Vol. 16, No. 1/2 127

Detection of Two-Quantum Nuclear Coherence by NuclearQuadrupole Induced Electric Polarization

David C. Newitt1 and Erwin L. Hahn

Physics DepartmentUniversity of California

Berkeley, California, 94720

Contents

I. Introduction

II. Nuclear Electric Resonance Detection

III. Experimental Results

IV. Conclusions

V. Acknowledgments

VI. References

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I. Introduction

The Nobel Prize in Chemistry for 1991 was con-ferred upon Richard Ernst for his development ofelegant NMR techniques and fundamental theoryapplicable to various types of physical and chem-ical analysis. These include in particular specificinnovations and extensions of pulse Fourier trans-form methods for NMR high resolution spectros-copy and MRI. We of the NMR community espe-cially salute and congratulate Ernst because thePrize brings honor as well upon all of us who havehad so much fun in a field that has yielded many in-novations over a time period much longer than manyof us expected. The field of NMR in its developmentreached a stage where one could hardly distinguishwhether chemists or physicists were doing NMR.Now the chemists, or rather the physical chemistsand biologists, have taken over the field of NMR,and they are doing most of the "physics" nowadays.As a consequence, because chemical technology ismore "up front" in the public and commercial eye,people know more about NMR than ever before.Also MRI has had an impact upon the public in the

1 Present address: MRSC, Department of Radiology,UCSF, San Francisco, California 94143

medical and health world, and in scientific researchthe analytic techniques made possible by NMR havebeen applied in one form or another to investigationsin many disciplines.

In contrast to the utilitarian revelations of theworks of Richard Ernst, the authors of this articlepresent in his honor the results of an experimentof the opposite sort, which they hope is acceptable,because most likely the reader will not find our ex-periment particularly useful. As will be seen in whatfollows, the electric resonance detection experimentis an interesting exercise in proving that nature willyield the reciprocal of a given effect if one goes tothe trouble to expose it. In the course of interpret-ing such experiments one is forced to improve hisunderstanding of things he thought he understoodbut in fact did not.

II. Nuclear Electric ResonanceDetection

Magnetic one-quantum signal detection of theevolution of multiquantum nuclear spin coherence

128 Bulletin of Magnetic Resonance

always requires the transfer of multiquantum su-perposition states into one-quantum superpositionstates (1). A directly observed two-quantum nuclearelectric quadrupole radiation signal is conceivabletheoretically but beyond observation experimentally(2), since it would be of the order of 10~~9 the sizeof an average NMR signal. In this gedanken exper-iment one would place the sample in a quadrupolecapacitor to detect the direct two-quantum nuclearquadrupole radiation signal. However, for nuclearquadrupole moments located at noncentrosymmet-ric crystal sites, as in the case of As and Ga inGaAs, net local electric-dipole moments are inducedin neighboring atoms by the electric quadrupole fieldof the nucleus. In this report we assume the "stickand ball model" (3). The electric field due to thenuclear quadrupole falls off as r~4, where r is the dis-tance between the point nuclear quadrupole and thepoint neighboring polarizable atom. The summa-tion of quadrupole induced dipole moments over theset of nearest-neighbor atoms and over the spin en-semble yields a detectable macroscopic polarization.The polarizability of the local atomic environmenteffectively magnifies (3) the otherwise unobservabledirect quadrupole radiation signal by a factor on theorder of 10+6, so that a direct electric signal may beobserved by placing the sample between the platesof a capacitor.

In a previous experiment (3), nuclear electricresonance detection (NERD) was first demonstratedusing the 30 MHz transition of 35C1 in NaC103. Inthat experiment a small magnetic field was appliedto remove the degeneracy of the m = ±1/2, ±3/2states. The resulting mixed m = ±1/2 states al-low the development of a detectable electric polar-ization FID signal after a single pulse. The elec-tric signal was characterized by a beating of differ-ent | Am | = 1,2 transition frequencies between themixed m = ±1/2 states and the m = ±3/2 states,distinguishable from the beat frequency signatureproduced by stray pickup of a direct nuclear mag-netic signal at the same Larmor frequency. In GaAsthe observation of a pure |Am| = 2 transition from75As at the sharply tuned frequency 2u is free ofany nuclear magnetic signal at frequency to. How-ever, there is a small reactive signal pickup transientfrom the transmitter pulses at frequency u>.

In general the observation of a one-quantum ortwo-quantum electric signal requires that the expec-

tation values

I±Iz> = , Tr{p(IzI(1)

be finite, which occurs only if the density matrixp is nonlinear in the nuclear spin operators. Thusin any system with equally spaced levels (includingthe degenerate NQR 35>37C1 levels in NaC103), it isnot possible to observe a NERD signal if the initialdensity matrix p is specified by a high temperaturepopulation distribution proportional to Iz, in whichcase the above traces of odd operators are zero. Therequirement of a nonlinear p is met in the case ofthe zinc-blende structure, including GaAs and mostother III—V semiconductors, by applying an exter-nal static electric field which produces a quadrupoleshift due to the linear Stark effect (4). In the spe-cial case of NaClO3 the electric FID signal is ob-served immediately after a single pulse because ini-tial signals which beat with one another, associatedwith different frequencies, have different amplitudes,which is not the case for GaAs where a minimum oftwo RF pulses is required.

The NERD-induced polarization effect may bedenned as the inverse of the linear Stark effect,where the former involves oscillating off-diagonalelements, and the latter involves on-diagonal el-ements which account for quadrupole frequencyshifts. Both effects are expressed by the followingHamiltonian perturbation in dyadic form (3): withelectric field gradients. The stick and ball descrip-tion assumes that a nuclear quadrupole point sourceelectric field induces polarization in nearest neighborpoint atoms. This cannot account for the polariza-tion spread over charge distributions and covalentbonds. Hence the definition of the "stick" distanceis a vague one. Although one can only predict ordersof magnitude of signal amplitudes by this approach,it is most valuable for predicting the signature andthe sample orientation dependence of pulse transientsignals which characterize the NERD phenomena.

A rough estimate of the NERD signal strength isobtained by expressing the local quadrupole electricfield

HQ = — P Q • E —Q Q • E Q (2)

The macroscopic polarization induced by the pre-cessing local quadrupole electric field EQ is given

Vol. 16, No. 1/2 129

by P Q = N/3a- E Q , where a is the atomic polariz-ability tensor. The Boltzmann factor j3 pertains toN participating quadrupole spins in the crystal. Theelectric field and corresponding oscillating voltage Von the capacitor plates are given by V = (E-n)d =47r(PQ-n)d, where n is normal to the capacitor plateand d is the plate separation. The alternative formof HQ, applicable to the dc linear Stark effect, de-fines E as the externally applied static electric field,where P E = N/3o;E for an isotropic polarizability a.Here one views the applied electric field as inducinginternal atomic dipole moments which interact withlocal nuclear quadrupole electric fields E Q .

The above model is a less rigorous equivalent ofthe usual analysis in terms of the quadrupole inter-action as EQ = eQS/r4, where S = 25 is taken as theSternheimer antishielding factor, the 75As gyromag-netic ratio 7 = 0.73 MHz/kG, Q = 0.3 • 10"24 cm2,a = 10™24 cm3 (assumed isotropic), ro = 1.2 • 10~8

cm (estimated as one-half the As—Ga lattice dis-tance, at the center of the covalent bond), andN = 1022 cm"3. Upon application of typical cir-cuit parameters, scaled from the estimate given indetail in ref. (3), one obtains an approximate ratio ofthe 75As NERD signal to the conventional magneticNMR signal in the present experiment of about 1/10to 1/100.

III. Experimental Results

The arrangement for a two-quantum NERD ex-periment consists of two independent resonant cir-cuits: an RF transmitter coil tuned to frequencyw/2ir, inside of which is contained the sample,housed between two plates of a receiver capacitortuned to frequency 2w/2vr = 13 MHz. The receiveris one used in a typical pulsed NMR system. Forour measurements a single crystal of semi-insulatingGaAs with dimensions 1.4 x 1.4 x 0.1 cm3 is placedwith the (111) crystal planes parallel to the capac-itor plates and transmitter coil axis. The circuitsare cooled to 77 K to reduce noise and ensure highresistivity of the GaAs sample.

Applying the stick and ball model to the struc-ture around an As atom shown in Figure 1, usingisotropic polarizabilities for the four nearest neigh-bors, results in an induced polarization

aeQ5 . 2Pind = " J ^ Q 8 1 1 1 6 £> + (£» (3)

for the (111) crystal orientation described above,where 0 is the angle between the static polarizingmagnetic field H o and the Stark field Eo. The po-larization P;nci is summed over the spin ensemble toobtain the macroscopic polarization P Q expressedin eqn. 2, which is proportional to the signal volt-age V.

A DC electric field on the order of |E0 =20 kV/cm is applied to the capacitor to providethe linear Stark shifts. The Stark-induced electricfield gradient for this orientation is given by eq =2R|Eo|/3, where R = 1.9 x 1010 cm"1 is the linearStark coefficient (4) for 75As. The resulting quad-rupole frequency shift is

3 cos 6 — 1 (4)

The angle 8 = 90° is chosen to provide the maximumNERD signal and an wq of the order of 7 kHz forthe 75As nuclei. Figure 1 shows the energy-leveldiagram and the NMR and NERD spectra for thissystem.

Figure 2 shows 75As two-quantum FID and echosignals and the interference between them for a twopulse 90x — r — 90~)-x sequence. The inverses of allRF pulse widths exceed level broadening and electricStark shifts. The second pulse alternates in phase by180° from one two-pulse sequence to the next. Thealternate data runs are added and subtracted fromthe signal average so that the nuclear signals add butthe reactive pickup signals from the constant phasetransmitter pulse cancel out. As the pulse spacingtime r is increased, the emergence of the electricquadrupole echo at t—2r may be seen, although itis rapidly attenuated by dephasing due to magneticinhomogenaeties and dipolar interactions.

The prediction of FID and echo signals may becarried out by use of the Majorana formula (5) toevaluate spin wave functions and expectation val-ues, or a density matrix technique such as that de-vised by Bowden and Hutchison (6) may be used.Since the spin levels are broadened by both elec-tric quadrupole and magnetic dipolar perturbations,we assume for simplicity that the broadenings dueto these perturbations are independent of one an-other. The echo refocusing of electric strain inho-mogeneous isochromats in this picture accounts for

130 Bulletin of Magnetic Resonance

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Figure 1: Tetrahedral structure of the four nearest neighbors in the zinc-blende structure, with the resultingenergy-level diagram and NMR and NERD spectra for a spin 1=3/2 nucleus subject to a Stark-inducedquadrupole shift. In the bottom plot the dotted line shows the spectrum of a normal NMR experiment onStark shifted 75As in GaAs, while the solid line shows the spectrum of a two-quantum NERD experiment onthe same sample.

observed NERD echo signals. As we are detectinga two-quantum coherence, the magnetic broadeningisochromats do not refocus at the same time as thequadrupole interactions for a two pulse sequence.Therefore the NERD echo amplitude lifetimes T2eremain very short, on the order of a millisecond, be-cause of the defocusing caused by inhomogeneousand homogeneous magnetic broadening. The quad-rupole echo is visible because the quadrupole de-phasing time, T£Q = 50 /xs, is significantly less thanthe magnetic dephasing time, TgM = 200 us.

The dotted lines in Figure 2 are fits to the dataaccording to this simple model of independent mag-netic and quadrupolar broadening, where only thepulse separation time t is varied. The inverse ofthe quadrupole broadening of the system varied be-tween T20 — 40 and 50 /zs during the data runsbecause of progressive damage to the sample causedby the applied high voltage. We believe this dam-age is caused by the migration of charged defectsor impurities, which over time produces changes inbulk strain and results in charge layers at the (111)

Vol. 16, No. 1/2 131

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surfaces. We have found that a fixed electric fieldranging up to lkV/cm caused by these charge layersopposes the applied field Eo. This fixed field can beremoved by warming the sample to room tempera-ture for several hours, which then yields an increaseof T^Q to 80-90 [is.

A clearer measurement of the two-quantumNERD echo may be obtained by observing theechoes produced by a three pulse sequence, as thisallows simultaneous or near-simultaneous refocus-ing of the magnetic and electric isochromats. Forspin 1 = 3/2 nuclei the first two pulses generateone-, two-, and three-quantum coherences amongthe spin levels, and the third pulse in turn trans-fers these prepared coherences among the levelsto provide observable two-quantum electric signals.Figure 3 shows two sets of data obtained from a90x — Ti — 90-tx ~ T2 ~ 90x sequence. In the toptrace T\ = 50 [is, which is relatively short comparedto the inhomogeneous dephasing time T^Q = 70 fis.A portion of the two-quantum coherence seen as an

FID following the second pulse is recreated after thethird pulse as an echo at time t = 2T2 = 1 /is. In thiscase both the electric and the magnetic isochromatsrefocus at approximately the same time. The bot-tom trace shows the multiple echoes formed by refo-cusing of the electric isochromats when T\ = 200 /xsand T2 = 600 [is are both longer than T^Q. Weobserve the three echoes labeled El, E2, and E3 attimes predicted by the inhomogeneous broadeningmodel, but cannot achieve a reasonable fit to thesignal amplitudes for the different echoes for thesethree pulse sequences.

IV. Conclusions

The direct detection of electric two-quantum co-herence signals from an I = 3/2 spin system hasbeen demonstrated without the need for an extraRF pulse to transfer unobservable two-quantum co-herence to one-quantum coherence as required in

132 Bulletin of Magnetic Resonance

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Figure 3: Two-quantum NERD FID and echo sig-nals from three 90° x-axis pulses. Pulses are appliedat t = 0, 50 and 550 /is for the top trace and t = 0,200 and 800 /xs for the bottom. The stronger sig-nals after the second pulse in each trace are scaleddown by a factor of five relative to the signals afterthe third pulse, and the lower trace is scaled up bya factor of two relative to the upper trace. Threequadrupole echoes, centered at t = 1000, 1200, and1400 /xs, labeled El, E2, and E3, are visible in thebottom trace.

NMR multiquantum methods. The time evolutionof the two-quantum coherence is mapped out in one"shot," whereas for NMR an additional inspectionpulse must be applied for successive times in re-peated pulse sequences to map out the two-quantumcoherence.

Our echo analysis does not take into account thecomplicated magnetic dipolar echo-dephasing effectscaused by the pulse reorientation of local dipolarfields. Many of the predicted echoes which occurfollowing three pulses not only interfere with FIDtransients, but interfere among themselves if theyare to be observed at all because the echo decay life-times are too short to always clearly resolve them.In some instances a predicted echo cannot be seen,

or an echo predicted to be canceled out by the ap-plied pulse sequence phase cycling is clearly visible.We do not understand this at present and hope toresolve it in a later report.

Given a fixed number of pulse excited spins overthe entire spin spectrum, the initial FID amplitudeof the electric polarization depends only upon thelocal distance and polarizability of atomic bondsand electrons in the vicinity of precessing nuclearquadrupole moments. On the other hand the initialFID signal in a conventional pulsed NMR experi-ment would be independent of these properties. Onemay conceive of experiments in which variations ofthese properties may be studied in terms of observedchanges in the initial electric signal. Applications ofexternal pressure, acoustic vibrations, and electricfields which induce charge layers in semiconductorstructures or charge density waves in special systemsare examples for future investigations.

Beyond the two-quantum case in NMR, an ad-vanced and rigorous review of multiple quantumNMR spectroscopy techniques is provided in ahighly comprehensive account (7) of modern NMRtechniques by Ernst and co-authors. Under onecover, this account as a book includes descriptionsof the important researches by Ernst and his collab-orators at the ETH, Zurich which were recognizedby the Nobel Prize.

V. Acknowledgments

We gratefully acknowledge helpful discussionswith John Keltner, Larry Wald, and Charles Pen-nington and the support of the National ScienceFoundation.

VI. References1C P. Slichter, "Principles of Magnetic Reso-

nance," 3rd ed., Chapter 9, Springer-Verlag, NewYork/Berlin 1990.

2M. Bloom and M. A. LeGros, Can. J. Phys.64, 1522 (1986).

3T. Sleator, E. L. Hahn, M. B. Heaney, C. Hilbertand J. Clarke, Phys. Rev. B 38, 8609 (1988).

4N. Bloembergen, in "Proceedings, XI ColloqueAmpere Conference on Electric and Magnetic Res-onance, Eindthoven, 1962" (J. Smidt, Ed.), p. 39

Vol. 16, No. 1/2 133

North-Holland, Amsterdam, 1963; F.A. Collins andN. Bloembergen, J. Chem. Phys. 40, 3479 (1961).

5E. Majorana, Nuovo Cimento 9, 43 (1932); F.Bloch and I. I. Rabi, Rev. Mod. Phys. 17, 237(1945).

6G. J. Bowden and W. D. Hutchison, J. Magn.Reson. 67, 403 (1966).

7R. R. Ernst, G. Bodenhausen and A. Wokaun,"Principles of Nuclear Magnetic Resonance in Oneand Two Dimension," Chapter 5, Clarendon Press,Oxford, 1990.