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13 C spin–lattice relaxation in natural type-IIb semiconducting diamond
27 August 1999
Chemical Physics Letters 310 Ž1999. 97–102 www.elsevier.nlrlocatercplett
13
C spin–lattice relaxation in natural type-IIb semiconducting diamond Cornelis J. Terblanche a , Eduard C. Reynhardt a
a,)
, Jan A. van Wyk
b
Department of Physics, UniÕersity of South Africa, P.O. Box 392, Pretoria 0003, South Africa b Department of Physics, UniÕersity of the Witwatersrand, Johannesburg, South Africa Received 27 May 1999; in final form 21 June 1999
Abstract 13
C spin–lattice relaxation times in the laboratory frame have been measured at 295 K in a field of 4.7 T for two natural type-IIb diamonds using nuclear magnetic resonance spectroscopy ŽNMR.. 13 C spin–lattice relaxation is mainly due to mobile hole carriers. The contribution of fixed paramagnetic impurities is at least an order of magnitude smaller. At 295 K, the carrier concentration is estimated to be 1.2 = 10 14 cmy3, resulting in an h , the electron density at the nucleus, of about 320 for natural type-IIb diamonds. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction The very long 13 C spin–lattice relaxation times in natural diamond Žseveral hours. w1x are mainly due to the low concentration of paramagnetic impurity centers, the low natural abundance Ž1.1%. of 13 C nuclei and the rigidity of the diamond lattice. From several studies performed on type-I and type-II diamond w1–3x, it seems that the main relaxation mechanism is the dipolar interaction between paramagnetic impurities and 13 C nuclei. However, in most of these studies, very little was known about the types and concentrations of paramagnetic centers present in the samples. Recently Terblanche et al. w4x investigated the 13 C spin–lattice relaxation behavior in natural type-Ia, -IIa and -Ib diamonds. EPR techniques were
) Corresponding author. Fax: q27-12-4293434; e-mail: [email protected]
used to characterize the paramagnetic impurities Žtype, concentration, electronic relaxation times, etc.. in each diamond. The knowledge that the samples contained either P1 and P2 or P1 and N3 or P1 and N2 centers resulted in definite conclusions regarding the relative effectiveness of paramagnetic impurity types in the relaxation of 13 C nuclei. Order of magnitude estimates of the 13 C spin–lattice times were obtained using existing paramagnetic relaxation models. Hoch and Reynhardt w2x investigated the 13 C spin–lattice relaxation over the temperature range 295–375 K of a 6.2 ct type-IIb diamond Žsample h. with a boron acceptor center concentration of 3 = 10 16 cmy3 , a detectable paramagnetic impurity concentration of below 10 15 cmy3 and a mobile hole concentration of about 10 14 cmy3 . The 13 C spin– lattice relaxation time ŽT1 s 4.1 h at 295 K. was ascribed to the interaction between the fixed paramagnetic hole centers and the nuclei.
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 7 5 8 - 7
98
C.J. Terblanche et al.r Chemical Physics Letters 310 (1999) 97–102
In this Letter, we discuss 13 C spin–lattice relaxation times observed for semiconducting type-IIb diamonds and show that the 13 C nuclei are actually relaxed by the mobile hole carriers. Continuous wave EPR was used to determine the concentrations of the fixed paramagnetic impurities in the diamonds, while electronic spin–spin and spin–lattice relaxation times were determined using pulse EPR. The mobile hole and fixed paramagnetic impurity contributions to the 13 C spin–lattice relaxation times are calculated and compared with the measured 13 C spin–lattice relaxation times.
2. Experimental and results Two natural type-IIb diamonds, sample b4 Ž3.12 ct. and sample b5 Ž2.06 ct., were used in this investigation. The Zeeman 13 C spin–lattice relaxation times were measured at 295 K on a Bruker CXP 200 pulsed NMR spectrometer operating at a Larmor frequency of 50 MHz for 13 C nuclei. Within experimental error, all relaxation data revealed only one relaxation time. Continuous wave Žcw. spectra were recorded at 295 K on an X-band Varian E-line Century Series EPR spectrometer. The impurity concentrations were determined using double integration techniques, a ruby reference sample and a type-Ib synthetic dia-
Fig. 1. cw EPR spectrum of sample b4 at 295 K, indicating the three ruby reference lines and the unknown paramagnetic impurity line which overlaps with the central ruby line.
Fig. 2. cw EPR spectrum of the ruby reference sample at 295 K.
mond with a known concentration of paramagnetic defects ŽP1 centers, S s 1r2.. Corrections were made for the change in the Q-factor of the cavity due to different sample sizes, etc. The cw EPR spectrum of sample b4 is shown in Fig. 1. The ruby reference sample gave rise to three resonance lines, a weak central line and two strong lines on each side of the central line, as depicted in Fig. 2. By comparing the central lines of Figs. 1 and 2, the presence of an unknown paramagnetic center which is not present in all type-IIb diamonds is evident. The EPR spectrum of this defect also has no hyperfine lines, making its identification extremely difficult. This defect could be a transition metal-type center which serves as a donor to compensate the holes. Assuming S s 1r2, the concentrations of these defects can be estimated Žsee Table 1.. Another interesting feature in the cw EPR spectrum of sample b4 ŽFig. 1. is the slope of the baseline. This phenomenon is common in type-IIb diamonds, and it seems that the gradient of the baseline gives an indication of the mobile hole concentration. Sample b5 revealed the same characteristics in its cw EPR spectra. The line structure of the central line corresponded with that observed for b4, suggesting the same type of paramagnetic center was present. The slope of the baseline was also approximately the same as that of sample b4, implying comparable mobile hole concentrations.
C.J. Terblanche et al.r Chemical Physics Letters 310 (1999) 97–102
99
Table 1 Measured 13 C spin–lattice relaxation times, paramagnetic impurity center spin–lattice ŽT1e . and spin–spin ŽT2e . relaxation times for samples b4 and b5. The corresponding values for sample h ŽRef. w2x. are also given Sample
ND Žcmy3 .
T1e Ž=10y3 s.
T2e Ž=10y6 s.
T1 Žexperimental. Žhours.
b4 b5 h
1.5 = 10 16 5.8 = 10 15 ; 10 15
3.6 " 0.2 10 " 5 0.1
16 " 1 11 " 1 –
3.7 " 0.7 5.1 " 1 4.1
An X-band Bruker ESP 380E pulsedrcw spectrometer was used to determine the electron spin–spin and spin–lattice relaxation times ŽT2e and T1e , respectively. of the paramagnetic impurity centers at 295 K. The apparatus and pulse sequences employed have been described elsewhere w5x. The results are listed in Table 1.
3. Nuclear relaxation mechanisms
and ^ s 12
bs
T1y1 , Ž 4 p ND Db 2 r R . =
Iy3 r4 Ž d . Ky3r4 Ž D . y Iy3r4 Ž D . Ky3r4 Ž d . Iy3 r4 Ž d . K 1r4 Ž D . q I1r4 Ž D . Ky3r4 Ž d .
,
Ž 1.
D s 12
b
ž / b
R
C
.
Ž 3.
1r4
ž /
.
D
Ž 4.
The average distance between two neighboring impurities is given by R s Ž 3r4p ND .
1r3
.
Ž 5.
The diffusion coefficient D of the estimated by D,
a2 50T2
,
13
C spins is
Ž 6.
where T2 is the 13 C spin–spin relaxation time and a is the average nearest-neighbor 13 C distance. The quantity C Žwith Cr2 r 6 the probability per unit time of a nuclear spin reorientation at a distance r from a paramagnetic electron due to electron relaxation. is given by C s 25
with
ž /
In Ž z . and Kn Ž z . are modified Bessel functions, ND is the concentration of paramagnetic impurities per unit volume, b the diffusion barrier radius ŽTable 2. and b can be interpreted as the distance from the paramagnetic impurity up to where direct nuclear relaxation is more effective than spin diffusion and is given by
3.1. Nuclear relaxation Õia paramagnetic impurities Natural diamonds contain low concentrations of paramagnetic impurities which can play an important role in the spin–lattice relaxation of the 13 C nuclei. A short overview will now be given of the singleparamagnetic-center relaxation theories of Bloembergen w6x, Khutshishvili w7x and Lowe and Tse w8x. Consider the spin–lattice relaxation of 13 C nuclei Ž I s 1r2. in a diamond lattice with a low concentration of paramagnetic impurities Ž S s 1r2. for the case where the Zeeman energy of the nuclear spin Ž " v 0 . is much larger than the interaction energy of the nuclear and impurity spins. The 13 C spin–lattice relaxation time T1 can be estimated by w8x
2
b
" 2ge2 S Ž S q 1 .
tc H02
,
Ž 7.
2
Ž 2.
where ge is the electronic gyromagnetic ratio, H0 is the constant external magnetic field strength and tc
C.J. Terblanche et al.r Chemical Physics Letters 310 (1999) 97–102
100
Table 2 Calculated and measured parameters for natural diamond at 4.7 T and 295 K Parameter
Value
a Žcm. T2 Žs. D Žcm2 sy1 . C Žcm6 sy1 . b Žcm. b Žcm.
Remarks y8
5.8=10 Ž1"0.2.=10y2 6.7=10y15 4.64=10y50tcy1 1.62=10y9tcy1r4 1.33=10y7
Eq. Ž5. Measured ŽNMR. Eq. Ž6. Eq. Ž7. Eq. Ž4. Ž "ge2 H0 rgn 2 kT .1r4 a
is valid. Furthermore, m1 , m 2 and m 3 represent the carrier anisotropic effective masses. Cyclotron resonance measurements w10x on the valence band of type-IIb diamonds showed degeneracy giving rise to effective hole masses of 2.1m e , 0.7m e and 1.06 m e , respectively, with m e the free electron mass. For a partially compensated semiconducting ptype diamond, with a sufficiently small acceptor concentration Ž NA . such that there is no degeneracy, the hole concentration p at a temperature T can be estimated by w11x p Ž p q ND .
is the electronic correlation time. The correlation time tc for "ge H0r2 kT ( 1 is given by 1
1 s
tc
1 q
T2e
T1e
.
Ž 8.
For most diamonds with low paramagnetic impurity concentrations in high fields T1e 4 T2e , and thus
tc , T2e .
Ž 9.
Therefore, it follows from Table 1 that the relaxation of 13 C nuclei is caused by the time-varying local magnetic field brought about by the spin–spin relaxation of the impurity centers. The exact solutions to Eq. Ž1. can be found by using Fortran routines for the modified Bessel functions. 3.2. Nuclear relaxation Õia mobile carriers Nuclear relaxation due to mobile carriers is caused by the scalar contact interaction, AI.S, of the hyperfine coupling between the nuclear spins and the spins of the mobile holes. The expression for the nuclear relaxation rate due to mobile carriers is given by w9x 1 T1mobile
s 329 plh 2ge2gn2 w 2 p m1 m 2 m 3 k B T x
1r2
,
Ž 10 .
where p is the total number of conduction holes per unit volume and l is the number of equivalent maxima in the valence band. ge and gn represent the gyromagnetic ratios of electron and nuclear spins, respectively, k B is the Boltzmann constant and T is the absolute temperature. If it is assumed that < f E Ž0.< 2 Žwith < f E Ž0.< 2 the normalized carrier density at the nucleus for a carrier of energy E . has a value h , independent of the energy, then the above expression
2 s
NA y ND y p
ga
ž
2 p mh k BT h
2
3r2
/
exp y
EA k BT
,
Ž 11 . where g a is the ground state degeneracy factor of the acceptor, m h is the density-of-states effective mass of the holes, EA is the acceptor ionization energy and ND is the donor concentration. For diamond g a should have a value of 4 up to about room temperature and 6 at higher temperatures.
4. Discussion and conclusions From the EPR spectrum ŽFig. 1. it is evident that two mechanisms can contribute to the relaxation of the 13 C nuclei, namely the fixed paramagnetic donors Žcentral resonance line. and the mobile holes Žslope of the baseline.. The spin–lattice relaxation rate of the 13 C nuclei can thus be written as w2x 1
1 s
T1
T1fixed
1 q
T1mobile
.
Ž 12 .
The fixed paramagnetic defects are not bound acceptor holes as reported by Hoch and Reynhardt w2x since such defects have only been observed in natural type-IIb diamonds at very low temperatures and under external uniaxial stress w12x. Bell and Leivo w13x suggested that the central resonance line in some semiconducting diamonds is due to unpaired spins at vacancy related defects. However, no resonance lines or hyperfine lines of any known nitrogen or vacancy related defects were observed in our samples. If it is assumed that all compensating donor
C.J. Terblanche et al.r Chemical Physics Letters 310 (1999) 97–102 Table 3 Calculated T1fixed and measured ŽT1Žexp..
13
101
C spin–lattice relaxation times, distances Ž R, b . and ratios Ž d , D . for samples b4 and b5
Sample
Impurity Žtype.
ND Žcmy3 .
R = 10 6 Žcm.
b = 10 9 Žcm.
d = 10 3
D = 10 6
T1fixed Žhours.
T1Žexp. Žhours.
b4 b5
S s 1r2 S s 1r2
1.5 = 10 16 5.8 = 10 15
2.5 3.5
26 28
19 22
54 32
3.4 = 10 3 6.5 = 10 3
3.7 " 0.7 5.1 " 1.0
centers are permanently ionized, then these centers are not expected to be S s 1r2 centers Žlike P1 centers which are common in most diamonds. since after donating their paramagnetic electrons they would not be paramagnetic and therefore not be detectable on the EPR spectra. Davies w14x suggested that the A-aggregate of nitrogen could act as donors in type-IIb diamonds. However, this would imply the formation of the nitrogen paramagnetic center W24 ŽN–Nq. w15x, which was also not observed. Consequently, it seems that the paramagnetic centers in our samples are transition metal like donor defects which result in only one resonance line in the EPR spectra. Transition metal defects occur in a wide variety of diamonds and in the present study it will be assumed that these donors have an effective spin of S s 1r2 after compensation and a concentration designated by ND . The parameters required for calculating the 13 C spin–lattice relaxation times due to fixed impurities are listed in Tables 1 and 2. Table 3 gives a comparison between the calculated T1fixed ŽEq. Ž1.. and experimental T1 values. The calculated T1fixed can only be considered as an order of magnitude approximation w6x since various factors, like the assumptions made in the relaxation theory, inaccuracies in the measured quantities Žimpurity concentration and electronic relaxation times. and inhomogeneous impurity center distributions, can contribute to its inaccuracy. Due to the low concentration of mobile holes in p-type diamond, Hoch and Reynhardt w2x considered their contribution to 13 C spin–lattice relaxation insignificant over the temperature range 295–375 K. Wedepohl w16x observed for a suite of five natural type-IIb diamonds that, for an activation energy of about 0.34 eV, a linear relationship between the logarithm of the mobile hole concentration Ž p . and the inverse of the absolute temperature in the range 200–400 K. Therefore, for the mobile carrier mechanism to be dominant in the relaxation of the 13 C there must be a linear relationship be-
tween log T1y1 and Ty1 q logT 1r2 Žfrom Eq. Ž10... This relationship holds for the results of Hoch and Reynhardt w2x, as depicted in Fig. 3, and contradicts their conclusion that the fixed center contribution is dominant. If the fixed impurity relaxation mechanism is to prevail over the mobile carrier mechanism, the nuclear spin–lattice relaxation time must be temperature-independent w17x. In order to apply the mobile carrier relaxation model ŽEq. Ž10.., estimates of p, l and h must be obtained. The concentration of the free holes can be estimated from the Hall coefficient Ž RX ., using p s rrRX e, where r is the ratio of the Hall mobility to the conductivity and e is the electronic charge. The quantity r depends on the scattering processes and in the range 200–400 K it is due to ionized impurities
Fig. 3. Plot of logT1y1 versus Ty1 qlogT 1r 2 for the 13 C spin– lattice relaxation data of Hoch and Reynhardt w2x. The temperature dependence suggests that the mobile carrier mechanism is dominant.
102
C.J. Terblanche et al.r Chemical Physics Letters 310 (1999) 97–102
w16x, giving rise to r s 315pr512 w18x. The value of RX is sample Žsmall differences in NA , ND , etc.. and temperature dependent and for natural IIb diamond is estimated to be 0.1 m3 Cy1 at 290 K w19x. Therefore, the carrier concentration at 295 K is about 1.2 = 10 14 cmy3 . Shulman and Wyluda w20x studied the spin– lattice relaxation of 29 Si as a function of carrier concentration in n- and p-type silicon at room temperature. It was observed that T1 is inversely proportional to the carrier concentration for high conductivity samples, but approaches an asymptotic value in purer samples. Concentrations of around 1.2 = 10 14 cmy3 holes in p-type silicon result in nearly constant 29 Si T1 values. It seems that the latter situation also prevails in p-type diamond, since the experimental 13 C T1 values do not differ much ŽTable 1.. Assuming l , 6, as for semiconducting silicon w10,20x, and using the average experimental 13 C T1 value of 4.3 h, it follows from Eq. Ž10. that h , 320 for a hole concentration of 1.2 = 10 14 cmy3 . Variations in boron concentration Ž NA . and the presence and concentration Ž ND . of fixed paramagnetic impurities, which influence the hole concentration, lead to small variations in the 13 C spin–lattice relaxation times among natural type-IIb diamonds. The main conclusion drawn from the results of this investigation is that the relaxation of the 13 C nuclei in natural type-IIb diamond is mainly due to mobile hole carriers. Since the concentration of carriers is usually low in these diamonds, the 13 C spin– lattice relaxation rate seems to be approximately the same for all samples.
Acknowledgements The Foundation for Research Development and the Research and Bursaries Committee of the Uni-
versity of South Africa are thanked for financial assistance. The Mineral Processing Division of De Beers Industrial Diamonds is thanked for supplying the samples. The authors also wish to thank Dr. S.A. Rakityansky for writing the Fortran program.
References w1x E.C. Reynhardt, C.J. Terblanche, Chem. Phys. Lett. 269 Ž1997. 464. w2x M.J.R. Hoch, E.C. Reynhardt, Phys. Rev. 37 Ž1988. 9222. w3x E. Shabanova, K. Schaumburg, J.P.F. Sellschop, J. Mag. Res. 130 Ž1998. 8. w4x C.J. Terblanche, E.C. Reynhardt, J.A. van Wyk, Solid State Nucl. Magn. Reson., submitted for publication. w5x E.C. Reynhardt, G.L. High, J. Chem. Phys. 109 Ž1998. 4090. w6x N. Bloembergen, Physica 15 Ž1949. 386. w7x G.R. Khutsishvili, Sov. Phys. Uspekhi 8 Ž1966. 743. w8x I.J. Lowe, D. Tse, Phys. Rev. 166 Ž1968. 279. w9x A. Abragam, The Principles of Nuclear Magnetism, Oxford University Press, London, 1961. w10x C.J. Rauch, in: A.C. Stickland ŽEd.., Proceedings of International Conference on the Physics of Semiconductors, Institute of Physics and the Physical Society of London, Exeter, UK, 1962, p. 276. w11x J.S. Blakemore, Semiconductor Statistics, Pergamon Press, Oxford, 1962. w12x C.A.J. Ammerlaan, Defects and Radiation Effects in Semicondutors, Institute of Physics, Bristol, UK, 1981, p. 81. w13x M.D. Bell, W.J. Leivo, J. Appl. Phys. 38Ž1. Ž1967. 337. w14x G. Davies, Chem. Phys. Carbon 13 Ž1977. 1. w15x J.A. van Wyk, J.H.N. Loubser, Diamond Conference, Cambridge, UK, 1979, p. 11.1. w16x P.T. Wedepohl, Proceedings of the Physical Society ŽLondon. B 70 Ž1957. 177. w17x N. Bloembergen, Physica 20 Ž1954. 1130. w18x S.M. Sze, Physics of Semiconductor Devices, Wiley, New York, 1981. w19x J.E. Field, The Properties of Natural and Synthetic Diamond, Academic Press, London, 1992, p. 687. w20x R.J. Shulman, B.J. Wyluda, Phys. Rev. 103 Ž1956. 1127.
Chemical Physics Letters 310 Ž1999. 97–102 www.elsevier.nlrlocatercplett
13
C spin–lattice relaxation in natural type-IIb semiconducting diamond Cornelis J. Terblanche a , Eduard C. Reynhardt a
a,)
, Jan A. van Wyk
b
Department of Physics, UniÕersity of South Africa, P.O. Box 392, Pretoria 0003, South Africa b Department of Physics, UniÕersity of the Witwatersrand, Johannesburg, South Africa Received 27 May 1999; in final form 21 June 1999
Abstract 13
C spin–lattice relaxation times in the laboratory frame have been measured at 295 K in a field of 4.7 T for two natural type-IIb diamonds using nuclear magnetic resonance spectroscopy ŽNMR.. 13 C spin–lattice relaxation is mainly due to mobile hole carriers. The contribution of fixed paramagnetic impurities is at least an order of magnitude smaller. At 295 K, the carrier concentration is estimated to be 1.2 = 10 14 cmy3, resulting in an h , the electron density at the nucleus, of about 320 for natural type-IIb diamonds. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction The very long 13 C spin–lattice relaxation times in natural diamond Žseveral hours. w1x are mainly due to the low concentration of paramagnetic impurity centers, the low natural abundance Ž1.1%. of 13 C nuclei and the rigidity of the diamond lattice. From several studies performed on type-I and type-II diamond w1–3x, it seems that the main relaxation mechanism is the dipolar interaction between paramagnetic impurities and 13 C nuclei. However, in most of these studies, very little was known about the types and concentrations of paramagnetic centers present in the samples. Recently Terblanche et al. w4x investigated the 13 C spin–lattice relaxation behavior in natural type-Ia, -IIa and -Ib diamonds. EPR techniques were
) Corresponding author. Fax: q27-12-4293434; e-mail: [email protected]
used to characterize the paramagnetic impurities Žtype, concentration, electronic relaxation times, etc.. in each diamond. The knowledge that the samples contained either P1 and P2 or P1 and N3 or P1 and N2 centers resulted in definite conclusions regarding the relative effectiveness of paramagnetic impurity types in the relaxation of 13 C nuclei. Order of magnitude estimates of the 13 C spin–lattice times were obtained using existing paramagnetic relaxation models. Hoch and Reynhardt w2x investigated the 13 C spin–lattice relaxation over the temperature range 295–375 K of a 6.2 ct type-IIb diamond Žsample h. with a boron acceptor center concentration of 3 = 10 16 cmy3 , a detectable paramagnetic impurity concentration of below 10 15 cmy3 and a mobile hole concentration of about 10 14 cmy3 . The 13 C spin– lattice relaxation time ŽT1 s 4.1 h at 295 K. was ascribed to the interaction between the fixed paramagnetic hole centers and the nuclei.
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 7 5 8 - 7
98
C.J. Terblanche et al.r Chemical Physics Letters 310 (1999) 97–102
In this Letter, we discuss 13 C spin–lattice relaxation times observed for semiconducting type-IIb diamonds and show that the 13 C nuclei are actually relaxed by the mobile hole carriers. Continuous wave EPR was used to determine the concentrations of the fixed paramagnetic impurities in the diamonds, while electronic spin–spin and spin–lattice relaxation times were determined using pulse EPR. The mobile hole and fixed paramagnetic impurity contributions to the 13 C spin–lattice relaxation times are calculated and compared with the measured 13 C spin–lattice relaxation times.
2. Experimental and results Two natural type-IIb diamonds, sample b4 Ž3.12 ct. and sample b5 Ž2.06 ct., were used in this investigation. The Zeeman 13 C spin–lattice relaxation times were measured at 295 K on a Bruker CXP 200 pulsed NMR spectrometer operating at a Larmor frequency of 50 MHz for 13 C nuclei. Within experimental error, all relaxation data revealed only one relaxation time. Continuous wave Žcw. spectra were recorded at 295 K on an X-band Varian E-line Century Series EPR spectrometer. The impurity concentrations were determined using double integration techniques, a ruby reference sample and a type-Ib synthetic dia-
Fig. 1. cw EPR spectrum of sample b4 at 295 K, indicating the three ruby reference lines and the unknown paramagnetic impurity line which overlaps with the central ruby line.
Fig. 2. cw EPR spectrum of the ruby reference sample at 295 K.
mond with a known concentration of paramagnetic defects ŽP1 centers, S s 1r2.. Corrections were made for the change in the Q-factor of the cavity due to different sample sizes, etc. The cw EPR spectrum of sample b4 is shown in Fig. 1. The ruby reference sample gave rise to three resonance lines, a weak central line and two strong lines on each side of the central line, as depicted in Fig. 2. By comparing the central lines of Figs. 1 and 2, the presence of an unknown paramagnetic center which is not present in all type-IIb diamonds is evident. The EPR spectrum of this defect also has no hyperfine lines, making its identification extremely difficult. This defect could be a transition metal-type center which serves as a donor to compensate the holes. Assuming S s 1r2, the concentrations of these defects can be estimated Žsee Table 1.. Another interesting feature in the cw EPR spectrum of sample b4 ŽFig. 1. is the slope of the baseline. This phenomenon is common in type-IIb diamonds, and it seems that the gradient of the baseline gives an indication of the mobile hole concentration. Sample b5 revealed the same characteristics in its cw EPR spectra. The line structure of the central line corresponded with that observed for b4, suggesting the same type of paramagnetic center was present. The slope of the baseline was also approximately the same as that of sample b4, implying comparable mobile hole concentrations.
C.J. Terblanche et al.r Chemical Physics Letters 310 (1999) 97–102
99
Table 1 Measured 13 C spin–lattice relaxation times, paramagnetic impurity center spin–lattice ŽT1e . and spin–spin ŽT2e . relaxation times for samples b4 and b5. The corresponding values for sample h ŽRef. w2x. are also given Sample
ND Žcmy3 .
T1e Ž=10y3 s.
T2e Ž=10y6 s.
T1 Žexperimental. Žhours.
b4 b5 h
1.5 = 10 16 5.8 = 10 15 ; 10 15
3.6 " 0.2 10 " 5 0.1
16 " 1 11 " 1 –
3.7 " 0.7 5.1 " 1 4.1
An X-band Bruker ESP 380E pulsedrcw spectrometer was used to determine the electron spin–spin and spin–lattice relaxation times ŽT2e and T1e , respectively. of the paramagnetic impurity centers at 295 K. The apparatus and pulse sequences employed have been described elsewhere w5x. The results are listed in Table 1.
3. Nuclear relaxation mechanisms
and ^ s 12
bs
T1y1 , Ž 4 p ND Db 2 r R . =
Iy3 r4 Ž d . Ky3r4 Ž D . y Iy3r4 Ž D . Ky3r4 Ž d . Iy3 r4 Ž d . K 1r4 Ž D . q I1r4 Ž D . Ky3r4 Ž d .
,
Ž 1.
D s 12
b
ž / b
R
C
.
Ž 3.
1r4
ž /
.
D
Ž 4.
The average distance between two neighboring impurities is given by R s Ž 3r4p ND .
1r3
.
Ž 5.
The diffusion coefficient D of the estimated by D,
a2 50T2
,
13
C spins is
Ž 6.
where T2 is the 13 C spin–spin relaxation time and a is the average nearest-neighbor 13 C distance. The quantity C Žwith Cr2 r 6 the probability per unit time of a nuclear spin reorientation at a distance r from a paramagnetic electron due to electron relaxation. is given by C s 25
with
ž /
In Ž z . and Kn Ž z . are modified Bessel functions, ND is the concentration of paramagnetic impurities per unit volume, b the diffusion barrier radius ŽTable 2. and b can be interpreted as the distance from the paramagnetic impurity up to where direct nuclear relaxation is more effective than spin diffusion and is given by
3.1. Nuclear relaxation Õia paramagnetic impurities Natural diamonds contain low concentrations of paramagnetic impurities which can play an important role in the spin–lattice relaxation of the 13 C nuclei. A short overview will now be given of the singleparamagnetic-center relaxation theories of Bloembergen w6x, Khutshishvili w7x and Lowe and Tse w8x. Consider the spin–lattice relaxation of 13 C nuclei Ž I s 1r2. in a diamond lattice with a low concentration of paramagnetic impurities Ž S s 1r2. for the case where the Zeeman energy of the nuclear spin Ž " v 0 . is much larger than the interaction energy of the nuclear and impurity spins. The 13 C spin–lattice relaxation time T1 can be estimated by w8x
2
b
" 2ge2 S Ž S q 1 .
tc H02
,
Ž 7.
2
Ž 2.
where ge is the electronic gyromagnetic ratio, H0 is the constant external magnetic field strength and tc
C.J. Terblanche et al.r Chemical Physics Letters 310 (1999) 97–102
100
Table 2 Calculated and measured parameters for natural diamond at 4.7 T and 295 K Parameter
Value
a Žcm. T2 Žs. D Žcm2 sy1 . C Žcm6 sy1 . b Žcm. b Žcm.
Remarks y8
5.8=10 Ž1"0.2.=10y2 6.7=10y15 4.64=10y50tcy1 1.62=10y9tcy1r4 1.33=10y7
Eq. Ž5. Measured ŽNMR. Eq. Ž6. Eq. Ž7. Eq. Ž4. Ž "ge2 H0 rgn 2 kT .1r4 a
is valid. Furthermore, m1 , m 2 and m 3 represent the carrier anisotropic effective masses. Cyclotron resonance measurements w10x on the valence band of type-IIb diamonds showed degeneracy giving rise to effective hole masses of 2.1m e , 0.7m e and 1.06 m e , respectively, with m e the free electron mass. For a partially compensated semiconducting ptype diamond, with a sufficiently small acceptor concentration Ž NA . such that there is no degeneracy, the hole concentration p at a temperature T can be estimated by w11x p Ž p q ND .
is the electronic correlation time. The correlation time tc for "ge H0r2 kT ( 1 is given by 1
1 s
tc
1 q
T2e
T1e
.
Ž 8.
For most diamonds with low paramagnetic impurity concentrations in high fields T1e 4 T2e , and thus
tc , T2e .
Ž 9.
Therefore, it follows from Table 1 that the relaxation of 13 C nuclei is caused by the time-varying local magnetic field brought about by the spin–spin relaxation of the impurity centers. The exact solutions to Eq. Ž1. can be found by using Fortran routines for the modified Bessel functions. 3.2. Nuclear relaxation Õia mobile carriers Nuclear relaxation due to mobile carriers is caused by the scalar contact interaction, AI.S, of the hyperfine coupling between the nuclear spins and the spins of the mobile holes. The expression for the nuclear relaxation rate due to mobile carriers is given by w9x 1 T1mobile
s 329 plh 2ge2gn2 w 2 p m1 m 2 m 3 k B T x
1r2
,
Ž 10 .
where p is the total number of conduction holes per unit volume and l is the number of equivalent maxima in the valence band. ge and gn represent the gyromagnetic ratios of electron and nuclear spins, respectively, k B is the Boltzmann constant and T is the absolute temperature. If it is assumed that < f E Ž0.< 2 Žwith < f E Ž0.< 2 the normalized carrier density at the nucleus for a carrier of energy E . has a value h , independent of the energy, then the above expression
2 s
NA y ND y p
ga
ž
2 p mh k BT h
2
3r2
/
exp y
EA k BT
,
Ž 11 . where g a is the ground state degeneracy factor of the acceptor, m h is the density-of-states effective mass of the holes, EA is the acceptor ionization energy and ND is the donor concentration. For diamond g a should have a value of 4 up to about room temperature and 6 at higher temperatures.
4. Discussion and conclusions From the EPR spectrum ŽFig. 1. it is evident that two mechanisms can contribute to the relaxation of the 13 C nuclei, namely the fixed paramagnetic donors Žcentral resonance line. and the mobile holes Žslope of the baseline.. The spin–lattice relaxation rate of the 13 C nuclei can thus be written as w2x 1
1 s
T1
T1fixed
1 q
T1mobile
.
Ž 12 .
The fixed paramagnetic defects are not bound acceptor holes as reported by Hoch and Reynhardt w2x since such defects have only been observed in natural type-IIb diamonds at very low temperatures and under external uniaxial stress w12x. Bell and Leivo w13x suggested that the central resonance line in some semiconducting diamonds is due to unpaired spins at vacancy related defects. However, no resonance lines or hyperfine lines of any known nitrogen or vacancy related defects were observed in our samples. If it is assumed that all compensating donor
C.J. Terblanche et al.r Chemical Physics Letters 310 (1999) 97–102 Table 3 Calculated T1fixed and measured ŽT1Žexp..
13
101
C spin–lattice relaxation times, distances Ž R, b . and ratios Ž d , D . for samples b4 and b5
Sample
Impurity Žtype.
ND Žcmy3 .
R = 10 6 Žcm.
b = 10 9 Žcm.
d = 10 3
D = 10 6
T1fixed Žhours.
T1Žexp. Žhours.
b4 b5
S s 1r2 S s 1r2
1.5 = 10 16 5.8 = 10 15
2.5 3.5
26 28
19 22
54 32
3.4 = 10 3 6.5 = 10 3
3.7 " 0.7 5.1 " 1.0
centers are permanently ionized, then these centers are not expected to be S s 1r2 centers Žlike P1 centers which are common in most diamonds. since after donating their paramagnetic electrons they would not be paramagnetic and therefore not be detectable on the EPR spectra. Davies w14x suggested that the A-aggregate of nitrogen could act as donors in type-IIb diamonds. However, this would imply the formation of the nitrogen paramagnetic center W24 ŽN–Nq. w15x, which was also not observed. Consequently, it seems that the paramagnetic centers in our samples are transition metal like donor defects which result in only one resonance line in the EPR spectra. Transition metal defects occur in a wide variety of diamonds and in the present study it will be assumed that these donors have an effective spin of S s 1r2 after compensation and a concentration designated by ND . The parameters required for calculating the 13 C spin–lattice relaxation times due to fixed impurities are listed in Tables 1 and 2. Table 3 gives a comparison between the calculated T1fixed ŽEq. Ž1.. and experimental T1 values. The calculated T1fixed can only be considered as an order of magnitude approximation w6x since various factors, like the assumptions made in the relaxation theory, inaccuracies in the measured quantities Žimpurity concentration and electronic relaxation times. and inhomogeneous impurity center distributions, can contribute to its inaccuracy. Due to the low concentration of mobile holes in p-type diamond, Hoch and Reynhardt w2x considered their contribution to 13 C spin–lattice relaxation insignificant over the temperature range 295–375 K. Wedepohl w16x observed for a suite of five natural type-IIb diamonds that, for an activation energy of about 0.34 eV, a linear relationship between the logarithm of the mobile hole concentration Ž p . and the inverse of the absolute temperature in the range 200–400 K. Therefore, for the mobile carrier mechanism to be dominant in the relaxation of the 13 C there must be a linear relationship be-
tween log T1y1 and Ty1 q logT 1r2 Žfrom Eq. Ž10... This relationship holds for the results of Hoch and Reynhardt w2x, as depicted in Fig. 3, and contradicts their conclusion that the fixed center contribution is dominant. If the fixed impurity relaxation mechanism is to prevail over the mobile carrier mechanism, the nuclear spin–lattice relaxation time must be temperature-independent w17x. In order to apply the mobile carrier relaxation model ŽEq. Ž10.., estimates of p, l and h must be obtained. The concentration of the free holes can be estimated from the Hall coefficient Ž RX ., using p s rrRX e, where r is the ratio of the Hall mobility to the conductivity and e is the electronic charge. The quantity r depends on the scattering processes and in the range 200–400 K it is due to ionized impurities
Fig. 3. Plot of logT1y1 versus Ty1 qlogT 1r 2 for the 13 C spin– lattice relaxation data of Hoch and Reynhardt w2x. The temperature dependence suggests that the mobile carrier mechanism is dominant.
102
C.J. Terblanche et al.r Chemical Physics Letters 310 (1999) 97–102
w16x, giving rise to r s 315pr512 w18x. The value of RX is sample Žsmall differences in NA , ND , etc.. and temperature dependent and for natural IIb diamond is estimated to be 0.1 m3 Cy1 at 290 K w19x. Therefore, the carrier concentration at 295 K is about 1.2 = 10 14 cmy3 . Shulman and Wyluda w20x studied the spin– lattice relaxation of 29 Si as a function of carrier concentration in n- and p-type silicon at room temperature. It was observed that T1 is inversely proportional to the carrier concentration for high conductivity samples, but approaches an asymptotic value in purer samples. Concentrations of around 1.2 = 10 14 cmy3 holes in p-type silicon result in nearly constant 29 Si T1 values. It seems that the latter situation also prevails in p-type diamond, since the experimental 13 C T1 values do not differ much ŽTable 1.. Assuming l , 6, as for semiconducting silicon w10,20x, and using the average experimental 13 C T1 value of 4.3 h, it follows from Eq. Ž10. that h , 320 for a hole concentration of 1.2 = 10 14 cmy3 . Variations in boron concentration Ž NA . and the presence and concentration Ž ND . of fixed paramagnetic impurities, which influence the hole concentration, lead to small variations in the 13 C spin–lattice relaxation times among natural type-IIb diamonds. The main conclusion drawn from the results of this investigation is that the relaxation of the 13 C nuclei in natural type-IIb diamond is mainly due to mobile hole carriers. Since the concentration of carriers is usually low in these diamonds, the 13 C spin– lattice relaxation rate seems to be approximately the same for all samples.
Acknowledgements The Foundation for Research Development and the Research and Bursaries Committee of the Uni-
versity of South Africa are thanked for financial assistance. The Mineral Processing Division of De Beers Industrial Diamonds is thanked for supplying the samples. The authors also wish to thank Dr. S.A. Rakityansky for writing the Fortran program.
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