples were cut from (100) wafers to rectangular dimensions of typically 1 x 0.4 x 0.06 cm. Ohmic indium contacts were soldered to a polished surface using an ultrasonic iron. It was not necessary to heat the contacts since their resistances were already much less than the crystal resistance.
At this point we have not introduced the explicit magnetic-field dependences of a. and a, themselves, which lead to the normal single-carrier magnetoresistances. This could be done by writing cr”= u&l - q+.‘B’) etc., where q” is a factor which is dependent upon the scattering mechanism[9]. In fact, the use of such substitutions in eqn (2) results in a formula for Ap/po, often quoted in the literature, which includes both single-carrier and mixed-conduction magnetoresistance effects to order p2B2[10]. However, we will ignore these 3. MXJB CONDUCl’lON EQUATIONS single-carrier effects because, for the holes qppP2B2s For conduction by both holes and electrons in a 0.001, and for the electrons Willardson and Duga (W-D) [ 1l] have shown that even a very small amount of magnetic field, of arbitrary strength, we have[B]’ ionized impurity scattering added to the lattice scattering B = R.u,2 f R,u,' t R.R,u.‘u;(R. + R,)B’ can drastically reduce q.. In fact, by using the W-D (on + up)*+ u.tu;(Rm + R,)*B* analysis we can calculate that q”p”*B*s0N11 for our sample, a negligible correction. Also, their analysis shows = R&+$ (1) that R, will be almost independent of field over our range n of B and T. If the impurity scattering centers are neutral, rather than ionized, then without doing a detailed analysis 1+ Yp,zB* (24 we would still expect qn to be small, because q” =O for u=““1t(X+Y)p.2B~ pure neutral-impurity scattering and we will find that the impurity scattering for electrons dominates the lattice or scattering over our temperature range. Thus, we believe that the field dependences of R and u will arise almost AP - * uo(u_l- uo-‘) = * (2b) entirely from mixed-conduction effects. n PO To proceed, we substitute cr, = nep,, uP = pep,, R. = -r,/ne, and R, = r,/pe into eqns (4)-(7), where r. and r, where are, respectively, the scattering factors for electrons and uo = 0" t a, (3) holes. The results are: 1t bc
(4)
xjL: =
u.u, (u.R. - upR, )’ (U” + u, )*
(5)
u~*u,‘(& + R, )’ (U”+ UP)*
(6)
yp.2 = ZjL”2=-
u.*u,‘R.R, (R. + R, ) u,*R. + u,‘R, ’
(7)
u0 = nietc, bdc R
0
&(I-ab*c)Vc
nie
(1 t bc)*
X=rzc(l+& ’ b(l+bc)’
y = r 2 (a - cl2 ’ (1 tbc)’
(8)
(9)
(IO)
(if)
1313
Mixed conduction in G-doped GaAs zcr2
h-c) ’ (ab*c - 1)
where (Y= r, /r,,, b = p. /pP, c = n/p, and ni is the intrinsic carrier concentration. These equations, using the same notation, are given by Hilsum and Barrie (H-B)[12] except that the scattering factors, m, r,,, and (I = r./r, were not included in their work. It is convenient to use two relationships obtained from eqns (1) and (2):
another advantage of these solutions is that ni, the intrinsic carrier concentration, is not required and, in fact, can itself be calculated: (ab - 1)4 spuo2 ” = %* cwb(1tab)’ e’(Rou$(l t R-‘)r’
(18)
From eqn (15), cub is independent of the scattering factors, so that ai a V(arz) = d/(r.r,). We will find that this factor is about 1.1 for our sample, and, in general, it will not be much different than unity. (13) To check the validity of eqn (l+( 17) we use data from H-B’s work on ptype InSb[lZ]. For their specimen No. 1, our results agree exactly, while for their specimen No. 6 and they get b =46 while we get b =74. However, for this sample, TIP is very close to 1 so that a small error in ++a.‘Y=p:X$=pz(Y+Z)j&. (14) either T or /3 will lead to a relatively large error in b, according to eqn (15). Since they determined b by an From eqn (13), a plot of R vs ApIp will give Ro and approximate method our value might be more accurate. In any case, our y’s agree, since CL,is not very dependent R = X/(Y + Z) as the ordinate and abcissa intercepts, respectively, and from eqn (14), a plot of l/B’ vs p~/Ap upon b when b * 1. Before analyzing our data we must determine m and r,. will give intercepts -l/B,* = pfizY and 8 = Y/X, and a plot of l/B* vs Ro/(Ro- R) will give intercepts -l/B,* and As discussed earlier, these quantities should be almost y = Y/( Y + Z), respectively. As will be seen later, our independent of R ; however, they can depend upon temperature since the relative strengths of the impurity data are not good enough to get accurate values of l/B,‘, S, and 7, but only the respective ratios, i.e. the slopes, and lattice scattering will depend upon temperature. It will turn out that p” = 3OOOcm*/V set, with only slight S, = -l/S&* = ,u;X, and & = -l/yB,’ = p,2( Y t Z). temperature dependence, and if we assume that the Fortunately, however, our solutions, given below, require impurities are ionized and that the lattice mobility is about only Ro, UO,and two of the three, S,, SR and /3. 10,000at 300°K and 6000 at 4OO”K,then the W-D analysis The solutions are: can be used to yield r. = 1.2 at 300°K and 1.1 at 400°K. If the scattering impurities are neutral, on the other hand, b =&{A +v(A2--4)) (15) then r, = 1.0-1.2 since both of the individual scattering factors (lattice and neutral impurity) will be in this range. For the holes, only lattice scattering will be important, where and, again, r, = l&1.2. It is thus clear that none of the quantities calculated by eqns (15)-( 18) will be in error by A = 12+ T(1-t P-*)1/(1- TIP), T E (RmJ2/Sp; (16) more than l&20% if we assume r. = r, = a = 1. As mentioned earlier, detailed measurements of R and Itab/ u vs B and T were carried out on sample E To ‘=b(cxb+R) demonstrate the method of calculation we present the data at 370°K. First, as shown in Fig. 1, R is plotted vs and Ap/pO and the ordinate and abscissa intercepts are, respectively, R. and /3. Then, in Fig. 2, l/B* is plotted vs (17) Ro/(Ro-R), and vs pO/Ap,and the slopes are, respectively, SR and S,. These slopes provide a check on /3,since Although eqn (15) has another solution, with a minus sign /3 = S,/S,.t In Fig. 3, Ro, uo, and Rouo are plotted as a in front of d(A* - 4), this is inadmissible for a sample that function of temperature, including values below 300°K to is ptype at low temperature and n-type at high show the change in sign of Ro. Finally, in Fig. 4, the temperature, i.e. a sample with b > 1. We note that b temperature dependences of b, c and CL,are shown, and depends upon the scattering factors in a very simple way in Fig. 5, p, n and ni, where p = uo/epn(c t b-l), n = cp (b 0~ a-‘) and that, as expected p,, Q (ar,)-’ = m-’ for and a = d(np). b-‘+a.
Besides the fact that no approximations are required, tActually, since we can measure all three, fi, S, and S,, and have no reason to favor S, over &, we use V/(&$,&) in place of S, in eqn (15). $This effect was present in all of our samples, and also appears in the data of Ref. 141.It is not an experimental problem since the performance of our apparatus should not be limited below lo”-10” t-l.
4. DISCUSSION
From Fig. 3 we see that at low temperatures u. flattens out.S The reasons for this are not entirely clear, but they may involve impurity conduction or surface conduction effects, which will be more thoroughly investigated later. At any rate, because of these effects the mixedconduction analysis may not be valid at these low temperatures. An indication of the range over which the
D. C
1314
SAMPLE 370%
F
Fig. 1. The Hall coefficient R vs the magnetoresistance Aplp,, for sample Fat 370°K. 3-
SAMPLE
F
Fig. 3. The temperature dependences of the Hall coefficient R,, the conductivity uOa, and their product R,uo, for sample Fat B = 0.
SAMPLE
0
5
200
300
IO
15
F
Fig. 2. The square of the inverse magnetic field strength l/B2 vs R&R, - R) andvs pO/Apfor sample F at 370°K. analysis should be applicable may be obtained from the behavior of ni, Fig. 5, which itself begins to flatten out below about 340°K. This is unreasonable behavior, and thus any of our calculated results below this temperature are suspect. Furthermore, the data plots, such as those shown in Figs. 1 and 2 are not as good below 340°K. For thermally activated holes, well below the exhaustion region (p <
IO/ 22
24
26
28
30
32
34
36
103/T(Wl
(19) where Na and ND are, respectively, the acceptor and donor concentrations, g, is the degeneracy factor, and N, = 2(2?rm~kT)“2/h’, the valence band density of states. Above 340°K our data in Fig. 5 are fitted by p = 3.15 x 10’5pq@@“T
(20)
from which, by assuming m$ = 0.5m, [14] and g, = 4, we tThe Brooks-Herring formula[l6] would give values of ND and NA about a factor four smaller, but the electron screening term employed in this formula is probably not accurate for the small carrier concentrations found in our sample.
Fig. 4. The temperature dependencies of the mobility ratio, b = pn/pp. the carrier concentration ratio, c -n/p, and the electron mobility pL.for sampleF.
obtain Na /ND = 8.4. To determine Na and ND, consider the data at 400°K; here we would expect an electron lattice mobility, pL = 6OOOcm*/V set, and since the measured mobility is e = 2600, impurity scattering must be an important effect. In fact, to within a factor two or so, the impurity scattering mobility should be given by /Lr=I(/._I - II,-‘)-‘=5000. If the impurities are ionized, the Conwell-Weisskopf formula[l5] can be employed to predict an ionized impurity concentration Ni = 5 x 10” cm-‘, and since Ni = 2No + p = 2ND (for p > n), we get No = 2 x lo”, NA =2x 10”cm-‘.t If the im-
1315
Mixed conduction in Cr-doped GaAs
SAMPLE
slightly as temperature increases, due to the competition between lattice and impurity scattering. In the temperature range 360-420”K the mobilities obey p. = 2700f lOOcm*/Vset, and pP = 350 + 50 cm’/V sec. These are quite reasonable values for this material. The measured mobility ratio, b = 8, is much lower than the value often quoted in the literature, b = 20[4,5]. In this regard two facts must be considered: (1) the room-temperature value for pure samples, b = 8000/400= 20, cannot simply be extended to GaAs: Cr since impurity scattering will limit p. much more effectively than pP,; and (2) from our experience, most values of b measured in GaAs : Cr at room-temperature are suspect because of effects not attributable to normal mixed conduction, as discussed earlier.
F
5. CONCLUSIONS
IO’,
This study has led to the following conclusions: (1) Chromium-doped GaAs can be either n-type or p-type at room temperature, and the very small values of &a0 IO" I often observed are due to mixed conduction. In some 22 24 26 28 30 32 34 36 samples, heating to 420°K affects the subsequent resistiv103/T d 1 ity at room temperature. The dominant acceptor level Fig. 5. The temperature dependencies of the carrier concentra- (presumably Cr) is about 0.69 eV from the valence band. tions, n and p, and the intrinsic concentration, ni = d(np). Impurity scattering limits the electron mobility to about half of the value found in pure samples, but the hole mobility is not affected nearly as much. (2) Mixedpurities are neutral, on the other hand, then the Erginsoy formula[l6] will predict N. = 2 x lOI cmm3, and since conduction data can be systematically analyzed from a N,=N,.,-ND-p=Na-ND, we get ND-3x10i5, knowledge of Ro, UO,and any two of the three, /?, S, and &, without any approximations and without knowledge of ni. Na =L2 x 1016.This value of Na is much more reasonable than the larger one predicted above since samples from However, the scattering factors may introduce small this manufacturer are generally found to have about uncertainties unless they are known, in which case they can be explicitly included. 1 ppm Cr. Above 34O”K,the intrinsic carrier concentration, Fig. 5, REFERENCES obeys I
/
ni = 2.27 x 101s~3’2e-*.48’kT.
(21)
By assuming that m i = 0*067m,[14], and m $2: 0.5m,, the theoretical relationship is ni = q(N,N,) e-E,12kT = 3.62 x 10’4T3’2 e-*ize e-E8012LT (22) thus giving a zero-temperature bandgap E,. = 1.48 eV and a temperature coefficient (Y= -3.2 x 10m4 eV/“K. This compares with Ego= 1.52 and (Y= -4.3 x 10m4 from optical data[l7]. Our experimental values of ni fall somewhat below those given in Sze’s book [ 181,although within a factor two. We now consider b, c and p,,, again above 340°K. As seen in Fig. 5, there is much scatter in the data for b and c, precluding detailed calculations. We would expect c to vary as exp [(2& - &)/kT] = exp (-O*l/kT)[4,13] and the data above 340°K are consistent with this. For b, the high temperature data seem to approach a constant value, b = 8. The values of p,, on the other hand, are not nearly as scattered, and p” appears to be decreasing only
1. Allen G. A., Brit. J. Appl. Phys. (J. Phys. D) 1, 593 (1968). 2. Heath D. R., Selway P. R. and Tooke C. C., Brit. J. Appl. Phys. (J. Phys. D) 1, 29 (1968). 3. Cronin G. R. and Haisty R. W., J. Electrochem. Sot. 111,874 (1964). 4. Inoue T. and Ohyama M., Sol. State Comm. 8, 1309(1970). 5. Philadelpheus A. Th. and Euthymiou P. C., J. Appl. Phys. 45, 955 (1974). 6. See e.g., Putley E. H., In The HaN E$ect and Semiconductor Physics Chap. 2. Dover Publications, New York (1968). 7. (a) Hemenger P. M., Rev. Sci. Instr. 44, 698 (1973); (b) Coleman D., Rev. Sci. Instr. 39, 1946(1968). 8. For the basic equations, see e.g., Ref. 6, Chap. 2, p. 88. 9. See e.g., Ref. [6], Chap. 3, p. 96. 10. See e.g., Sze S. M., In The Physics of Semiconductor Devices Chap. 1, p. 46. John Wiley, New York (1969). 11. Willardson R. K. and Duga J. J., Proc. Phys. Sot. (London) 75, 280 (1960). 12. Hilsum C. and Barrie R., Proc. Phys. Sot. (London) 71,676 (1958). 13. See e.g., Ref. [6], Chap. 4, pp. 122-127. 14. See e.g., Ref. [IO], Chap. 1, p. 57. IS. Conwell E. and Weisskopf V. F., Phys. Rev. 77, 388 (1950). 16. See e.g., Ref. [6], Chap. 4, p. 146. 17. Hilsum C. and Rose-Innes A. C., In Semiconducting III-V Compounds Chap. 7, p. 172. Pergamon Press, New York (1961). 18. Ref. [lo], Chap. 1, p. 28.
Pergamon Press.
Printed in Great Britain
MIXED CONDUCTION IN_ Cr-DOPED GaAst D. C. LOOK Physics Department, University of Dayton, Dayton, OH 45469,U.S.A. (Received 18January 1975;accepted 18April 1975) Abstract-Hall-effect and magnetoresistance measurements have been carried out in GaAs:Cr as functions of magnetic field strength (B = O-18kG) and temperature (T = 125420°K). Independent solutions for the mobilities, CL. and ppr and the carrier concentrations, n and p, are obtained from the basic mixed-conductivity equations. These quantities, as well as the intrinsic carrierconcentration,B, are then calculatedas a function of temperaturefor one sample,and subsequentanalysisyieldsthe followingvalues in the range T = 360-420”K:an acceptor (presumably Cr) energy E, = 0.69 2 0.02 eV (from the valence band); the bandgap energy E, = Ego+ crT,with Ego= 1.48 f 0.02 eV, (I = 3.2 X lo-’ eVpK; p” = 2700+ 100cm*/V set, decreasing slightly with temperature; pLp= 350+ 50 cm’/V set; and an acceptor-to-donor concentration ratio, &IN, =8. The electron mobility appears to be limited by neutral impurity scattering, with N, = 2 x lOI cmb3. Several other samples tiere also investigated but as a function of temperature only (at E = 0). At room temperature both positive @type) and negative (n-type) Hall coefficients were observed. 1. INTRODUCTION
It is well-known that GaAs doped with chromium is semi-insulating and thus is useful as a substrate for devices produced by diffusion, implantation, or epitaxial growth. Because of this its properties have been rather extensively explored by several techniques, including Hall-effect, photoconductivity, and optical absorption measurements [ l-51. Although the Hall coefficient (R) has always, in the past, been found to be negative, it has nevertheless been apparent that holes also contribute appreciably to the conductivity, precluding the usual simple calculation of the electron concentration (n) and mobility (h) from the zero-magnetic-field values of Hall coefficient (Ro) and conductivity (~0). Inoue and Ohyama[4] have studied R and u as a function of temperature (but not magnetic field) and have fitted their data by using the mixed-conductivity model with assumed values of CL,,, cup,and Ego, the bandgap at T = 0. They find a deep acceptor level at 066 eV from the valence band, and an acceptor-to-donor concentration ratio of about 2.1. In another recent study, at room temperature only, Philadelpheus and Euthymiou[S] combined Ro and u. with the magnetic-field dependence of u to obtain CL.,CL~, n and p ; however, their calculations were only approximate because the field dependence of R was too small to be measured. In our study we have investigated six GaAs : Cr samples from various manufacturers. Three of these were p-type (i.e. positive Hall coefficient) at room temperature, a phenomenon not reported before, to our knowledge. Furthermore, each of these three changed to n-type upon heating, thus indicating mixed conduction. The roomtemperature data are summarized in Table 1, including the “activation energy” of the resistivity (p memEdkr).Some tWork performed at Aerospace Research Laboratories, Wright-Patterson Air Force Base, under Contract No. F33615-71C-1877. Sin fact, within a few degrees of the Hall-zero (i.e. the temperature at which the Hall coefficient changes sign) R can even change sign as a function of B; however, this was not the case for any of these samples at room temperature.
of the Hall coefficients listed in the table were measured at B = 18kG and thus the values of Ro/pomay be somewhat different than the true Ro/po.SAlso listed in Table 1 is the number of the temperature run, each run extending from 300-420”K. It is observed that heating the sample to 420°K in an He atmosphere can, in some cases, appreciably affect the subsequent room-temperature values of p. and Ro. In one case, sample C, p. decreased by a factor 20 to a value low enough to possibly make it unfit for use as a substrate. For some of the samples the electrical properties annealed at room temperature, such that after several days they were about the same as those before heating. Another observation from Table 1 is the wide variation in values of p. and E,, not unrepresentative of those reported in the literature. Thus, it appears that the electrical properties of this material are not well understood and we have undertaken a more complete investigation by measuring a,,, in sample F, from 125-420”K, R. from 195-420°K, and the field dependences of R and u from 300420°K. To determine CL.,h,,, n and p above 300°K we have written independent solutions for these quantities starting from the basic mixed-conductivity equations. Then their temperature dependences make possible a determination of the acceptor activation energy and the concentrations of donors and acceptors. 2. EXPERIMENTAL
1311 IPCS Vol. 36 No. 12-A
CONSIDERATIONS
A conventional five-contact configuration[6] was employed for the Hall measurements, with each of the three voltage leads connected to an electrometer with triaxial cable. To reduce effective cable capacitances, the unity-gain output of each electrometer was connected to the middle shield of its respective cable [7]. Furthermore, the current through the sample was passed to ground through another electrometer operated in the “fastfeedback” mode. This scheme dramatically reduces the time constant for attainment of equilibrium, such that it is only about 20sec at 10”R. The sample was surrounded by a He exchange gas of about 50 mm pressure in a Pyrex dewar system which was capable of rapid, stable
1312
D. c.
LOOK
Table 1. Hall-effectand resistivitydata for several GaAs:Cr samples.Eachrun extends from 300420°K
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D
operation from 5.5-6OO’K.The maximum magnetic field available was 18 kG. The samples were cut from (100) wafers to rectangular dimensions of typically 1 x 0.4 x 0.06 cm. Ohmic indium contacts were soldered to a polished surface using an ultrasonic iron. It was not necessary to heat the contacts since their resistances were already much less than the crystal resistance.
At this point we have not introduced the explicit magnetic-field dependences of a. and a, themselves, which lead to the normal single-carrier magnetoresistances. This could be done by writing cr”= u&l - q+.‘B’) etc., where q” is a factor which is dependent upon the scattering mechanism[9]. In fact, the use of such substitutions in eqn (2) results in a formula for Ap/po, often quoted in the literature, which includes both single-carrier and mixed-conduction magnetoresistance effects to order p2B2[10]. However, we will ignore these 3. MXJB CONDUCl’lON EQUATIONS single-carrier effects because, for the holes qppP2B2s For conduction by both holes and electrons in a 0.001, and for the electrons Willardson and Duga (W-D) [ 1l] have shown that even a very small amount of magnetic field, of arbitrary strength, we have[B]’ ionized impurity scattering added to the lattice scattering B = R.u,2 f R,u,' t R.R,u.‘u;(R. + R,)B’ can drastically reduce q.. In fact, by using the W-D (on + up)*+ u.tu;(Rm + R,)*B* analysis we can calculate that q”p”*B*s0N11 for our sample, a negligible correction. Also, their analysis shows = R&+$ (1) that R, will be almost independent of field over our range n of B and T. If the impurity scattering centers are neutral, rather than ionized, then without doing a detailed analysis 1+ Yp,zB* (24 we would still expect qn to be small, because q” =O for u=““1t(X+Y)p.2B~ pure neutral-impurity scattering and we will find that the impurity scattering for electrons dominates the lattice or scattering over our temperature range. Thus, we believe that the field dependences of R and u will arise almost AP - * uo(u_l- uo-‘) = * (2b) entirely from mixed-conduction effects. n PO To proceed, we substitute cr, = nep,, uP = pep,, R. = -r,/ne, and R, = r,/pe into eqns (4)-(7), where r. and r, where are, respectively, the scattering factors for electrons and uo = 0" t a, (3) holes. The results are: 1t bc
(4)
xjL: =
u.u, (u.R. - upR, )’ (U” + u, )*
(5)
u~*u,‘(& + R, )’ (U”+ UP)*
(6)
yp.2 = ZjL”2=-
u.*u,‘R.R, (R. + R, ) u,*R. + u,‘R, ’
(7)
u0 = nietc, bdc R
0
&(I-ab*c)Vc
nie
(1 t bc)*
X=rzc(l+& ’ b(l+bc)’
y = r 2 (a - cl2 ’ (1 tbc)’
(8)
(9)
(IO)
(if)
1313
Mixed conduction in G-doped GaAs zcr2
h-c) ’ (ab*c - 1)
where (Y= r, /r,,, b = p. /pP, c = n/p, and ni is the intrinsic carrier concentration. These equations, using the same notation, are given by Hilsum and Barrie (H-B)[12] except that the scattering factors, m, r,,, and (I = r./r, were not included in their work. It is convenient to use two relationships obtained from eqns (1) and (2):
another advantage of these solutions is that ni, the intrinsic carrier concentration, is not required and, in fact, can itself be calculated: (ab - 1)4 spuo2 ” = %* cwb(1tab)’ e’(Rou$(l t R-‘)r’
(18)
From eqn (15), cub is independent of the scattering factors, so that ai a V(arz) = d/(r.r,). We will find that this factor is about 1.1 for our sample, and, in general, it will not be much different than unity. (13) To check the validity of eqn (l+( 17) we use data from H-B’s work on ptype InSb[lZ]. For their specimen No. 1, our results agree exactly, while for their specimen No. 6 and they get b =46 while we get b =74. However, for this sample, TIP is very close to 1 so that a small error in ++a.‘Y=p:X$=pz(Y+Z)j&. (14) either T or /3 will lead to a relatively large error in b, according to eqn (15). Since they determined b by an From eqn (13), a plot of R vs ApIp will give Ro and approximate method our value might be more accurate. In any case, our y’s agree, since CL,is not very dependent R = X/(Y + Z) as the ordinate and abcissa intercepts, respectively, and from eqn (14), a plot of l/B’ vs p~/Ap upon b when b * 1. Before analyzing our data we must determine m and r,. will give intercepts -l/B,* = pfizY and 8 = Y/X, and a plot of l/B* vs Ro/(Ro- R) will give intercepts -l/B,* and As discussed earlier, these quantities should be almost y = Y/( Y + Z), respectively. As will be seen later, our independent of R ; however, they can depend upon temperature since the relative strengths of the impurity data are not good enough to get accurate values of l/B,‘, S, and 7, but only the respective ratios, i.e. the slopes, and lattice scattering will depend upon temperature. It will turn out that p” = 3OOOcm*/V set, with only slight S, = -l/S&* = ,u;X, and & = -l/yB,’ = p,2( Y t Z). temperature dependence, and if we assume that the Fortunately, however, our solutions, given below, require impurities are ionized and that the lattice mobility is about only Ro, UO,and two of the three, S,, SR and /3. 10,000at 300°K and 6000 at 4OO”K,then the W-D analysis The solutions are: can be used to yield r. = 1.2 at 300°K and 1.1 at 400°K. If the scattering impurities are neutral, on the other hand, b =&{A +v(A2--4)) (15) then r, = 1.0-1.2 since both of the individual scattering factors (lattice and neutral impurity) will be in this range. For the holes, only lattice scattering will be important, where and, again, r, = l&1.2. It is thus clear that none of the quantities calculated by eqns (15)-( 18) will be in error by A = 12+ T(1-t P-*)1/(1- TIP), T E (RmJ2/Sp; (16) more than l&20% if we assume r. = r, = a = 1. As mentioned earlier, detailed measurements of R and Itab/ u vs B and T were carried out on sample E To ‘=b(cxb+R) demonstrate the method of calculation we present the data at 370°K. First, as shown in Fig. 1, R is plotted vs and Ap/pO and the ordinate and abscissa intercepts are, respectively, R. and /3. Then, in Fig. 2, l/B* is plotted vs (17) Ro/(Ro-R), and vs pO/Ap,and the slopes are, respectively, SR and S,. These slopes provide a check on /3,since Although eqn (15) has another solution, with a minus sign /3 = S,/S,.t In Fig. 3, Ro, uo, and Rouo are plotted as a in front of d(A* - 4), this is inadmissible for a sample that function of temperature, including values below 300°K to is ptype at low temperature and n-type at high show the change in sign of Ro. Finally, in Fig. 4, the temperature, i.e. a sample with b > 1. We note that b temperature dependences of b, c and CL,are shown, and depends upon the scattering factors in a very simple way in Fig. 5, p, n and ni, where p = uo/epn(c t b-l), n = cp (b 0~ a-‘) and that, as expected p,, Q (ar,)-’ = m-’ for and a = d(np). b-‘+a.
Besides the fact that no approximations are required, tActually, since we can measure all three, fi, S, and S,, and have no reason to favor S, over &, we use V/(&$,&) in place of S, in eqn (15). $This effect was present in all of our samples, and also appears in the data of Ref. 141.It is not an experimental problem since the performance of our apparatus should not be limited below lo”-10” t-l.
4. DISCUSSION
From Fig. 3 we see that at low temperatures u. flattens out.S The reasons for this are not entirely clear, but they may involve impurity conduction or surface conduction effects, which will be more thoroughly investigated later. At any rate, because of these effects the mixedconduction analysis may not be valid at these low temperatures. An indication of the range over which the
D. C
1314
SAMPLE 370%
F
Fig. 1. The Hall coefficient R vs the magnetoresistance Aplp,, for sample Fat 370°K. 3-
SAMPLE
F
Fig. 3. The temperature dependences of the Hall coefficient R,, the conductivity uOa, and their product R,uo, for sample Fat B = 0.
SAMPLE
0
5
200
300
IO
15
F
Fig. 2. The square of the inverse magnetic field strength l/B2 vs R&R, - R) andvs pO/Apfor sample F at 370°K. analysis should be applicable may be obtained from the behavior of ni, Fig. 5, which itself begins to flatten out below about 340°K. This is unreasonable behavior, and thus any of our calculated results below this temperature are suspect. Furthermore, the data plots, such as those shown in Figs. 1 and 2 are not as good below 340°K. For thermally activated holes, well below the exhaustion region (p <
IO/ 22
24
26
28
30
32
34
36
103/T(Wl
(19) where Na and ND are, respectively, the acceptor and donor concentrations, g, is the degeneracy factor, and N, = 2(2?rm~kT)“2/h’, the valence band density of states. Above 340°K our data in Fig. 5 are fitted by p = 3.15 x 10’5pq@@“T
(20)
from which, by assuming m$ = 0.5m, [14] and g, = 4, we tThe Brooks-Herring formula[l6] would give values of ND and NA about a factor four smaller, but the electron screening term employed in this formula is probably not accurate for the small carrier concentrations found in our sample.
Fig. 4. The temperature dependencies of the mobility ratio, b = pn/pp. the carrier concentration ratio, c -n/p, and the electron mobility pL.for sampleF.
obtain Na /ND = 8.4. To determine Na and ND, consider the data at 400°K; here we would expect an electron lattice mobility, pL = 6OOOcm*/V set, and since the measured mobility is e = 2600, impurity scattering must be an important effect. In fact, to within a factor two or so, the impurity scattering mobility should be given by /Lr=I(/._I - II,-‘)-‘=5000. If the impurities are ionized, the Conwell-Weisskopf formula[l5] can be employed to predict an ionized impurity concentration Ni = 5 x 10” cm-‘, and since Ni = 2No + p = 2ND (for p > n), we get No = 2 x lo”, NA =2x 10”cm-‘.t If the im-
1315
Mixed conduction in Cr-doped GaAs
SAMPLE
slightly as temperature increases, due to the competition between lattice and impurity scattering. In the temperature range 360-420”K the mobilities obey p. = 2700f lOOcm*/Vset, and pP = 350 + 50 cm’/V sec. These are quite reasonable values for this material. The measured mobility ratio, b = 8, is much lower than the value often quoted in the literature, b = 20[4,5]. In this regard two facts must be considered: (1) the room-temperature value for pure samples, b = 8000/400= 20, cannot simply be extended to GaAs: Cr since impurity scattering will limit p. much more effectively than pP,; and (2) from our experience, most values of b measured in GaAs : Cr at room-temperature are suspect because of effects not attributable to normal mixed conduction, as discussed earlier.
F
5. CONCLUSIONS
IO’,
This study has led to the following conclusions: (1) Chromium-doped GaAs can be either n-type or p-type at room temperature, and the very small values of &a0 IO" I often observed are due to mixed conduction. In some 22 24 26 28 30 32 34 36 samples, heating to 420°K affects the subsequent resistiv103/T d 1 ity at room temperature. The dominant acceptor level Fig. 5. The temperature dependencies of the carrier concentra- (presumably Cr) is about 0.69 eV from the valence band. tions, n and p, and the intrinsic concentration, ni = d(np). Impurity scattering limits the electron mobility to about half of the value found in pure samples, but the hole mobility is not affected nearly as much. (2) Mixedpurities are neutral, on the other hand, then the Erginsoy formula[l6] will predict N. = 2 x lOI cmm3, and since conduction data can be systematically analyzed from a N,=N,.,-ND-p=Na-ND, we get ND-3x10i5, knowledge of Ro, UO,and any two of the three, /?, S, and &, without any approximations and without knowledge of ni. Na =L2 x 1016.This value of Na is much more reasonable than the larger one predicted above since samples from However, the scattering factors may introduce small this manufacturer are generally found to have about uncertainties unless they are known, in which case they can be explicitly included. 1 ppm Cr. Above 34O”K,the intrinsic carrier concentration, Fig. 5, REFERENCES obeys I
/
ni = 2.27 x 101s~3’2e-*.48’kT.
(21)
By assuming that m i = 0*067m,[14], and m $2: 0.5m,, the theoretical relationship is ni = q(N,N,) e-E,12kT = 3.62 x 10’4T3’2 e-*ize e-E8012LT (22) thus giving a zero-temperature bandgap E,. = 1.48 eV and a temperature coefficient (Y= -3.2 x 10m4 eV/“K. This compares with Ego= 1.52 and (Y= -4.3 x 10m4 from optical data[l7]. Our experimental values of ni fall somewhat below those given in Sze’s book [ 181,although within a factor two. We now consider b, c and p,,, again above 340°K. As seen in Fig. 5, there is much scatter in the data for b and c, precluding detailed calculations. We would expect c to vary as exp [(2& - &)/kT] = exp (-O*l/kT)[4,13] and the data above 340°K are consistent with this. For b, the high temperature data seem to approach a constant value, b = 8. The values of p,, on the other hand, are not nearly as scattered, and p” appears to be decreasing only
1. Allen G. A., Brit. J. Appl. Phys. (J. Phys. D) 1, 593 (1968). 2. Heath D. R., Selway P. R. and Tooke C. C., Brit. J. Appl. Phys. (J. Phys. D) 1, 29 (1968). 3. Cronin G. R. and Haisty R. W., J. Electrochem. Sot. 111,874 (1964). 4. Inoue T. and Ohyama M., Sol. State Comm. 8, 1309(1970). 5. Philadelpheus A. Th. and Euthymiou P. C., J. Appl. Phys. 45, 955 (1974). 6. See e.g., Putley E. H., In The HaN E$ect and Semiconductor Physics Chap. 2. Dover Publications, New York (1968). 7. (a) Hemenger P. M., Rev. Sci. Instr. 44, 698 (1973); (b) Coleman D., Rev. Sci. Instr. 39, 1946(1968). 8. For the basic equations, see e.g., Ref. 6, Chap. 2, p. 88. 9. See e.g., Ref. [6], Chap. 3, p. 96. 10. See e.g., Sze S. M., In The Physics of Semiconductor Devices Chap. 1, p. 46. John Wiley, New York (1969). 11. Willardson R. K. and Duga J. J., Proc. Phys. Sot. (London) 75, 280 (1960). 12. Hilsum C. and Barrie R., Proc. Phys. Sot. (London) 71,676 (1958). 13. See e.g., Ref. [6], Chap. 4, pp. 122-127. 14. See e.g., Ref. [IO], Chap. 1, p. 57. IS. Conwell E. and Weisskopf V. F., Phys. Rev. 77, 388 (1950). 16. See e.g., Ref. [6], Chap. 4, p. 146. 17. Hilsum C. and Rose-Innes A. C., In Semiconducting III-V Compounds Chap. 7, p. 172. Pergamon Press, New York (1961). 18. Ref. [lo], Chap. 1, p. 28.