Influence of excited states of a deep substitutional dopant on majority-carrier concentration in semiconductors

Transcription

1 Infuence of excited states of a deep substitutiona dopant on majority-carrier concentration in semiconductors Hideharu Matsuura* Department of Eectronic Engineering Computer Science, Osaka Eectro-Communication University, 8-8 Hatsu-cho, Neyagawa, Osaka , Japan Received 20 June 2006; revised manuscript received 20 October 2006; pubished 26 December 2006 The density N A energy eve E A of an acceptor in a p-type wide-b-gap semiconductor e.g., SiC, GaN, diamond are determined by a east-squares fit of the charge neutraity equation to the temperature dependence of the hoe concentration pt using the Fermi-Dirac distribution function for acceptors that does not consider the infuence of the excited states of the acceptor. The N A obtained this way is, however, much higher than the concentration of acceptor atoms determined by secondary ion mass spectroscopy. Because E A is far from the vaence-b maximum E V the Fermi eve is between E A E V, the infuence of the excited states of the acceptor on pt shoud not be ignored. A distribution function incuding the infuence of excited states, which is derived from the viewpoint of the microcanonica ensembe, not the gr canonica ensembe, eads to reiabe N A E A. The situation in n-type Te-doped A 0.6 Ga 0.4 Sb is the same, because the Te donor eve is far from the conduction-b minimum for arge A moe fraction. Finay, the excited states of a substitutiona dopant with a deep energy eve are found to enhance the ionization of the dopant, even though they were expected to suppress the ionization because they acted as a trap according to the distribution function derived from the viewpoint of the gr canonica ensembe. DOI: 0.03/PhysRevB PACS numbers: 7.55.i, 73.6.Le I. INTRODUCTION The excited states of a substitutiona dopant in a semiconductor have been theoreticay discussed using the hydrogenic mode or the effective-mass approximation, 3 the existence of excited states of the dopant e.g., B or P in Si or Ge was experimentay confirmed from infrared absorption measurements at very ow temperatures. However, the infuence of the excited states on the majority carrier concentration in Si or Ge was not experimentay confirmed, 2,4 partiay because the excited state eves of the dopants in Si or Ge were too shaow partiay because the Fermi eve E F T was deeper than the dopant energy eve in the temperature range above 77 K. Therefore, by a east-squares fit of the charge neutraity equation to the temperature dependence of the majority-carrier concentration experimentay obtained by Ha-effect measurements, the vaues of dopant density, dopant energy eve, compensating density can be determined using the Fermi-Dirac FD distribution function, which does not incude the infuence of the excited states of the dopant. The FD distribution functions for donors acceptors are expressed as 5 f FD E D = + exp E FT E D g D,FD f FD E A = +g A,FD exp E FT E A, 2 respectivey, where E D E A are the donor acceptor eves, respectivey, g D,FD g A,FD are the degeneracy factors for donors acceptors, respectivey, k is the Botzmann constant. Because in p-type wide-b-gap semiconductors e.g., SiC, GaN, diamond the experimentay obtained vaues of E A have been reported to be deep, 6 E F T is often between E A the vaence-b maximum E V. Furthermore, because the excited-state eves of acceptors in these semiconductors are as deep as a B acceptor eve i.e., ground-state eve in Si, 7 E F T is cose to the excited-state eves. The excited states of the acceptor must, therefore, affect the temperature dependence of the hoe concentration pt. Using a distribution function incuding the infuence of the excited states of an acceptor derived from the viewpoint of the gr canonica ensembe, 2,8,9 an acceptor density N A was determined by fitting the simuation pt to the experimenta pt. The N A determined this way was, however, much higher than the concentration of acceptor atoms determined by secondary ion mass spectroscopy SIMS. 0 This same situation occurred in the case of the FD distribution function. 0 2 In this paper, we report on our investigation of the optimum distribution function for determining the density energy eve of a deep substitutiona dopant in p-type wideb-gap semiconductors i.e., B-doped diamond, A-doped 6H-SiC, A-impanted 4H-SiC, Mg-doped GaN n-type Te-doped A 0.6 Ga 0.4 Sb, where a Te donor eve in A 0.6 Ga 0.4 Sb has been reported to be deep. 3 II. EXPERIMENT A.73-m-thick singe crystaine B-doped p-type diamond epiayer was grown at 830 C on the 00 surface of a synthesized singe-crysta b diamond substrate area of.52.0 mm 2, thickness of 0.3 mm, resistivity of /2006/7424/ The American Physica Society

2 HIDEHARU MATSUURA cm by a microwave pasma assisted chemicavapor-deposition method. The B concentration C B in the epiayer, measured by SIMS, was approximatey cm 3. The detais of the sampe preparation were reported earier. 4,5 A 400-m-thick heaviy A-doped p-type 6H-SiC wafer in which the A concentration C A was approximatey cm 3 is referred to as a heaviy doped 6H-SiC, whie a 4.9-m-thick A-doped p-type 6H-SiC epiayer C A of approximatey 60 5 cm 3 on an n-type 6H-SiC substrate is referred to as a ighty doped 6H-SiC. These sampes were purchased from Cree Res. Inc. The detais of these sampes were reported earier. 6 A ions were impanted at 000 C in a 5-m-thick n-type 4H-SiC epiayer with N atoms at cm 3 on an n-type 4H-SiC 000 substrate cut 8 off toward the 2 0 direction. To obtain a box profie of the concentration of A atoms, sevenfod A ion impantation was carried out at different energies onto the SiC epiayer surface tited to 7 to norma. After the impantation, the sampe was anneaed at 575 C for h in an Ar atmosphere. The.3-m-thick ayer was removed from the surface by reactive ion etching. The C A in the impanted ayer was approximatey 0 9 cm 3. The detais of the sampe preparation were reported earier. 7 A2-m-thick Mg-doped p-type GaN epiayer was grown at 025 C by metaorganic chemica vapor deposition on undoped GaN/sapphire, anneaed at 800 C in N 2 for 20 min. The Mg concentration C Mg in the epiayer, measured by SIMS, was approximatey 20 9 cm 3. The detais of the sampe preparation were reported earier. 8 A2-m-thick Te-doped n-type A 0.6 Ga 0.4 Sb epiayer was grown on a semi-insuating GaAs00 substrate at 470 C by water-cooed moecuar-beam epitaxy. The temperatures of Knudsen ces for Ga A were C, respectivey, the temperature of a crysta ce for Sb 4 was 460 C. For doping Te into the epiayer during the deposition, the temperature of the Knudsen ce for Ga 2 Te 3 was 330 C. The Te concentration C Te in the epiayer was approximatey 0 7 cm 3. The detais of the sampe preparation were reported earier. 3 The pt vaues for 4H-SiC, 6H-SiC, diamond, GaN, the temperature-dependent eectron concentration nt for A 0.6 Ga 0.4 Sb, were obtained by Ha-effect measurements in the van der Pauw configuration at a magnetic fied of.4 T using a modified MMR Technoogies Ha system. III. RESULTS Figure shows pt E F T for the B-doped diamond. Here, E F T was derived from pt = N V Texp E FT E V, 3 where N V T is the effective density of states in the vaence b, given by FIG.. Temperature dependence of hoe concentration Fermi eve for B doped diamond. N V T =2 2m h * h 2 3/2, 4 m h * is the hoe effective mass in the semiconductor, h is Panck s constant. Because the E A of B in diamond is approximatey E V ev, the E F T vaues were ower than E A over the measurement temperature range. N A E A of B acceptors, a donor density N D, can be determined by fitting the simuation pt to the experimenta pt using Eq. 3 pt = N A FE A N D, where nt is much ess than pt over the measurement temperature range, FE A is the distribution function for acceptors. By repacing FE A with f FD E A, the vaues of N A, E A, N D were determined are isted in Tabe I. In the tabe, f FD, f MC, f GC represent the FD distribution function the distribution functions incuding the infuence of the excited states of a dopant. In the foowing section, these distribution functions wi be derived from the viewpoint of the microcanonica MC gr canonica GC ensembes, respectivey. The N A of cm 3, which was determined by the curve-fitting procedure described previousy, is approximatey five times higher than the C B of approximatey cm 3. Since N A is the density of B atoms at the attice sites, N A shoud be ess than or equa to C B. Therefore, f FD E A seems inappropriate to a distribution function for B acceptors in diamond. Figure 2 depicts pt E F T for the heaviy A-doped 6H-SiC wafer, whie Fig. 3 shows pt E F T for the ighty A-doped 6H-SiC epiayer. The E F T vaues for the heaviy doped 6H-SiC were beow E A over the measurement temperature range, whereas the E F T vaues for the ighty doped 6H-SiC were above E A over amost a the measurement temperature range. By the curve-fitting procedure using f FD E A, the vaues of N A, E A, N D were estimated are isted in Tabe I. The N A obtained for the heaviy doped 6H-SiC is

3 INFLUENCE OF THE EXCITED STATES OF A DEEP TABLE I. Resuts for each distribution function. B-doped diamond Heaviy doped 6H-SiC Lighty doped 6H-SiC A-impanted 4H-SiC Mg-doped GaN Te-doped A 0.6 Ga 0.4 Sb f FD N A cm N D E A ev E V E V +0.8 E V +0.9 E V +0.6 E V +0.5 E D E C 0.09 N D cm N A.70 7 f MC N A cm N D.40 7 E A ev E V E V +0.8 E V E V +0.8 E V +0.6 E D E C 0.3 N D cm N A f GC N A cm N D Doping Density E A ev E V E V +0.8 E V +0.9 E V +0.7 E V +0.7 E D E C 0.0 N D cm N A C cm cm 3, which is approximatey six times higher than the C A. On the other h, the N A obtained for the ighty doped 6H-SiC is cm 3, which is neary equa to the C A. This suggests that f FD E A can be used ony in the ighty doped sampe. The reason for this is that in the ighty doped case there are few hoes ocated at the excited states of the acceptor due to the arge difference in energy between E F T the excited-state eve. As is cear from Tabe I, in the A-impanted 4H-SiC ayer, the N A of cm 3 determined by the curvefitting procedure is approximatey five times higher than the C A, in the Mg-doped GaN epiayer the N A of cm 3 is approximatey four times higher than the C Mg. The E F T vaues for the A 0.6 Ga 0.4 Sb epiayer were ower than E D over amost the entire measurement temperature range. This is because the Te donor in A x Ga x Sb changed from a shaow to a deep donor with the increase in A moe fraction x. 3 By the curve-fitting procedure using f FD E D, the vaues of N D, E D, N A were determined are isted in Tabe I. The N D of cm 3 is approximatey three times higher than the C Te. Moreover, the ratio of N A to N D is 0.54, which seems too high because the residua acceptor density in our undoped p-type A 0.6 Ga 0.4 Sb was of the order of 0 5 cm 3. 3 IV. DISTRIBUTION FUNCTION FOR A DEEP SUBSTITUTIONAL DOPANT A. The number of configurations of the system We now consider the microcanonica ensembe. Eectrons hoes in semiconductors are fermions, which obey the Paui excusion principe. Because of this, in the aowed bs, the mutipicity function W B E i for n e E i eectrons arranged in DE i states at a given energy E i is expressed as 8 DE i! W B E i = DE i n e E i!n e E i!, 6 where DE i is the number of degenerate states per unit voume at E i n e E i is the number of eectrons per unit voume at E i. In a forbidden b, on the other h, the mutipicity function for n D eectrons arranged in N D donors is quite dif- FIG. 2. Temperature dependence of hoe concentration Fermi eve for heaviy A-doped 6H-SiC. FIG. 3. Temperature dependence of hoe concentration Fermi eve for ighty A doped 6H-SiC

4 HIDEHARU MATSUURA ferent from Eq. 6, where n D is the number of eectrons bound to donors per unit voume. When spin degeneracy as we as the excited states of the donor is negected, the mutipicity function W D for the n D eectrons arranged in the N D donors is given by N D! W D = N D n D!n D!. 7 Each state of the ground excited states consists of a spin-up state a spin-down state. When the difference in energy between the two states in a magnetic fied is denoted by E spin, the mutipicity function W D2 is given by W D2 = + exp E spinnd. 8 When the magnetic fied is zero or very weak E spin 0, W D2 2 n D. 9 In a neutra donor, furthermore, ony an eectron is ocated at one state of the ground excited states of the donor. The mutipicity function W D3 is given by W D3 =g + g r exp E r,d E D nd, 0 where E r,d is the r th excited-state eve r2 of the donor, g is the ground-state degeneracy factor of, g r is the r th excited-state degeneracy factor of r 2, the vaue of is the highest order of the excited states considered here. Consequenty, the mutipicity function W D for the n D eectrons arranged in the N D donors is expressed as W D = W D W D2 W D3. Finay, the tota number of configurations of the system W is obtained from the product of these mutipicities as W = W D W Bi. i B. Therma equiibrium configuration 2 Therma equiibrium configuration occurs when the entropy S = k n W 3 reaches a maximum vaue under the foowing two conservation aws: i the tota number n tota of eectrons in the system is conserved, that is, n tota = n D + ne i = const, 4 i ii the tota energy E tota of eectrons in the system is conserved, i.e., E tota = E D Tn D + E i ne i = const, 5 i where E D T is the average donor eve given by E D T = E D + E ex,d T, 6 E ex,d T is the ensembe average of the ground- excited-state eves of the donor, measured from E D, is given by E ex,d T = E r,d E D g r exp E r,d E D + g r exp E r,d E D. 7 Under these conditions, the distribution functions for eectrons incuding the infuence of the excited states of the donor f MC E D can be derived, 0 f MC E D n D = N D + g D,MC T exp E FT E D, 8 where g D,MC T is here caed the effective degeneracy factor for donors given by g D,MC T =2+g r exp E r,d E D exp E ex,dt. 9 C. The hydrogenic donor case A neutra donor can be approximatey described as a hydrogen atom, that is, a positivey ionized donor with an eectron in orbit about the ionized donor. In this case, E r,d is given by 2,3 q 4 * m E r,d = E C e 8h 2 2 s 2 0 r 2 = E C E,D r 2 20 E,D = 3.6 m * e 2 ev, 2 m 0 s where q is the eectron charge, m 0 is the free-space eectron mass, s is the semiconductor dieectric constant, 0 is the free-space permittivity. Because the Bohr radius a * of the ground state is very sma, E D is expressed as E D = E,D E CCC, 22 where E CCC is the centra ce correction induced due to a strongy ocaized potentia. Therefore, the ground-state eve does not obey the hydrogenic mode. However, because the wave-function extension of the r th excited state is of the order r 2 a *, the excited-state eves i.e., E r,d with r2 are expected to foow the hydrogenic mode

5 INFLUENCE OF THE EXCITED STATES OF A DEEP D. Distribution function for acceptors In the case of acceptors, different from those in the conduction b, there are two degenerate vaence bs i.e., a ight-hoe b a heavy-hoe b, indicating that there is an acceptor state for the ight-hoe b as we as an acceptor state for the heavy-hoe b. When the density of eectrons occupied at acceptors n A is considered, therefore, f MC E A is derived as 0,6 f MC E A n A = N A +g A,MC Texp E FT E A, 23 where g A,MC T is here caed the effective degeneracy factor for acceptors given by g A,MC T =4+ g r exp E A E r,a exp E ex,at. 24 Here, E ex,a T is the ensembe average of the ground- excited-state eves of the acceptor, measured from E A, is given by E ex,a T = E A E r,a g r exp E A E r,a + g r exp E A E r,a, 25 E r,a is the r th excited-state eve of the acceptor, expressed as E r,a = E V + q 4 m h * 8h 2 s r 2 = E V + E,A r 2, 26 E,A = 3.6 m h * 2 ev. 27 m 0 s The average acceptor eve E A T the acceptor eve are expressed as E A T = E A E ex,a T E A = E,A + E CCC. E. Distribution function derived from the gr canonica ensembe The gr canonica ensembe is an ensembe of the same subsystems, partices can transfer from one subsystem into another subsystem, indicating that the number of partices in the subsystem N can change. The partition function for the gr canonica ensembe is given by 9 where = z N Z N, N 30 z = exp E FT. 3 Z N is the partition function for the canonica ensembe for a given N, which is expressed as n j E j j Z N = exp. 32 This summation is carried out over a the sets of n j under the condition N = n j, 33 j where E j n j are the energy eve number of eectrons at a jth state in the subsystem, respectivey. A donor is a subsystem. Each donor has one eectron or no eectron i.e., N=0 or. The eectron is ocated at one state of the ground excited states i.e., n j =0 or. Moreover, the number of spin states degenerate excited states shoud be taken into account. Therefore, Finay, = Z 0 = Z =2exp E D + N=0 + g r exp E r,d. z N Z N = Z 0 + zz =+2exp E D E F T g r exp E r,d E F T. 36 The mean number N of eectrons in the subsystem in therma equiibrium is given by N=0 Nz N Z N N =. 37 Therefore, the distribution function incuding the infuence of the excited states of a donor is derived as where f GC E D N = + g D,GC T exp E FT E D,

6 HIDEHARU MATSUURA g D,GC T =2+ g r exp E r,d E D, 39 which coincides with the reported distribution functions. 2,4,9 On the other h, the distribution function for acceptors is derived as where f GC E A = g A,GC T =4+ +g A,GC Texp E FT E A, 40 g r exp E A E r,a. 4 F. Comparison between three distribution functions If the infuence of the excited states of a donor or an acceptor coud be ignored i.e., =, f MC E D or f MC E A woud coincide with f FD E D or f FD E A. The reason why f GC E D is different from f MC E D is discussed beow. Because an eectron can be at a higher excitedstate eve at eevated temperatures, the energy of an eectron bound to a donor increases with increasing T. Therefore, the average donor eve shoud increase with T, which is consistent with Eqs If, on the other h, eectrons were ocated at the ground-state eve at a temperatures, Eq. 5 coud be repaced by E tota = E D n D + E i ne i = const. 42 i In this case, the distribution function for donors derived from the microcanonica ensembe viewpoint woud coincide with f GC E D. This suggests that f GC E D is correct ony under the assumption that a the eectrons bound to donors have E D at a temperatures. In the same way as iustrated for f GC E D, if E A T coud be assumed to be E A i.e., E ex,a T=0, f MC E A woud coincide with f GC E A. V. DETERMINATION OF RELIABLE DENSITY AND ENERGY LEVEL OF DEEP DOPANT The ground-state eve i.e., acceptor eve of E A does not obey the hydrogenic mode because a * is too sma. However, the vaues of E r,a with r2 are expected to foow the hydrogenic mode, because the wave-function extension of the r th excited state is of the order r 2 a *. In B-doped diamond, by fitting the simuation pt to the experimenta pt using f MC E A under a simpe assumption i.e., E,A =0.39 ev, the vaues of N A, E A, N D were determined as cm 3, E V ev, cm 3, respectivey. The highest excited state considered here was the sixth excited state i.e., =7, under which the best curve fitting was achieved. Because the radius of the sixth excited state is approximatey 5 nm the attice constant for diamond is nm, the number of C atoms in a sphere of radius 5 nm is approximatey Because the C density is cm 3 the C B is approximatey cm 3, there is, on the other h, one B atom in approximatey 0 6 C atoms. These suggest that the condition i.e., =7 is not so bad. In the discussion above, E,A is ambiguous. When E,A was taken as 0.30 ev, N A, E A, N D were determined as cm 3, E V ev, cm 3, respectivey. When E,A was taken as 0.50 ev, N A, E A, N D were determined as cm 3, E V ev, cm 3, respectivey. This resut suggests that N A, E A, N D have ony a sma dependence on the excitedstate eves. In Tabe I, therefore, E A, N A, E D, N D are expressed in doube figures. Thonke reported that from their study of photoconductivity, the energy eves of ong-ived excited states of B in diamond were 0.200, 0.240, ev from E A, respectivey. 7 On the other h, the vaues of E A E 2,A, E A E 3,A, E A E 4,A we obtained are 0.22, 0.27, 0.29 ev, respectivey, when E,A =0.39 ev. Because the vaues obtained are cose to the reported vaues, the excited-state eves considered here are reasonabe. N A, E A, N D determined by the curve-fitting procedure using f GC E A are aso shown in Tabe I. In Tabe I, a the E A vaues determined using three distribution functions seem reasonabe. However, the N A for f GC E A is highest, whie the N A for f MC E A is owest. The N A obtained using f MC E A is cosest to the C B. Therefore, f MC E A is suitabe for determining N A from pt. Tabe I shows N A, E A, N comp for the A-impanted 4H-SiC ayer determined by the curve-fitting procedure using f MC E A or f GC E A for E,A of 0.46 ev. The highest excited state considered here was the fourth excited state i.e., =5, which was ess than in the B-doped diamond. This is because the A-doping density in the 4H-SiC ayer was higher than the B-doping density in the diamond epiayer. The N A obtained using f MC E A is cosest to the C A among the three distribution functions. As is cear from Tabe I, ony f MC E A ed to reiabe N A, E A, N D for the heaviy A-doped 6H-SiC, whereas a three distribution functions ed to reasonabe vaues for the ighty A-doped 6H-SiC epiayer. This suggests that the excited states of the A acceptor do not affect pt very much in the ighty A-doped 6H-SiC epiayer whose E F T is far from the excited-state eves. Therefore, f MC E A is determined to be suitabe for both cases. In Mg-doped GaN or Te-doped A 0.6 Ga 0.4 Sb where E F T was shaower than the energy eve of the dopant, f MC is suitabe for determining the density energy eve of the dopant from the temperature dependence of the majoritycarrier concentration. Three vaence bs exist cose to the point at the center of the Briouin zone. As discussed in Sec. IV D, in diamond zinc-bende structures the ight-hoe heavyhoe vaence bs are degenerated at, the other vaence b is ocated ower than the degenerate vaence bs by spin-orbit spitting SO. Since SO for Si GaAs are mev, 22 respectivey, amost a of the hoes exist in the two degenerate vaence bs at. There

7 INFLUENCE OF THE EXCITED STATES OF A DEEP FIG. 4. Temperature dependence of simuated degeneracy factors for acceptors for 6H-SiC. fore, g A,FD of 4 in f FD E A a factor of 4 in g A,MC T of f MC E A are quite reasonabe. SO for diamond is, however, 6 mev. 22 If a three vaence bs can be assumed to contribute equay to the hoe conduction, both g A,FD the factor in g A,MC T shoud be changed from 4 to 6. The vaues of N A obtained for the B-doped diamond epiayer using f FD E A f MC E A are cm 3, respectivey. The N A obtained using f MC E A is cose to the C B. Therefore, N A is probaby between cm 3. In a wurtzite structure hexagona poytypes of SiC e.g., 6H-SiC 4H-SiC, three vaence bs are a spit due to both spin-orbit crysta-fied interactions at. 20,2 The highest vaence b is the heavy-hoe vaence b, the other vaence bs are ocated ower than the heavy-hoe vaence b by SO CF, respectivey, where CF is the energy separation due to the crysta-fied interaction. The vaues of SO CF for 6H-SiC are reported to be mev, respectivey. 2,22 If a the hoes can be assumed to exist ony in the heavy-hoe vaence b, both g A,FD the factor in g A,MC T shoud be changed from 4 to 2. In this case, the vaues of N A obtained for the heaviy A-doped 6H-SiC wafer using f FD E A f MC E A are cm 3, respectivey. The N A obtained using f MC E A is cose to C A. Therefore, N A is probaby between cm 3. Judging from the above discussions, the distribution function derived from the microcanonica ensembe viewpoint is most appropriate, athough the N A obtained depends sighty on the vaence-b structures. The reason a reasonabe dopant density can be obtained using Eq. 8 or Eq. 23 is discussed from the viewpoint of the effective degeneracy factors. Figure 4 shows g A,FD, g A,MC T, g A,GC T for 6H-SiC, denoted by broken, soid, dotted ines, respectivey. g A,MC T decreases from 4 with increasing T, whereas g A,GC T increases. As the effective degeneracy factor for acceptors decreases, the distribution function for acceptors approaches at the same T, indicating that in the case of f MC E A, the ionization efficiency of acceptors is highest at eevated temperatures. FIG. 5. Temperature dependence of simuated ionized acceptor densities for heaviy A-doped 6H-SiC. Figure 5 depicts the ionized acceptor densities N A T simuated using N A of cm 3, E A of E V +0.8 ev, N D of cm 3, the effective degeneracy factors for acceptors shown in Fig. 4, for f FD E A, f MC E A, f GC E A, denoted by broken, soid, dotted curves, respectivey. As is cear from the figure, the N A T for f MC E A is highest at eevated temperatures. The N A 450 for f MC E A is, for exampe, approximatey twice higher than that for f FD E A. In the case of f MC E A, therefore, the N A required to satisfy the experimentay obtained pt is much ess than that determined using f FD E A or f GC E A. Another interpretation of Fig. 5 is as foows. As the temperature increases, the possibiity that a hoe bound to the acceptor is ocated at a higher excited-state eve increases. That is why the acceptor can more easiy emit a hoe to the vaence b at eevated temperatures. This coincides with Eq. 28. Therefore, it is cear that the excited states of the acceptor enhance the ionization of the acceptor at eevated temperatures. Figure 6 shows N A T simuations for the ighty A-doped 6H-SiC using the effective degeneracy factors for acceptors shown Fig. 4. Here, N A, E A, N D used in the simuation FIG. 6. Temperature dependence of simuated ionized acceptor densities for ighty A-doped 6H-SiC

8 HIDEHARU MATSUURA were cm 3, E V ev, cm 3, respectivey. In the figure, the broken, soid, dotted ines represent the N A T simuations for f FD E A, f MC E A, f GC E A, respectivey. Athough the effective degeneracy factors for acceptors in the ighty doped case are the same as those in the heaviy doped case, N A T for f FD E A, f MC E A, f GC E A are simiar to each other in the ighty doped case. This is because the effective degeneracy factor for acceptors has itte effect on f FD E A, f MC E A, f GC E A, when E F T is far from the acceptor eve, that is, exp E FT E A 43 in Eqs. 2, 23, 40 is neary 0. To determine N A from pt, therefore, f MC E A is the most appropriate among them. VI. CONCLUSION A distribution function suitabe for determining the density of a deep substitutiona dopant from the temperature dependence of the majority-carrier concentration was discussed. The foowing three distribution functions were investigated: i the Fermi-Dirac distribution function, which does not consider the infuence of the excited states of the dopant, ii the distribution function that incudes the infuence of excited states derived from the microcanonica ensembe viewpoint, iii the distribution function incuding the infuence of excited states derived from the gr canonica ensembe viewpoint. The distribution function derived from the microcanonica ensembe was determined to be the most appropriate among them for determining the density energy eve of a deep substitutiona acceptor in heaviy doped p-type wide-bgap semiconductors e.g., A-doped SiC, Mg-doped GaN, B-doped diamond whose Fermi eves were between E V E A. This was because the excited states of the acceptor enhanced the ionization of the acceptor. This same situation was observed in n-type A 0.6 Ga 0.4 Sb with deep Te donors. ACKNOWLEDGMENTS The author woud ike to thank H. Sugimoto of Mitsubishi Eectric Corp. for the A-impanted 4H-SiC sampe preparation, A. Namba, T. Imai, T. Takebe of Sumitomo Eectric Industries for the diamond sampe preparation, T. Suzuki, S.-W. Kim, T. Shibata of Nippon Institute of Technoogy for the GaN sampe preparation, W. Susaki of Osaka Eectro-Communication University for the A 0.6 Ga 0.4 Sb sampe preparation. He woud ike to acknowedge K. Nishikawa, K. Sugiyama, S. Kagamihara, H. Iwata, other members of Matsuura Laboratory for the Ha-effect measurements. This work was partiay supported by the Academic Frontier Promotion Projects of the Ministry of Education, Cuture, Sports, Science Technoogy MEXT in *Eectronic address: matsuura@isc.osakac.ac.jp P. Y. Yu M. Cardona, Fundamentas of Semiconductors: Physics Materias Properties, 2nd ed. Springer, Berin, 999, pp B. Sapova C. Hermann, Physics of Semiconductors Springer-Verag, New York, 993, pp J. Singh, Semiconductor Devices: An Introduction McGraw-Hi, New York, 994, p.0. 4 R. A. Smith, Semiconductors, 2nd ed. Cambridge University Press, Cambridge, 978, p S. M. Sze, Physics of Semiconductor Devices, 2nd ed. Wiey, New York, 98, Chap.. 6 O. Meung, Semiconductors: Data Hbook, 3rd ed. Springer, Berin, 2004, pp., 60, K. Thonke, Semicond. Sci. Techno. 8, S K. F. Brennan, The Physics of Semiconductors with Appications to Optoeectronic Devices Cambridge University Press, Cambridge, 999, p N. W. Ashcroft N. D. Mermin, Soid State Physics Hot Rinehart Winston, Phiadephia, 976, pp H. Matsuura, New J. Phys. 4, T. Troffer, M. Schadt, T. Frank, H. Itoh, G. Pens, J. Heind, H. P. Strunk, M. Maier, Phys. Status Soidi A 62, N. Schuze, J. Gajowski, K. Semmeroth, M. Laube, G. Pens, Mater. Sci. Forum , H. Matsuura K. Nishikawa, J. App. Phys. 97, H. Shiomi, Y. Nishibayashi, N. Fujimori, Jpn. J. App. Phys., Part 30, H. Matsuura, T. Morizono, Y. Inoue, S. Kagamihara, A. Namba, T. Imai, T. Takebe, Jpn. J. App. Phys., Part 45, H. Matsuura, J. App. Phys. 95, H. Matsuura, K. Sugiyama, K. Nishikawa, T. Nagata, N. Fukunaga, J. App. Phys. 94, H. Matsuura, D. Katsuya, T. Ishida, S. Kagamihara, K. Aso, H. Iwata, T. Aki, S.-W. Kim, T. Shibata, T. Suzuki, Phys. Status Soidi C 0, C. V. Heer, Statistica Mechanics, Kinetic Theory, Stochastic Processes Academic Press, New York, 972, p H. Morkoç, Nitride Semiconductors Devices Springer, Berin, 999, p W. J. Choyke, Siicon Carbide, edited by W. J. Choyke, H. Matsunami, G. Pens Springer, Berin, 2004, p S. Adachi, Properties of Group-IV, III-V II-VI Semiconductor Wiey, Chichester, 2005, p