PHYSICAL REVIEW B 15 SEPTEMBER 1997-IIVOLUME 56, NUMBER 12

Accurate valence band width of diamond

I. Jiménez, L. J. Terminello, D. G. J. Sutherland, and J. A. CarlisleLawrence Livermore National Laboratory, Livermore, California 94550

E. L. ShirleyNational Institute of Standards and Technology, Optical Technology Division, Gaithersburg, Maryland 20899

F. J. HimpselDepartment of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706

~Received 12 May 1997!

An accurate width is determined for the valence band of diamond by imaging photoelectron momentumdistributions for a variety of initial- and final-state energies. The experimental result of 23.060.2 eV2 agreeswell with first-principles quasiparticle calculations~23.0 and 22.88 eV! and significantly exceeds the local-density-functional width, 21.560.2 eV2. This difference quantifies effects of creating an excited hole state~with associated many-body effects! in a band measurement vs studying ground-state properties treated bylocal-density-functional calculations.@S0163-1829~97!05136-9#

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I. INTRODUCTION

The bandwidth may be considered the single most imptant parameter characterizing the electronic structure osolid. The ratio of bandwidthW and Coulomb repulsionUdetermines how correlated (U/W.1) or delocalized (U/W,1) an electron system is. Some of the most interessolids straddle the boundary between localized and deloized systems, e.g., high-temperature superconductors. Mband calculations available today are based on local-denfunctional ~DF! theory.1,2 Even though Kohn-Shameigenvalues2 from that theory do not represent the outcomof a photoemission band-mapping experiment, they aremarkably similar to band energies mapped experimentaStrictly speaking, one should use a formulation addressthe solid’s many-body effects on the energy required to cate a hole in photoemission. Quasiparticle~QP! calculationsof band energies, which use Dyson’s equation, include smany-body corrections via the electron self-energy.3–7

Angle-resolved photoemission experiments have becoaccurate enough to see many-body effects on the bandwby comparing data with DF and QP calculations. For sodiua prototype metal, the measured occupied bandwidthsmaller than the DF value but is close to the QP result.6,8,9

On the other hand, the measured bandwidth in LiF, a protype insulator, is larger than the DF value, and in agreemwith the QP result.4,10 Between such extremes, one may cosider many-body effects on bandwidths in semiconductBecause their properties are intermediate between thosmetals and insulators, it is not clear whether many-bodyfects should narrow or widen bands. Calculations withinjellium model6 indicate that metals having an electron desity comparable to semiconductors could have DF-like bawidths, particularly for diamond withr s'1.3, whereas truesemiconductors also exhibit a wide variety of band gapsvarying degrees of nearly free-electron character, averager s ,and ionicity.

Comparing DF and QP results gives a band widening

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semiconductors, just as in wide-gap insulators, but the wening is too small to be confirmed with the available daThe predicted band widening is only 0.05 eV in germaniuand 0.08 eV in silicon4,7 ~both systems havingr s'2.0), i.e.,widenings smaller than the typical uncertainty in a valenband minimum determined by angle-resolved photoemissDiamond is a better-suited system wherein to observe a wening because of many-body effects, featuring a 1.5 eV wening as predicted by QP calculations.4,5,7 Diamond is a use-ful prototype semiconductor because of its low atomnumber and large bandwidth, which has made it a long-tifavorite for testing band theory. Yet, the two experimenvalues of the bandwidth of diamond have error bars of61eV and differ by 3.2 eV,11,12 which is more than twice thepredicted widening of 1.5 eV.

To obtain an accurate~valence! bandwidth for diamond,we use a band-mapping method that collects isoenergy ptoelectron momentum distributions instead of the usualergy distributions. This approach has undergone extenexperimental and theoretical tests in determining the bawidth of lithium fluoride.10 An efficient, imaging photoelec-tron spectrometer13 is coupled with a state-of-the-art undulator beam line at the Advanced Light Source~ALS! to permitcollecting a large number of data sets. Because it takesa few seconds to take an image of the photoelectrons eminto an 84° full cone, we could use photon energies as has 350 eV, where the photoemission cross section fromvalence band is quite low, but photoelectrons behave frelectron-like before leaving the solid. This makes it mueasier to locate the origin of the interband transitions in mmentum space.

The result of our band mapping is a bandwidth of 2360.2 eV, which agrees well with QP results by Hybertsand Louie4 ~23.0! and by Rohlfing, Kruger, and Pollmann7

~22.88 eV!. The band widening of 1.5 eV~7%! compared tothe DF bandwidth~21.67 eV by Hybertsen and Louie, 21.3eV by Rohlfing, Kruger, and Pollmann! is clearly visible.The origin of this widening might be understood as an

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creasing localization of electron states towards the bottomthe band, together with the fact that energies of such loized states tend to be too high in DF calculations.

II. EXPERIMENT

Our data are images of photoelectron momentum distrtions emitted with given kinetic energies. Such images wtaken for'102 combinations of electron and photon enegies. The basic idea is outlined in Fig. 1. By selecting sofinal-state energy, one defines a momentum sphere wintersects many isoenergy surfaces in the band structurescribing electron initial states~to use the standard terminoogy which describes photoemission in an independeelectron picture!. Conceivably, the intersection cacorrespond to closed contours which shrink to a pointinitial states at band extrema, and observing such shrinkgives an accurate bandwidth. The data in Fig. 2 illustratemethod. While isoenergy contours are rather complex witthe band, they become simpler near band edges, shrinkinpoints at the maximum~defined as the energy zero! andminimum ~around223.0 eV!.

Before taking a set of images versus initial energy, suas in Fig. 2, the final-state~kinetic! energy of the photoelectrons needs to be adjusted. In so doing, the momentummal to the surface,kz , is set to a value equivalent to theGpoint in the bulk Brillouin zone, where the valence-band etrema are located, cf. Fig. 3.@The coordinate system fo(kx ,ky ,kz) is indicated in Fig. 1, whereas the~111! crystal-lographic direction corresponds to thekz direction.# Then, a

FIG. 1. Schematic of the experiment in (E,ki) space. The planesindicate initial-state isoenergy surfaces, and the paraboloid thestructure near the valence-band minimum. The intersectionplane and the paraboloid forms a loop that is visible in the expmental image. At the band minimum, the loop shrinks to a pointappropriatekz . Note thatkz is defined implicitly by the band dispersion of the final state.

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series of images is obtained for several initial-state eneracross the valence band by varying photon energy, cf. Figand 4.

The imaging electron spectrometer used for this workan improved version of a display analyzer describpreviously.13 It combines an ellipsoidal electron-repellinmirror as low pass with a spherical retarding grid as hpass. Photoelectrons emitted within an 84° full cone arouthe sample normal are imaged onto a channel plate andout optically. A new feature is a segmentation of the ellsoidal mirror into five elements with three slightly differebias voltages, which compensates for the change in angincidence along the elliptical direction of the electron mirroThis produces a threefold-higher resolving power or, altertively, better homogeneity of the band pass across the fielview. The instrument is operated at an undulator beam linthe ALS ~5-cm period!, where the data-acquisition time ireduced to less than 30 s per image, compared to aboumin at a second-generation light source. The intensity is nmalized to the transmission of the analyzer, as measufrom the emission of secondary electrons. The latter are pduced at higher photon energy while preserving the analysettings. Threefold symmetrization suppresses inhomogities and polarization effects, although all features discushere are clearly visible in the raw data. The normalized iages are converted into momentum distributions numericausing the analyzer geometry and reference data fromsharp Fermi surface of graphite for calibration.

A hydrogen-terminated diamond~111!-131 surface wasprepared from a naturally boron-doped, type-IIb diamo

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FIG. 2. Images of photoelectron momentum distributionsseveral initial-state energies across the entire valence band ofmond. The kinetic energy~final-state energy! is kept constant at 142eV, corresponding to a point nearG (222) ~Ref. 16!. When the bandextrema are approached, the emission pattern shrinks to a pothe center~normal emission!. The initial-state energies are given,eV, relative to the valence-band maximum. Each frame cov7.6 Å2137.6 Å21.

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56 7217ACCURATE VALENCE BAND WIDTH OF DIAMOND

FIG. 3. ~a! Images of (kx ,ky) photoelectron momentum distributions for hydrogen-terminated diamond~111!-131. The initial-stateenergy is fixed at the valence-band maximum, while the final-state energy is varied, in order to vary the perpendicular momentukz . AtG (333) , the emission pattern shrinks to a point corresponding to normal emission. This defines theG (333) point for electrons with final-statekinetic energy of 308 eV. Each frame covers 12 Å21312 Å21. ~b! Schematic of transitions in~a! in (kx ,kz) plane~bottom! and (ky ,kz)plane~top!. Points of typeA andB produce the triangular spots surrounding the central spot in~a! when the final-state energy is higher thaG (333) and lower thanG (333) , respectively, and points of typeC produce the hexagonal pattern at high energies.~c! Schematic of~a! in the(kx ,ky) plane. The circles represent the reciprocal space monitored by the analyzer~corresponding to a subtended full-cone angle of 84!.

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cleavage surface as described earlier.11 Heating in ultrahighvacuum to 900 K sharpened the C(1s) core-level spectrumwithout destroying the hydrogen termination of brokbonds. The clean 231 surface obtained after further heatinto 1300 K showed a well-resolved surface component of

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C(1s) signal and introduced surface states into the valenband spectra. Such states hamper identification of bulktures and blur angular distributions by introducing extra mmentum transfer via the 231 surface lattice. The hydrogenterminated 131 surface does not exhibit such surface sta

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7218 56I. JIMÉNEZ et al.

FIG. 4. Images of (kx ,ky) photoelectron momentum distributions for initial states near valence-band maximum~top row! and minimum~bottom row!. The final-state~kinetic! energy is kept constant at 308 eV, corresponding to theG (333) point. Initial-state energies are given ieV relative to the valence-band maximum. Each frame covers 12 Å21312 Å21.

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at the band edges, as evidenced by photoemission11 andby band calculations14 for the isostructural, hydrogenterminated Si~111!-131 surface. Therefore, only thhydrogen-terminated surface was used for determiningbulk bandwidth.

III. EXPERIMENTAL DETERMINATIONOF THE BAND WIDTH

Sample momentum distributions are given in Figs. 2, 3~a!,and 4, with regions of high electron-emission intensshown dark, because representation of images as negaenhances visibility of the weakest features. In Fig. 3~a! thefinal-state energy of the photoelectrons is varied for fixinitial-state energy, but in Figs. 2 and 4 the initial-state eergy is varied for fixed final-state energy. As stated earlthese are the two main steps of the bandwidth measurem

In the first step@Fig. 3~a!#, the final-state energy is adjusted to ensure that a transition can be observed at thGpoint of the bulk Brillouin zone (k50). For the final statesthere is a series ofG points along the~111! sample normal@Fig. 3~b!# corresponding to the~111!, ~222!, ~333!, etc.reciprocal-lattice vectors~in units of 2p/a with the latticeconstanta53.567 Å!. In a free-electron band, theseG pointsare reached at energies of 35.5, 142, and 319 eV abovebottom of the band, respectively. In Fig. 3~a!, we considerthe G (333) point and determine a~vacuum! kinetic energy of308 eV for it, implying an inner potential of211 eV. This isfound by taking images at a series of final-state energwhile keeping the initial-state energy at the valence-bamaximum. Transitions to theG (333) point are seen as pointlike emission at the center of the images in Fig. 3~a!. At 308eV, the intensity of this emission reaches a maximum retive to neighboringG points, e.g., the points markedA andBin Figs. 3~a!, 3~b!, and 3~c!. For kinetic energies below thG (333)-point transition~e.g., 260 eV!, one observes a trianglof G points equivalent toB with a corner on the right-handside. Above theG (333) transition, there is similarly a trianglewith a corner on the left~e.g., at 370 eV!, plus an outerhexagon. As shown in Fig. 3~b! ~bottom!, the additional trip-let spotsA and B correspond toG points located on~111!planes 1.02A21 above and below the plane containinG (333) . The outer hexagon of points of typeC correspond to

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otherG points located in the~333! plane@Fig. 3~b!, top#. Ananalogous procedure yields a kinetic energy of about 131for the G (222) point. After determining the kinetic energy fothe final-stateG point, the kinetic energy is fixed in all further images.

In the second step, the initial-state energy is varied to sacross the valence band in constant-final-state mode. Fi2 shows images that span the entire valence-band withfinal-state energy fixed near theG (222) point. We observe asharp spot at the center (ki50) near the top of the band. Fodecreasing energy, the central spot opens up into a circle,the circle splits further because of light- and heavy-hbands. Near the middle of the valence band, the topologthe momentum distributions becomes rather complex, busimplifies again near the bottom of the band, which mirrothe behavior at the top, with a circular feature shrinking topoint at the very bottom.

Figure 4 contains images similar to Fig. 2, but with tfinal state fixed at theG (333) point instead of near theG (222)point. They are taken at finer energy intervals to allowaccurate determination of the valence-band extrema. Atenergy, the photoemission cross section at the valence-bminimum is larger than near theG (222) point, making theimages in Fig. 4 more suitable for this determination~Figs. 5and 6!. Intensity profiles along theky direction in Fig. 4 aredisplayed in Fig. 5 for states near the valence-band mmum. They are decomposed into Gaussian profiles to bedistinguish various valence-band features, which is critinear the band edges, where transitions on either side ozone center are superposed. The distance between twosolved peaks can be determined unambiguously by fixingwidth of the Gaussians to the sharpest overall width,tained at the band extrema. A similar analysis is performfor the dispersion near the valence-band minimum~notshown!. Plotting the momentaky of the features at variousinitial-state energies, we obtain the energy bands nearmaximum and minimum in Fig. 6. Then, by fitting parabolto the band edges, we obtain accurate values for the valeband extrema and hence for the valence band width~dashedlines in Fig. 6!.

IV. EXPERIMENTAL VS QUASIPARTICLE BANDS

The diamond bulk band structure has been extensivstudied, and the data may be compared with QP res

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56 7219ACCURATE VALENCE BAND WIDTH OF DIAMOND

which were given by Hybertsen and Louie,4 Godby, Schu¨ter,and Sham,5 and Rohlfing, Kruger, and Pollmann.7 These re-sults yield a 1.5-eV enhancement in bandwidth compareDF results. Indeed, QP results for diamond valence baexhibit a uniform 7% ‘‘stretch’’ compared to DF resultThis facilitates modeling electron band states readily ussuitably amended, convenient DF results, which we did ifashion greatly accelerated by the choice of an optimal bset,15 to treat band dispersion and simulate photoelectmomentum distributions. Regarding further details ontreatment of one-electron band states, the calculations wsimilar to those in Ref. 15.

In calculated momentum distributions, weightingsC(2s) and C(2p) orbitals in initial-state wave functionwere determined and used to compute matrix elementsphotoexcitation into several nearly degenerate outgoplane waves, one of which corresponds to the vacuum plawave part of the final state. Couplings between such plwaves were computed within the pseudopotential/plawave framework which was used, including effects of nolocal pseudopotential projectors. We assumed a meanpath of 10 Å, and due preferential weighting of orbitalsthe atoms closer to the surface was taken into account.necessary integration overkz , results presented includesampling of 30kz values over a 1 Å

21 range centeredaround^kz&[(K

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FIG. 5. Set of radial intensity profiles along they direction inFig. 4 ~vertical!. Gaussian fits with constrained width are usedprecisely determine momenta where the transitions occur for vous photon energies.

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pseudo wave functions, but plane-wave matrix elemewere corrected using known differences between full apseudo wave functions in atomic core regions. A simpapproximation worked well in graphite and LiF, but mocare may be required in diamond, which exhibits stronadmixture of atomic 2s and 2p states. Calculated momentumdistributions for final states near16 the G (222) point are en-couraging, featuring theoretical counterparts to severalperimental features. They are shown in Fig. 7 and correspto experimental images of Fig. 2.

While QP calculations predict the bandwidth correctone may also desire insight into the shortcomings oftheory regarding predicted bandwidths. In the following, wpropose a qualitative argument to explain the differencestween DF results and the true band structure. We note opsite trends for wide-gap insulators LiF and the semicondtor diamond vs the nearly free-electron metal, sodiuRather than formulating a distinction between metallicnonmetallic systems, we note that many-body effectselectron states involve exchange, which tends to widbands, and dynamical correlation, which can widen or nrow bands. For low-density metals, correlation dominaover exchange, leading to an overall band narrowiwhereas exchange dominates at very high densities, evejellium. Therefore, the many-body effects on such a bawidth involve two largely canceling parts. The net effect c

ri-FIG. 6. Interband transitions in an (E,ky) diagram. A parabolic

fit to data near band edges is used to determine the valence-maximum and minimum. It gives a bandwidth of 23.060.2 eV,substantially larger than the density-functional result, 21.560.2 eV.

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be narrowing, widening, or neither.In diamond and wide-gap insulators, nonjellium charac

of such systems causes electron states to be more atolike, causing DF results to produce an artificial band narroing. Also, localization can enhance the shortcomings of atreatment regarding exchange effects, which cancel spur‘‘self-interactions’’ present in many DF treatments, as illutrated most severely for the semicore 3d states of zinc17 andfor a localized hydrogen orbital on Si~111!-131:H.14 Also, aDF treatment ‘‘envisions’’ the final-state hole in a photemission experiment as experiencing the same potewhether it is at the valence-band minimum or maximum~orat the Fermi surface in a metal!. However, metallic screeningtends to screen a hole at the band minimum more efficienan effect not realized in a DF treatment. Therefore, whcorrelation effects dominate, as they do in sodium, DFsults can give too large a bandwidth. For the semimegraphite, on the other hand, the DF bandwidth is'14%smaller than the QP bandwidth.18

FIG. 7. Simulated photoelectron momentum distributions cosponding to experimental images of Fig. 2. The calculationsbased on quasiparticle bands.

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Recently, a modification to the local-density approximtion, the local mass approximation,19 was designed to obtainthe correct bandwidth in simple metals. It also predictscorrect bandwidth for diamond~22.9 eV!, where the differ-ence between experiment and local-density theory goethe opposite direction.

V. CONCLUSIONS

In summary, we demonstrate how an extensive set of ptoelectron momentum distributions can be used to determa bandwidth with unsurpassed accuracy. This resultother successful applications of photoelectron momendistributions10,20–22suggest that viewing photoemission dain two- or three-dimensional momentum space rather thaone-dimensional energy space can be advantageous. Fomond, we obtain a bandwidth of 23.060.2 eV, allowing oneto judge relative merits of local-density-functional and qusiparticle results. The experimental bandwidth agrees wthe quasiparticle result, and it is 7% larger than the locdensity-functional result. This also illustrates limitationsground-state calculations such as local-density-functiotheory for determining certain properties of systems, wherquasiparticle calculations address many-body effectselectron-removal energies probed in photoemission. Formond and other wide-gap materials, we have also providemotivation for why local-density-functional theory underetimates the bandwidth, which involves the relatively potreatment by local-density-functional theory of the morecalized states at the valence-band minima.

ACKNOWLEDGMENTS

We are indebted to N. D. Lang and S. G. Louie for elightening discussions on local-density-functional theory,R. Hill for technical support, and to the ALS staff for genersupport. This work has been supported by the DivisionMaterials Sciences, Office of Basic Energy Science, andformed under the auspices of the U.S. Department of Eneby Lawrence Livermore National Laboratory under ContraNo. W-7405-ENG-48, and the Advanced Light SourcLBNL under Contract No. DE-AC03-76SF00098. One of tauthors~I.J.! acknowledges financial support from the Spaish Ministerio de Educacio´n y Ciencia.

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211 eV would be 131 eV, not 142 eV. The differences in timages are minor.

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