11048
J . Phys. Chem. 1992, 96, 11048-11054
Atomic Distribution in Zn, Cdl, GeP, Semiconductor Alloys. 31Pand li3Cd Magic-Angk Spinning, 31P Spin-Echo, and 31P-113CdHeteronuclear Double Quantum Filtering MAS-NMR Studies Deanna Franke, Kesha Banks, and Hellmut Eckert* Department of Chemistry, University of California, Santa Barbara, California 93106 (Received: August 28, 1992; In Final Form: September 30, 1992)
Cation substitution in the Zn,Cd,-,GeP, (0 Ix I1) system results in homogeneous solid solutions which crystallize in the chalcopyrite structure. The distribution of local (nearest-neighbor)environments in these materials has been examined on the basis of complementary solid-state NMR experiments, including 31Pand lI3Cdmagic-angle spinning (MAS), spin-echo, and 31P-113Cd heteronuclear double quantum filtering (HDQF) MAS-NMR. 'I3Cd and 31P MAS-NMR spectra show poorly resolved resonances whose chemical shifts change monotonically with increasing x. 31Pspin-echo decay data reveal that the 31P_31P homonuclear dipole coupling is stronger than calculated based solely on inhomogeneous dipolar broadening. 31P-113Cd HDQF MAS-NMR experiments offer site discrimination attributable to specific local nearest-neighbor configurations and thus provide useful peak deconvolution constraints. The results obtained using these techniques suggest a nonstatistical population distribution of the nearest-neighbor environments, in which mixed PGe2CdZnsites are disfavored.
Introduction
The distribution of local environments in binary and ternary semiconductor alloys has become an important issue in the attempt to understand their physicochemical properties. Experimental and theoretical evidence for bimodal bond length distributions, clustering, and long-range-order phenomena has been presented for several 111-V systems, suggesting that such configurations are more stable than corresponding random arrangements.'V2 To test such hypotheses, locally selective spectroscopic techniques are needed, and it has been previously demonstrated that nuclear magnetic resonance is a well-suited technique for this purpose. For instance, in CdTe-based semiconductor alloys, the IzsTe chemical shifts, measured by the technique of magic-angle spinning (MAS)-NMR differentiate sensitively between the various possible nearest-neighbor configurations present. In the system Zn,Cd,-,Te, these site populations are close to those predicted by statistical pr~bability.~ Most recently, a similar result has been obtained for alloys with the compositions Gao.51no.sPand Ga1,414ts,&4.' In contrast, the population distribution was reported to be nonstatistical in some Hg,Cd,-,Te alloys.5 The present study is devoted to substitution effects in a closely related family of compounds, based on the 11-IV-V, semiconductors, which crystallize in the tetragonal chalcopyrite structure.6~' Specifically, we discuss here the effect of cation substitution in the semiconductor alloy Zn,Cd,-,GeP, on the basis of detailed composition-dependent and Il3Cdsingle and double-resonance NMR experiments. NMR spectroscopy offers a variety of experimental approaches for studying the issue of the distribution of local environments. In the present system, the detailed atomic distribution will affect 31Pand I l T d chemical shifts and the strength of homonuclear 31P-31Pdipole-dipole couplings and of the heteronuclear 31P-1!3Cdinteractions. The frequently observed lack of complete site resolution in semiconductor alloy^'^^^^ necessitates the use of spectral editing techniques for arriving at peak assignments. In the present contribution, we demonstrate that this objective can be achieved by heteronuclear double quantum filtering. This technique originally developed by L. MUller,lo and subsequently developed further" for detecting I3C, 15N,and other rare-spin nuclei via IH NMR in the liquid state, exploits J coupling for excitation of heteronuclear multiple quantum coherences between abundant and rare spins. Recent work demonstrates that this method can also be used in crystalline inorganic solids with 31Pas the abundant high-y nuclei.12 Here, we combine indirect detection schemes known in liquid-state NMR with solid-state MAS in order to provide spectral editing that utilizes scalar coupling between the high-y nuclei 3 1 Pand the lower-y nuclei ll3Cd present in Zn,Cdl-,GeP2 alloys. 0022-3654/92/2096-11048$03.00/0
Experimental Section Sample Preparation and Characterization. Crystalline samples
in the Znl-xCd,GeP2 system were prepared from stoichiometric amounts of the elements (Aldrich; Zn, 99.99% Cd, 99.95%; Ge, 99.99%; red P, 99.999%) in evacuated (10-3-Torr) silica glass ampules. The temperature was increased slowly to 450 OC, held for 3-5 h, and then increased to 850 OC. After 24 h or more at this temperature, the samples were cooled slowly (30-60 OC/h) to room temperature. The resting period at 450 OC is crucial in order to avoid too rapid sublimation of phosphorus that causes explosions and fires. Attempts to make glass by quenching ampules into ice water failed. Initial products were contaminated with Zn,CD2-J'207 and ~~-Zn3(P04)2 impurities (10% or less of the phosphorus content as determined by 31PNMR). Preparations that showed high degrem of meta1 phosphate impurities and broad X-ray powder diffraction lines were repeated until sharp clean X-ray patterns were produced, confirming that phase-pure homogeneous materials were formed. Formation of the phosphate impurity could be minimized by purifying the Cd and Zn in a 9% Ar/5% H2 gas stream at 175 OC to reduce surface metal oxide. In addition, graphitizing the silica ampules eliminated any reaction between the sample and ampule materials and provided a nonstick surface for ease of sample removal. Although the final materials are not air-sensitive, sample preparation steps and subsequent manipulations were generally carried out in a nitrogen glovebox. Table I lists the best fit lattice parameters determined from X-ray powder diffraction data of the ZnxCdl,GeP2 samples studied by NMR and the LATCON fitting procedure. These were fit to the tetragonal chalcopyrite unit cell, space group Z42d,637 and all significant lines were included in the calculations. The unit cell volume decreases monotonically with x, confirming that cation substitution in this system results in homogeneous solid solutions. Solid-state NMR Studies. 31Pand l13Cd MAS-NMR studies were carried out at frequencies near 121.65 and 66.71 MHz, respectively, using a General Electric GN-300 spectrometer. The measurementsemployed 5-mmor 7-mm MAS-NMR probes from Doty Scientific. Typical conditions were 90° pulse length 8 MS and spinning speed 6-9 kHz. Isotropic chemical shifts are reported in Table I relative to 85% H3P04and liquid (CH,),Cd. The phosphorus-31 spin-echo and the 31P-113CdHDQF experiments employed a 7-mm doubly broad band tuned MASNMR probe. For each sample, the resonance frequencies were carefully adjusted to minimize resonance offset effects. The spin-echo experiments were conducted on nonspinning samples, under quantitative conditions necessitating delays of 3 0 6 0 min. The spin-echo height at zero evolution time was obtained from a polynomial fit to the experimental data. 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No.26, 1992 11049
Zn,Cdl-,GeP2 Semiconductor Alloys
TABLE I: Compositions, Lattice Parameters, and Average Chemical Shifts for Single-Pulse MAS-NMR,” 31P-113cd HDQF,b FWHM; a d Calclrlrted md Merswed Avenge 31PMpolar Second Momentsd 3 Ip-31 p 5
a, b, A
0.0oO
5.744 5.730 5.724 5.643 5.603 5.563 5.526 5.505 5.473
0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.O00
c, A 10.771 10.755 10.755 10.775 10.758 10.744 10.717 10.694 10.720
b(”’Cd), PPm
a(’lP), PPm
”P FWHM, PPm
237 238 242 272 278 300 319 330
-32 -32 -33 -37 -42 -47 -54 -58 -60
3.4 4.8 4.9 25.3 27.9 26.5 20.8 11.0 1.6
31P[’13Cd] HDQF, PPm
HDQF FWHM, ppm
-3 3
4.8
-48 -54 -6 5 -70
24.6 22.6 15.8 11.1
M2(expY
M2(cak)d
15.4 15.3 14.8 15.0 15.3 16.5 18.8 19.2 21.1
11.2 11.1 11.4 11.7 12.7 12.5 13.2 13.1 13.7
M,(exp)l ~~(calc) 1.38
1.38 1.30 1.28 1.20 1.32 1.42 1.47 1.54
‘fl ppm vs liquid (CH’),Cd and 85% H3P04. bHeteronuclear double quantum filtering, fl ppm. ‘Full width at half-maximum, A1 ppm. d M 2 , in units of lo6 rad2/s2, *lo%. #Fraction of zinc.
A
Zn~Cdl.~GePn X
O.OO0 Ii3Cd -rn t2
Figure 1. Pulse sequence for 31P-113Cdheteronuclear double quantum filtering (HDQF) MAS-NMR.
0.125
Heteronuclear double quantum filtering (HDQF) MAS-NMR experiments were carried out using the pulse sequence shown in Figure 1, with 90° pulses of 9-1 1 ps for 31Pand 8-9 ps for l13Cd, respectively. Recycle delays ranged from 60 to 120 s. The delay time 1/25 was optimized experimentally, following an initial estimate of the heteronuclear scalar coupling constant from the 31P single quantum signal of CdGeP2. A value of 1.45 ms was chosen. Double quantum coherence was allowed to evolve for ca. 30 ps. The timing of the pulse sequence was such that the 180° refocusing pulse on the 31Pchannel and the start of the data acquisition coincided with multiples of the rotor period, spinning speeds 4-5 kHz.
0.250
0.375
0.500
Results, Data Analysis, end Interpretation “Fd MAS-NMR. Figure 2 shows representative Il3Cd MAS-NMR spectra of various crystalline Zn,Cdl-xGePz samples and shows a monotonic chemical shift trend downfield with increasing zinc content. The compositional dependence of the isotropic chemical shifts is included in Table I. In each sample, the cadmium atoms are located in a tetrahedral site surrounded by four phosphorus atoms. Although this nearest-neighbor environment is preserved upon substitution of Cd for Zn, the chemical shifts of the different compositions vary over a surprisingly wide range of 93 ppm. The change in chemical shift reflects next-nearest-neighborsubstitution effects. The second coordination shell from the cadmium atom comprises 8 germanium atoms, n cadmium atoms, and 4 - n zinc atoms, where 0 In I 4. The l13Cd resonances of these five sites are not resolvable from each other, however, resulting in the observation of simple broad peaks only. 3*PMASNMR. Figure 3 shows the results from 31PMASNMR on the ZnxCd,-,GeP2 samples. The spectra are broad, are mostly unresolved, and shift upfield with increasing x. The monotonic Il3Cd and 31Pchemical shift dependences on x confii, in agreement with X-ray diffraction data, that Zn,Cdl-xGeP2 samples arc homogeneous solid solutions. The spectra also contam minority components which have shorter relaxation delays than the major peak, causing unrepresentative relative peak areas at short delays €or the ,IP spectra as shown in Figure 4. Longer delays of approximately 15 min allowed quantification of the results. This difference in relaxation times suggests that the minor peaks arise from a separate phase. 31PMAS-NMR of pure Zn2PZO7 and Cd2PZO7, which were prepared and checked by X-ray powder diffraction, resulted in sharp peaks at -16.0, -19.2, and
0.625
0.750
0.875 400
350
300
250
200
150
PPM
Figure 2. Compositional depcndence of the Il3Cd single-pulse MASNMR spectra in crystalline alloys Zn,Cdl-,GeP2.
-21.3 ppm and at -2.5 and -4.7 ppm, respectively. It is, therefore, assumed that these additianal peaks are caused by phosphate impurities including mixed Zn,Cd2-,P2O7 phases. In addition, samples with x = 0.875 and 1.0 show a weak peak at 3.9 ppm, assignable to an impurity of CY-Z~~(PO,)~. We note that no impurities were detected by X-ray powder diffraction, presumably either due to microcrystalline character or low overall concentration of these oxidic phases. Spia-Echo NMR. Dipolar spectroscopy offers a powerful approach to disordered systems, because the spectroscopic information can often be calculated from first principles. As discussed in detail previously,Icl6 9O0,-tI-18O0, spin-echo experiments with incrementation of the pulse delay tl offer, in principle, selective information concerning homonuclear dipole-dipole couplings among 31Pspins. For multispin systems, a Gaussian decay of spin-echo intensity with time evolution 24 is expected,
11050 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992
n
Franke et al.
ZnxCdi-xGePn X
0.000
0.125
0.250
0.375
0.500
0.625
0.750
0.875
1.ooo 40
2G
0
-20
-40
-60
-80
-100
-120
PPM
Figure 3. Compositional dependence of the 31Pone-pulse MAS-NMR spectra in the crystalline alloys Zn,Cdl-,GeP2. Spinning side bands are marked by asterisks.
yielding a homodipolar second moment M2(31P-31P)from the semilogarithmic plot defined by
-
-150
PPM
.
.-E 0.6 0.0
01
P
Here I and 7 are the spin quantum number and the gyromagnetic ratio of the nucleus under consideration, h is Planck's constant, N is the number of nuclei for which M2 is calculated, and d,, are the distances between the nuclei under consideration and the surrounding nuclei generating dipolar fields. This analysis assumes that the chemical shift difference between dipolarly interacting nuclei quenches the flip-flop term in the homonuclear dipolar Hamiltonian, which is, in fact, a necessary condition for the complete refocusing of the chemical shift by the spin-echo sequence. In simple w,eq 2 provides an excellent opportunity for testing possible atomic distribution models against experimental data. This principle has been previously exploited for investigation of the distribution of fluorine and phosphorus dopants in silica glass,I8 amorphous hydrogenated silicon,I6J9and a variety of non-oxide glasses and crystalline systems.2w25 Figure 5 shows a rcpramtative 31Pspin-ccho NMR decay for the system under study. Table I compares experimental M2(31pJ)1P)values (eq 1) with calculated average values. The latter were obtained from eq 2 by assuming complete occupancy of all
-100
Zno.sCdo.sGeP2 31P Spin Echo
1.0 -
r
;' C
-.-8 The M 2 values characterize the average strength of the homonuclear dipolar interactions under consideration. Theoretical second-moment values can be calculated from the van Vleck equation:"
-50
-L
Figure 4. 31P single-pulse MAS-NMR spectra of Znoa2sCdo375GeP2as a function of relaxation delay.
0.4-
0.2
-
0
0.0
0.2
0.4 0.6 2t (ms)
0.8
1.0
Figure 5. ,IP spin-echo decay of Zn,,sC&.sGeP2. The spin-echo height is plotted as a function of the evolution time 2tl. The solid curve is the spin-echo decay using the experimentally determined lattice parameters and assuming entirely heteronuclear character of the 31P-31Pdipoledipole coupling. the anionic sites in the chalcopyrite lattice and considering P P distances, within a lo-A radius derived from the experimentally determined lattice constants. Figure 6 shows the compositional trend of the 31P-31P homodipolar second moments, illustrating the expected increase due to the unit cell contraction with increasing x. Note, however, that for all compositions considered, the experimental M2(3'P-31P) values are 2044% higher than calculated from eq 2. While the calculation assumes that the dipolar 31P_31Pinteraction is only heteronuclear in character, this assumption is most likely inappropriate in chalcopyrites: The P atoms are nearly chemically equivalent and the chemical shift anisotropy is rather small. This situation increases the likelihood that neighboring P atoms are effectively isochromatic at many crystal orientations. In this case, the flip-flop term in the Hamiltonian representing the 31P21Pdipolar coupling is not quenched, and therefore, the refocusing of the chemical shift evolution during 22, is incomplete. For a more quantitative analysisof this situation in the chalcopyrite structure, single-crystal NMR data are nec-
Zn,Cdl,GeP2
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 11051
Semiconductor Alloys
TABLE II: Population Distributions of PGe2Zn,Cd2-, Sites in Zn,Cdl-,CeP2 AUoys According to the Bimodal Distribution Model Assuming Only PCe2Cd2( a = 0) and PGe2Zn2(n = 2) Sites, the Ordered Distribution Model Assuming Maximum Amount of PGe2ZaCd ( n = 1) Sites, a d the Random Distribution Model Expected from Equation 3 and Comprrisoa with Experiwatpl Site Distributions (Error *2%) X
model bimodal
0.875 87.50% 0% 12.50% 75.00% 25*00w 0% 76.56% 21.88% 1.56% 82% 18%
n
ordered
2 1 0 2
random
0 2
1
1
0 2 0, 1
exptl
1
2o 18
25.00% 23.44%
)
e
1
e experimental *
.
e
* 0
12
g
I " '
.
0.0
1.6 1.5
12.50%
) )
1
22
N U
0.750
o I
0.2
o o calculated
o
0.8
1.0
o -
0.6
x, fraction of zinc
e e e
1.3
e
1
0.0
e
0.2
0.625 62.58
)
25'0096
0%
50'00%
)
50.00%
56.25% 37.50%
)
43.75%
25.00% 50.00%
6.25% 68% 32%
)
37.50% 25.00% 0% 7500%) 39.06% 46.884% 14.06% 48% 52%
)
31.50%
75.00% 60.94%
due to all other 31Pisotopomers and non-Cd-bonded P atoms are phase cycled out. (For khese reasons, the signal-to-noise ratio in the HDQF spectra is generally lower here, due to the low natural abundance of l13Cd.) F w e 7 contrast the 31PMAS spectra with the 31P-113CdHDQF MAS spectra for selected compositions, while Table I lists the chemical shifts of the corresponding peak maxima observed in both experiments. As expected, the disparities in these two spectra increase with decreasing cadmium content in the samples under study.
I
0.4
11
1.44
7 5 .OO% 0%
0.4
0.6
0.8
1.0
x, fraction of zinc Figure 6. Compositional dependence of (a, top) calculated and experimentally determined M 2values (in units of lo6 rad2/s2)and (b, bottom) the M2(exp)/M2(calc) ratio in Zn,Cd,-,GeP, alloys.
w r y in order to localize the 31Pshielding tensors for neighboring P atoms relative to the crystal axes, and the orientational statistics in the magnetic field need to be considered. Evidently, even in the mixed phases, the structural perturbations introduced by substituting Zn for Cd are not large enough to quench the flip-flop transitions completely. However, Figure 6 and Table I illustrate that the ratio M2(exp)/M2(calc) goes through a relative minimum at x = 0.5, the alloy with the maximum possible extent of disordering effects.
HetaonuclearDoubleQuantumFcltering(HDQF)MAS-NMR. HDQF as applied to the present materials involves double quantum excitation within pairs of the highly abundant 31Pspins and the rare spin I3Cd (natural abundance 12.26%). The pulse sequence (Figure 1)utilizes the presence of heteronuclear J coupling between the two nuclei and acts as a heteronuclear double quantum filter, providing heteronuclear correlation and spectral editing for peak assignments: The 90' pulse pair on the two separate channels, separated by the time interval 1/W(where J is the scalar coupling constant in Hz), generates a heteronuclear double quantum coherence between the observed nucleus and the rarespin nucleus. This double quantum coherence is allowed to evolve for a short period of time, under the sum of the resonance offsets of both nuclei, and is then back-transferred afterwards to observable single quantum coherence by a 90' pulse on the insensitive-nucleus channel. Acquisition of the observe-nucleus magnetization commences after an additional period of 1/25during which the Jcoupling evolution is reversed. In these experiments, only the signals due to Wd-bonded 31Patoms will be detected, while those
Discussion and Conclusions In the chalcopyrite lattice, the metal cations (Zn, Cd, Ge) are coordinated to four pnictogen anions (P in this series) and all pnictogen atoms are coordinated to two Ge and two metal atoms (Zn, Cd).*v9 Partial substitution of Cd by Zn in CdGeP2 is expected to generate three distinct PGe2Zn,Cd2-, environments, where 2 1 n 1 0. We can envision three extreme scenarios for the distribution of these environments: (a) a bimodal model, in which only sites with n = 0 and 2 occur; (b) an ordered model, in which the number of sites with n = 1 is maximized and only a maximum of two sites occur at a given composition; and (c) a random (statistical) model, where the probability P(n,x)of a P atom to be surrounded by n zinc atoms and 2 - n cadmium atoms is given by
Figure 8 summarizes the compositional dependences of the PGe2Zn,Cd2-, site populations predicted from these three scenarios. In principle, one expects the sites to have distinct 31Pchemical shifts. Thus, as the Zn/Cd ratio is varied, multiple-peak spectra are expected, whose area ratios should reflect site populations. However, only the 31P MAS-NMR spectnun of Zn,.875Cd,,lzsGeP2 reveals partial site resolution for individual phosphorus nearestneighbor configurations. Considering that there is a 28 ppm chemical shift difference between pure CdGeP2and pure ZnGeP2, one would expect more site resolution for the other compositions as well, but evidently, the chemical shift differences between the individual PGe2Zn,Cd2-, sites are not large enough compared to the extent of inhomogeneous line broadening in the MAS-NMR spectra. Evidence for local contributions to the chemical shifts in these alloys comes from the comparison of the single-pulse 31PMASNMR spectra (Figure 7, top traces) with the 31P-113CdHDQF MAS-NMR spectra (Figure 7, bottom traces). While the 31P MAS-NMR spectra reflect superpositions of the resonancesarising from the various PGe2Zn,Cd2-, sites, the HDQF spectra discriminate against the nonCd-bonded PGe2Zn2sites. Although we expect the HDQF spectra to be comprised of two line-shape components (due to the two Cd-bonded P sites with n = 0 and l),no such resolution is observed experimentally, and thus, all of the HDQF spectra are fitted reasonably well to simple Gaussians. The peak positions and widths thus determined for the Cd-bonded P sites are then used as constraints in the peak
11052 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992
1
0
'
'
~
-20
I
~
'
'
I
'
'
-60
-40
'
l
-80
V
~
-100
b, ,&, -15
~
'
l
PPM
Franke et al.
~ 2 0'
'
I , , , , I , ,
-20
-25
-30
-35
I
0~
'
-20
,
-40I
I
I
,
-60
I
I
L-JLJ -80
-100
-120
PPM
".;--40
-45
-50
PPM
Figure 7. Comparison between the "P single-pulse MAS-NMR spectra (a) and their corresponding 31P-''3CdHQDF spectra (b) for several alloys in the system Zn,Cd,,GeP,
deconvolution of the single-pulse MAS-NMR spectra. For the samples with the three highest Zn concentrations, Table I1 compares the experimental results with the predictions from the three models discussed above. For the alloy Zn,,875C&,125GeP2, Figure 7 (left top), the comparison between the single-pulse MAS and the HDQF spectra suggests that the shoulder at -70 ppm must be assigned to Cdbonded P atoms, whereas all of the remaining spectral intensity is due to non-Cd-bonded P atoms. Figure 9a shows a peak deconvolution of the 31PMAS-NMR spectrum for this alloy, where the line-shape parameters for the -70 ppm site have been constrained by the HDQF experiment. A satisfactory fit can only be obtained if at least two additional line-shape components at -58 and -50 ppm are assumed, both of which must be assigned to PGezZnzsites. These two components most likely reflect P atoms with different third-nearest-neighbor configurations. The number of Cd-bonded P atoms falls in-between the predictions
based on the statistical and the bimodal distribution model, respectively. Figure 7 (right top) contrasts the single-pulse MAS (top trace) and the 31P-113Cd HQDF MAS-NMR (bottom trace) spectra for the alloy Zno.7sCdo.25GePz, The single pulse spectrum shows a peak maximum at -54 ppm with a barely resolved shoulder on the upfield side. Using the 31P-113Cd HDQF detection method, this shoulder is detected selectively at -65 ppm. With the position and tine width of the HDQF-detected resonance constrained, the entire line shape of the single-pulse spectra can be simulated as shown in Figure 9b. Again, the area ratios determined experimentally fall in-between those expected for a random distribution and a bimodal distribution and clearly rule out the ordered model. Table I1 illustrates further that the same conclusion is reached for the composition x = 0.625. As the Cd content is increased further, the site resolution is lost in the 31P MAS-NMR spectra. Although the 31P-113Cd
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 11053
Zn,Cd,-,GeP2 Semiconductor Alloys
Bimodal Distribution Model e . . . .
/' I
I
0.0
\
I
0.2
0.4
0.6
0.8
1 .o
x, fraction of zinc
Ordered Distribution Model
I
0.0
#
0.2
0.4 0.6 x, fraction of zinc
0.8
1 .o
Random Distribution Model
0.0
0.2
0.4 0.6 x, fraction of zinc
0.8
1 .o
Figure 8. Quantitative distribution of FGe2Zn,,Cd2-,, site populations for the three scenarios discussed in the text: (a) bimodal model, (b) ordered model, and (c) random (statistical) model.
HDQF spectra meal that the Cd-bonded P atoms resonate upfield from the PGe2Zn2sites, it becomes more difficult here to distinguish between the three models. The significant difference between the single-pulse MAS and the HDQF spectra for x = 0.5 demonstrates, however, that the ordered model is not applicable. Finally, in Zno.2sCdo.7sGeP2, where there is a very high probability (93.75% according to eq 3) of having phosphorus bound to either one or two cadmium atoms, the peak positions and widths for both the 31PMAS-NMR and the 31P"3cd HDQF experiment are virtually identical, which argues against the bimodal model for this composition. All of the results suggest that the concentration of the mixed PGe2ZnCdsites in these alloys is generally substatistical, but not zero as predicted from a bimodal model. That a purely bimodal scenario is not applicable is also evident from the poorly resolved MAS-NMR spectra and particularly from the spin-echo NMR data shown in Figure 6 . The decrease of the ratio M,(exp)/ M2(calc)from both sides toward the composition x = 0.5 is a clear manifestation of chemical disorder, resulting in a lower extent of 3 1 P 1 Pflip-flop transitions among equivalent neighboring spins. In contrast, for a strictly bimodal scenario, one would expect the M2(exp)/M2(calc) ratio for the alloys to correspond to the weighted average of the respective ratios for the end members. Finally, it is interesting to note that at any composition x within the entire series studied, the Cd-bonded P sites resonate upfield of the PGqZn, sites. This chemical shift distinction is the opposite
ppm
Figure 9. Best fit simulations for the single-pulse I'P MAS-NMR spectra holding the position and line width of the HDQF detected resonance constrained for (a, top) Zn0.~7~Cd,,,,~CieP~ and (b, bottom) Zno,7s-
Cdo,2sGeP2. of that expected based on the overall chemical shift trend observed with x in the Zn,Cd,-,GeP, system. Evidently, the latter trend is dominated by nonlocal effects such as charge delocalization and lattice contraction effects, which oppose the effect of local substitution on the chemical shift. Most likely, this "counteractive" behavior explains the rather poor spectroscopic site resolution in this case. By contrast, for the anion resonances in the systems Zn,Cdl,Te and Ga,Inl-,P, the local nearest-neighbor substitution effects and the nonlocal origins of compositional chemical shift trends reinforce each other, resulting in very well-resolved spectra for individual sites. In conclusion, the present study illustrates that NMR spectroscopy can clearly differentiate between ordered, random, and bimodal site population scenarios in Zn,Cd,,GeP2 alloys. The results rule out an ordered population distribution and suggest that the reality lies in-between a statistical and a bimodal distribution. Applications of the spectroscopic strategy outlined here appear feasible in a wide range of other materials with order/ disorder phenomena, including 11-VI semiconductors, zeolitic materials, and amorphous systems. Acknowledgment. Support from the National Science Foundation, Grant No. DMR-89-13738, is gratefully acknowledged.
J. Phys. Chem. 1992,96, 11054-1 1065
11054
We thank Kyung-Ah Park for technical assistance with the sample preparation. References and Notes (1) Martins, J. L.; Zunger, A. fhys. Reu. 1984, 830, 6217. (2) Akimoto, K.; Mori, Y . ; Kojiman, C. fhys. Rev. 1987, 835, 3799. (3) Beshah, K.; Zamir, D.; Becla, P.; Wolff, P.A.; Griffin, R. G. fhys. Reu. 1987,836,6420. (4) Tycko, R.; Dabbagh, G.; Kurtz, S. R.; Goral, J. P. fhys. Rev. 1992, 845, 13452. ( 5 ) Zax, D.; Vega, S.; Yellin, Y.; Zamir, D. Chem. fhys. Lett. 1987, 138, 105. (6) Vaipolin, A. A. Izu. Akud. Nuuk. SSSR Neorg. Mater. 1967, 3, 260. (7) Girault, B. Moter. Res. Bull. 1978, 13, 457.
(8) Zamir, D.; Beshah, K.; &la, P.;Wolff, P. A.; Griffin, R. G.; Zax, D.; Vega, S.; Yellin, Y. J . Vuc. Sci. Technol. 1988, A6, 2612. (9) Vieth, H. M.; Vega, S.; Yellin, N.; Zamir, D. J . Phys. Chem. 1991, 95, 1420. (10) Muller, L. J . Am. Chem. Soc. 1979, 101, 4481. (11) Bax, A.; Griffey, R. H.; Hawkins, B. L. J . Magn. Reson. 1983, 55, 301.
(12) Franke, D.; Hudalla, C.; Eckert, H. Solid Stare NMR 1992, 1 , 33. (13) Maxwell. R.; Franke, D.; Lathrop, D.; Eckert. H. Angew. Chem., Inr. Ed. Engl. 1990, 101, 884. (14) Engelsberg, M.; Norbert, R. E. fhys. Reu. 1972, 85, 3395. (15) Boden, N.; Gibb, M.; Levine, Y.K.; Mortimer, M. J . Mug.Reson. 1974, 16, 471. (16) Reimer, J. A.; Duncan, T. M. fhys. Rev. 1983, 827, 4895. (17) Van Vleck, J. H. Phys. Reo. 1948, 74, 1168. (18) Douglas, D. C.; Duncan, T. M.; Walker, K. L.; Csencsits, R. J. Appl. fhys. 1985, 58, 197. (19) Boyce, J. B.; Ready, S. E. fhys. Reu. 1988,838, 11008. (20) Lathrop, D.; Eckert, H. J . Am. Chem. Soc. 1989, 111, 3536. (21) Tullius, M.; Lathrop, D.; Eckert, H. J . fhys. Chem. 1990,94,2145. (22) Lathrop, D.; Eckert, H. fhys. Rev. 1991, 843, 7279. (23) Franke, D. R.; Maxwell, R.; Lathrop, D. A.; Eckert, H. J. Am. Chem. SOC.1991, 113, 4822. (24) Eckert, H.; Franke, D.; Lathrop, D.; Maxwell, R.; Tullius, M. Muter. Res. Soc. Symp. froc. 1990,172, 193. ( 2 5 ) Franke, D.; Banks, K.; Maxwell, R.;Eckert, H. J . Phys. Chem. 1992, 96, 1906.
Thermodynamic and Stochastlc Theory of Transport Processes Far from Equilibrium John Ross,* Xiaolin Cbu, AUen Hjelmfelt, Department of Chemistry, Stanford University, Stanford, California 94305
and Manuel C.Velarde Facultad de Ciencias, UNED, Apartado Correos 60141, Madrid 28071, Spain (Received: August 24, 1992;
In Final Form: October 2, 1992)
We develop the thermodynamics of the transport processes of diffusion, thermal conduction, and viscous flow at a macroscopic level for the simplest cases of one-dimensional transport in fluids for individual linear and nonlinear processes approaching a stationary non-equilibrium state. We restrict ourselves to systems with single or multiple stationary states. As in earlier work on chemical kinetics by Ross,Hunt, and Hunt, we defme a macroscopic function 0 and show that it is related to changes in thermodynamicfunctions appropriate for each transport process; it is ‘‘excess work” of moving a system away from a stationary state; it is the thermodynamic potential rdriving force”) for each process, an extremum at each stationary state, and a Liapunov function; its second derivative gives criteria of stability of each stationary state; its time derivative is a component of the dissipation in the relaxation to a stationary state; and it determines the stationary distribution of a stochastic equation. For reaction and diffusion,the excess work @ determines the stationary distribution, in the thermodynamic limit, of a birthdeath master equation; for thermal conduction it is the stationary distribution given by a Fokker-Planck equation with temperature-dependent probability coefficient; and for viscous flow it is the stationary distributiongiven by a Fokker-Planck equation with constant probability diffusion coefficient for linear systems and with state-dependent probability diffusion coefficient for nonlinear flow. For reaction, diffusion, and thermal conduction the excess work is related to thermodynamic functions; for viscous flow the excess work is mechanical and it is the kinetic energy of the fluid in excess of that in the stationary state. In the presence of externally imposed fluctuations larger in magnitude than the thermal fluctuations but still exceedingly small compared to macroscopic averages, the excess work 0 is transformed in all cases to an integral of the kinetic flux, so that the driving force to the stationary state is proportional to the flux, both for linear and nonlinear transport. For all transport proceases then the stationary probability distribution can be reduced to a generalized Landau-Ginzburg or Schlbgl form.
1. Introduction
In a series of articles’-’ Ross, Hunt, and Hunt (RHH) have developed a globally valid themodynamia of nonlinear chemical systems, in particular systems with multiple stationary states, and the connection with stochastic descriptions. Consider a simple example of a reaction with one chemical intermediate, species X, either a linear reaction mechanism kl
A-X kl
which can have only a single stationary state, or a mechanism4
which may have three stationary states in a range of ratio B/A. The reaction is allowed to occur under the constraints as shown in Figure 1: reactant A (product B) is at constant pressure p A (PB)in volume 1 (3); the reaction occurs only in the Tied volume 2. The appropriate thermodynamic function for changes in state of this reaction is dM = dG, + d A 2 + dG, (1.3) where G ( A ) is the Gibbs (Helmholtz) free energy of the regions
0022-3654f 9212096-1 1054$03.00/0 0 1992 American Chemical Society
