Physica B 302–303 (2001) 88–100

Magnetospectroscopy of acceptors in ‘‘blue’’ diamonds

Hyunjung Kima,*, A.K. Ramdasb, S. Rodriguezb, Zdenka Barticevicc,M. Grimsditchd, T.R. Anthonye

aAdvanced Photon Source, Argonne National Laboratory, Experimental Facilities Division, 9700 S. Cass Avenue,

Argonne, IL 60439, USAbDepartment of Physics, Purdue University, West Lafayette, IN 47907, USA

cDepartamento de F!isica, Universidad T !ecnica Federico Santa Mar!ia, Valpara!iso, ChiledMaterials Science Division, Argonne National Laboratory, Argonne, IL 60439, USAeGeneral Electric Corporate Research and Development, Schenectady, NY 12309, USA

Abstract

Naturally occurring, nitrogen-free, p-type diamond}now known to be boron-doped}as well as man-made

diamonds deliberately doped with boron display an electronic Raman transition, D0, originating in the 1sðp3=2Þ :G8ground state of the acceptor and terminating in its 1sðp1=2Þ :G7 spin–orbit partner. With magnetic field B along ½0 0 1�,½1 1 1�, or ½1 1 0�, the electronic Raman spectrum displays eight Zeeman transitions and four Raman lines ascribed totransitions between the Zeeman sublevels of G8 (Raman-electron-paramagnetic-resonance: Raman-EPR). They exhibitpolarizations expected from the polarizability tensors formulated in terms of the Luttinger parameters g1; g2, and g3characterizing the top of the valence band. The selection rules and the relative intensities of the Zeeman components as

well as of the Raman-EPR lines, observed in diverse polarization configurations and scattering geometries, have led to:assignments of magnetic quantum numbers; the level ordering of the Zeeman sublevels, or equivalently, the magnitudesand signs of g1 and g2, the orbital and spin g-factors of the acceptor-bound hole; the extreme mass anisotropy asreflected in the ratio ðg2=g3Þ ¼ 0:08� 0:01. Magnetic-field-induced mixing of zero field states, time reversal symmetry,and the diamagnetic contributions which characterize the different sublevels are fully taken into account in theinterpretation of the experimental results. These include the striking mutual exclusion of the Stokes spectrum from itsanti-Stokes counterpart in specific polarization configurations. # 2001 Elsevier Science B.V. All rights reserved.

PACS: 71.70.E; 71.55.A; 78.55.A

Keywords: Zeeman effect; Acceptors; Diamond

1. Introduction

Substitutional acceptors and donors in tetra-hedrally coordinated group IV semiconductors are

perhaps the best delineated point defects incrystals [1]. Their electronic states find a preciseformulation in terms of the band structure of thehost in which they are embedded. The symmetryand mass parameters of the band extremum withwhich they are associated and the screenedCoulomb potential which binds the charge carrierallow calculations of the eigenfunctions and

*Corresponding author. Tel.: +1-630-252-9141; fax: +1-

630-252-3222.

E-mail address: hjk@aps.anl.gov (H. Kim).

0921-4526/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 4 1 1 - 2

eigenvalues of these solid state analogs of theH-atom. The predictions of this so-called effectivemass theory (EMT) have provided the theoreticalbases for the interpretation of the experimentalresults obtained with some of the most powerfulexperimental techniques [2]. The departures fromEMT revealed in the experiments have beenexplored}and understood to various degrees}interms of deviations from the screened Coulombpotential in the immediate vicinity of the impurity;the mismatch of the electronic core of thesubstitutional impurity and that of the host atomit replaces; and several other factors collectivelyreferred to as ‘‘central cell corrections’’. Indeed,they provide a strong motivation for investigationsof such point defects in both elemental andcompound semiconductors. The experimentaltechniques which have been applied with greatsuccess in the discovery and description of theelectronic states of acceptors and donors areLyman lines observed in the infrared as absorption[3,4] and photothermal ionization spectra (SeeRefs. [5,6], for the application of photothermalionization spectroscopy to the Lyman spectrum ofboron in diamond), Raman spectroscopy [7,8],photoluminescence, including donor–acceptor pairspectra [9,10] in the visible and near infrared, andelectron-paramagnetic-resonance (EPR) in micro-wave spectroscopy [11].The focus of the present paper being the

electronic states of acceptor-bound holes inelemental semiconductors, in particular substitu-tional boron in diamond, it is instructive tocompare the relevant band parameters of dia-mond, silicon, and germanium. The spin–orbitsplitting, D, of the valence band maximum at thecenter of the Brillouin Zone; the Luttinger para-meters, g1, g2, and g3; and, D0, the spin–orbitsplitting of the ground state of the boron acceptor,are displayed in Table 1, the sign convention forthe Luttinger parameters being that for holeenergies.In the EMT, the electronic levels of the

acceptor-bound hole are described by wave func-tions which are linear combinations of products ofBloch states at the G-point and envelope functionssatisfying the Schr .odinger equation, H0c ¼ Ec.Here c is a six-component column vector and H0

in terms of hole energies is

H0 ¼1

m

1

2g1p

2 � 3g2X3i¼1

p2i I2i �1

3I2

� �

�3g3Xi5j

pipjfIi; Ijg

!þ VðrÞ: ð1Þ

The operators I1; I2; I3 are angular momentumoperators with angular momentum I ¼ 1 in therepresentation generated by the components of avector on the cubic axes. The numbers g1; g2, andg3 are the Luttinger parameters, VðrÞ is thepotential of the hole in the field of the ionizedacceptor. The operator H0 can be expressed as asum of a spherically symmetric part and anotherhaving only cubic symmetry as first demonstratedby Baldereschi and Lipari [16,17]. Assuming thecubic part to be small compared to the spherical,an algorithm can be developed to solve theeigenvalue problem starting from the solutionsfor the spherical part. In a subsequent paper theymodified the Coulomb potential by introducing awave vector dependent dielectric constant [18]thereby deducing the central cell corrections tothe ground state binding of different group IIIacceptors and estimating ðD0=DÞ (see also Serranoet al. [19]).Custers [20,21] made the startling discovery that

there are extremely rare specimens of type IIdiamonds which are highly conducting, some withresistivities as low as 25 O cm, whereas acceptor-free specimens typically exhibit resistivities well inexcess of 1016 O cm (resistivities as low as 5 O cmhave since been encountered). The low resistivitydiamonds have been designated as type IIb,whereas the nitrogen-free, high resistivity speci-mens are classified as type IIa. Hall effect studies[22,23] showed that type IIb diamonds are p-type.Further, the infrared spectra of type IIb diamondsexhibit Lyman transitions characteristic of thebound states of the acceptor-bound holes [24]. InFig. 1 we display the Lyman spectra of boronacceptors in a natural diamond. The inset showsschematically the 1sðp3=2Þ :G8 ground state and its1sðp1=2Þ :G7 spin–orbit-split counterpart and a setof Lyman transitions. Even after the successfulsynthesis of diamond with the high-pressure–

H. Kim et al. / Physica B 302–303 (2001) 88–100 89

high-temperature technique (HPHT) and withchemical vapor deposition (CVD) [25,26], only p-type doping with substitutional group III boronimpurities has been successful (See Refs. [27–29]1).

Boron-doped diamonds are bluish to distinctly bluein color as a result of the absorption in the red dueto the photo-ionization continuum of holes boundto boron acceptors.Raman spectroscopy has enabled the discovery

and delineation of the unique, Raman allowed andinfrared forbidden, electronic excitation at D0,shown in the inset of Fig. 1 [14,30]. Diamond,with its unparalleled transparency and scatteringpower, is ideally suited for Raman spectroscopy.The Stokes and the anti-Stokes Raman lines at D0

in natural and 13C boron-doped diamonds areshown in Fig. 2. The experimental results on thesplitting of the D0 Raman line in a magnetic fieldand the underlying theory (a comprehensivepresentation on the subject is given in Ref. [31])are presented in this paper.

2. Magnetospectroscopy: theory

The site symmetry Td of a substitutionalacceptor in the group IV elemental semiconductorrequires the states of a single hole bound to it tobelong to one of the irreducible representations G6,G7, or G8 of the double group %Td [32].

2 The orbitalground state belongs to the G5 irreducible repre-sentation of Td. Taking the spin of the hole intoaccount this level belongs to the G5 � G6 repre-sentation of %Td, split by the spin–orbit interactioninto a quadruplet G8 and a doublet G7. Theangular momentum states of an atomic p-level alsogenerate the G5 � G6 representation of %Td. Thismakes possible the assignment of quantum num-bers to the states of the hole in one-to-one

Fig. 1. The Lyman transitions of holes bound to boron

acceptors in a natural diamond recorded at T ¼ 79:4 and5:0 K. The arrows identify the new lines which appear at highertemperatures. Note the difference between the two lines labeled

with the same number (unprimed and primed) is 2 meV. The

inset shows the Lyman transitions from the ground state

associated with the p3=2 valence band as well as those associated

with the p1=2 valence band. D is the difference between the

maxima of the two bands and D0 that between the two ground

states, associated with the p3=2 and the p1=2 valence bands,

respectively.

Table 1

Band parameters of the valence band maximum at the center of

the Brillouin Zone in the elemental group IV semiconductorsa

Band parameters C Si Ge

ðDÞ (meV) 6 44 290

g1 3.61 4.27 13.3

g2 0.09 0.32 4.29

g3 1.06 1.56 5.66

D0 Boron (meV) 2.07b 22.77c }

aFrom Landolt–B .ornstein [12] except for D0. Willatzen et al.

[13] have calculated D ¼ 13 meV for diamond.bH. Kim et al. [14]. The value given is for natural diamond

ð12C0:89 13C0:11Þ; for 13C diamond it is 2:01 meV.cH.R. Chandrasekhar et al. [15].

1On the basis of a critical examination of the literature and

their own investigations these authors concluded that the

‘‘acceptor’’ in semiconducting diamond is substitutional boron.

2Wherever possible we follow the notation and conventions

in this reference. We depart in the assignment of quantum

numbers by taking expð�ij#n � JÞ for the rotation operation byj about an axis parallel to #n. According to this rule, with

B jj ½0 0 1�, i.e., for %S4, c3=2 belongs to G5 of %S4, c�3=2 to G6,c�1=2 and f�1=2 to G7, and c1=2 and f1=2 to G8. For B jj ½1 1 1�the appropriate symmetry group is %C3 and c�1=2 and f�1=2belong to G4 of %C3, c1=2 and f1=2 to G5, and c3=2 and c�3=2 to

G6. When B jj ½1 1 0�, the group being %Cs, c�3=2; c1=2 and f1=2belong to G3 of %Cs while c3=2;c�1=2 and f�1=2 belong to G4. Weremark that the characters of the element %S4 in the group %S4 in

this reference should be corrected to o3; � o; � o3, and ofor the double valued representations G5, G6, G7, and G8,respectively.

H. Kim et al. / Physica B 302–303 (2001) 88–10090

correspondence with those of the atomic states.The G8 states will be denoted here by c3=2, c1=2,c�1=2, and c�3=2 where the sub-indices correspondto the rows of the representation G8 generated byp3=2 atomic states. The G7 states are labeled f1=2and f�1=2 in correspondence with p1=2 atomicstates. These states are obtained using theClebsch–Gordan coefficients for the doublegroup %Td.Even though the Hamiltonian of the bound hole

lacks spherical symmetry, the quantum numbersascribed to these states have well defined physicalmeanings whose significance will be shown in moredetail later. We call them ‘‘pseudo angular momen-tum quantum numbers’’.The G8 and G7 representations of %Td are of type

c in the classification of Frobenius and Schur[33,34] so that the subspace spanned by fcMg ðM¼ �3=2;�1=2Þ and that generated by the time-reversed states fYcMg are identical. The same istrue for ffmg and fYfmg ðm ¼ �1=2Þ.In the presence of a magnetic field B, the new

symmetry group contains the common elements of%Td and %C1h, the axis of rotation of the latter beingalong B. For B parallel to ½0 0 1�, ½1 1 1�, or ½1 1 0�the symmetry group is %S4, %C3 or %Cs, respectively.

For a direction of B other than a high symmetrydirection of the crystal the group is %C1.Application of a magnetic field splits the G8 and

G7 levels into non-degenerate states. The wavefunctions cM and fm are approximate eigenfunc-tions of the Zeeman interaction for B parallel to½0 0 1�. Fig. 3 shows the selection rules for transi-tions between the Zeeman levels caused by aninteraction in the form of a second rank symmetrictensor as, for example, is appropriate for Ramanor electric quadrupole interactions. The order ofthe levels shown in Fig. 3 is that deducedunambiguously from an appeal to the experimen-tal results reported in Section 3. The left part of thefigure shows transitions from initial states cM tofinal states fm; those for which d ¼ m�M ¼0;�2 are labeled 1, 2, 3, and 4 and those for d ¼�1 are designated by the symbols 10; 20; 30, and40. The right-hand side of the figure shows theselection rules for transitions within the G8 multi-plet, i.e., the Raman-electron-paramagnetic-reso-nance transitions. The lines corresponding to achange in quantum number M by DM ¼ �2 aredesignated E1 and E2 while those for which DM ¼�1 are labeled E10 and E20. It is important to notethat the transitions 3=2$ �3=2 and 1=2$ �1=2

Fig. 2. The Stokes and anti-Stokes components of the 1sðp3=2Þ ! 1sðp1=2Þ Raman transition labeled D0 in a natural type IIb (D1)

diamond at 46 K recorded in the right-angle scattering geometry x0ðzzþ zx0Þy0; x0 jj ½1 1 0�; y0 jj ½1 %1 0�, and z jj ½0 0 1� and a 13C

diamond at 37 K, in backscattering. Both spectra were excited with the 4765 A Arþ laser line.

H. Kim et al. / Physica B 302–303 (2001) 88–100 91

within the G8 multiplet are forbidden by timereversal symmetry. We emphasize that thisrestriction holds only for transitions within asingle multiplet belonging to an irreducible repre-sentation of type c, not for transitions betweendistinct multiplets. A relaxation of this selectionrule due to mixing of G8 and G7 states by amagnetic field has negligible consequences.Valuable information can be extracted from a

microscopic model of the energy levels of anacceptor in an elemental group IV semiconductor.To make use of this model we derive the form ofthe Raman scattering cross section in the limit inwhich the energy of the incident photon is muchlarger than the ionization energy of the scatteringobject while its wavelength is long compared to thesize of the acceptor. These conditions are clearlysatisfied for shallow acceptors in diamond byincident radiation in the visible region.In this regime the differential scattering cross

section for a transition from a state jn0i of theacceptor to a state jni while a photon of angularfrequency o and polarization #e is absorbed andreplaced by one with angular frequency o0 and

polarization #e0 is

dsdO0 ¼

n2

"2c2

� �2 o0

o

� �3

� n ½#e0* � d; ½#e � d;H�� þO1

o

� ���������n0

� ���������2

: ð2Þ

The Hamiltonian operator for the acceptorstates in the field B is

H ¼ H0 þH 0; ð3Þ

where H0 is given in Eq. (1) and

H 0 ¼1

3D0 �

2

3D0I � S þ mBðg1I þ g2SÞ � B

þ Q1ðmBBÞ2 þQ2m2BðB � IÞ2

þQ3m2BX3i¼1

B2i I2i : ð4Þ

Here S is the spin operator of the hole, mB theBohr magneton, g1 and g2, are orbital and spin g-factors, respectively, and Q1; Q2; Q3 are con-stants (of dimensions equal to reciprocal energy).The Hamiltonian H ¼ H0 þH 0 contains all possi-ble contributions linear and quadratic in B and pconsistent with the site symmetry.Evaluation of the double commutator in Eq. (2)

with the Hamiltonian in Eq. (3) gives (notingo0 o)

dsdO0 ¼

n2e2

mc2

� �2jhnj#e0* � a � #ejn0ij2; ð5Þ

where the tensor a is

a ¼ 3ðg3 � g2Þð2Ið2Þ0 X

ð2Þ0 þ ðI ð2Þ2 þ I ð2Þ�2ÞðX

ð2Þ2 þ X ð2Þ

�2ÞÞ

� 6g3X2k¼�2

ð�1ÞkI ð2Þk Xð2Þ�k: ð6Þ

Here I ð2Þk ðk ¼ 0;�1;�2Þ are the irreduciblecomponents of the second-rank tensor operatorfIi; Ijg, and X ð2Þ

k are the corresponding tensors forthe dyadics xixj þ xjxi.The last term in Eq. (6), proportional to g3

alone, is the product of 6g3 and the scalar productof the tensors I ð2Þ and X ð2Þ and is thus sphericallysymmetric, a result which proves advantageous inthe calculation of the Raman tensors for severaldirections of B. The first term, proportional toðg3 � g2Þ, has only cubic symmetry, so that the

Fig. 3. Schematic energy level diagram showing the Zeeman

effect of the D0 transition ½1sðp3=2Þ :G8 ! 1sðp1=2Þ :G7�, of anacceptor-bound hole in a group IV semiconductor. The figure

shows the Raman allowed transitions ð1; 2; 3; 4; 10; 20; 30; 40Þ fromthe M ¼ 3=2; 1=2;�1=2;�3=2 Zeeman sublevels of G8 to them ¼ 1=2;�1=2 sublevels of G7 corresponding to d ¼m�M ¼ 0; � 1, and �2. Also shown are the Raman-EPRtransitions (E1; E2; E10, and E20) allowed within the G8multiplet. The spacings for the G8 multiplet have been drawn torepresent those observed for B jj ½0 0 1�.

H. Kim et al. / Physica B 302–303 (2001) 88–10092

departure from isotropy is measured by thedifference of the Luttinger parameters g2 and g3.Eqs. (5) and (6) establish that the only prominentRaman transitions in the high frequency limit arethose between levels belonging to the G5 � G6 ¼G8 þ G7 ground states. Experiments demonstratethat for boron in diamond, the extreme anisotropyof the valence band of diamond, i.e., g34g2, allowsus to distinguish between symmetry allowed strongand weak transitions.We define a total ‘‘pseudo angular momentum’’

operator

J ¼ I þ S; ð7Þ

which corresponds to levels with J ¼ 3=2ðG8Þ andJ ¼ 1=2ðG7Þ. The interaction H 0, to first order in Bbecomes

H 0 � 54D

0 � 13D

0JðJ þ 1Þ þ gJmBB � J : ð8Þ

Application of the Wigner–Eckart theorem showsthat

g3=2 ¼13ð2g1 þ g2Þ ð9Þ

and

g1=2 ¼13ð4g1 � g2Þ: ð10Þ

The functions cM and fm, however, are noteigenvectors of H 0. For B jj ½0 0 1�, the states c1=2and f1=2 (belonging to G8 of %S4) and c�1=2 andf�1=2 (belonging to G7 of %S4) are mixed by H 0.A detailed analysis shows that the difference in

the energies of lines 30 and 20 is �2mBBg1 while thecorresponding difference between lines 40 and 1 is�mBBð2g1 þ g2Þ. These results are used in thedetermination of the g-factors from the experi-mental data.The Raman tensors for the eight transitions 1;

2; 3; 4; 10; 20; 30; 40 for B jj ½0 0 1� are given in Table 2and the factors f�; g�; h� and k takeinto account the mixing of the G8 and G7 statesby the Zeeman interaction [31]. They are equalto unity for B ¼ 0 and detailed numerical evalua-tion show they range between 0.85 and 1.25 at B ¼6 T [35].

3. Experimental results and discussion

3.1. General considerations and level ordering

In order to interpret the experimental results,the level ordering of the Zeeman sublevels of G8and G7 hole states as displayed in Fig. 3 is adopted.The rotational invariance of the Zeeman interac-tion mBðg1I þ g2SÞ � B shows that the level order-ing of the states characterized by M ¼ 3

2;12;

�12; � 32 and m ¼ 1

2; � 12 is independent of the

direction of B with respect to the cubic axesh1 0 0i. The quadratic terms in B in Eq. (4) aresmall and modify only slightly the separation ofthe sublevels. With the sign convention in Eq. (4),it is expected that g2 is approximately equal to �2,the intrinsic g-factor for a free hole; similarly, g1,the orbital g-factor, is also expected to be negative.

Table 2

The Raman tensors for transitions from the Zeeman levels of

the G8 manifold to those of the spin–orbit split G7 states. Themagnetic field B is along ½0 0 1� and the components of thetensors are referred to the cubic axes. The tensors appear in

pairs of simply related tensors. In each case the upper and lower

signs correspond to the first and second lines displayed on the

left column

Transition Raman tensor

1, 4ffiffiffi6

pf�

�g2 ig3 0ig3 �g2 00 0 0

0@

1A

2, 3 �ffiffiffi2

pg2g�

�1 0 00 �1 00 0 2

0@

1A

10, 40 g3h�ffiffiffiffiffiffiffiffiffiffiffið3=2Þ

p 0 0 10 0 �i1 �i 0

0@

1A

20, 30 �ð3=ffiffiffi2

pÞkg3

0 0 10 0 �i1 �i 0

0@

1A

E1;E2 �ffiffiffi3

ph�

g2 ig3 0ig3 �g2 00 0 0

0@

1A

E01;E

02 �

ffiffiffi3

pg3f�

0 0 10 0 i1 i 0

0@

1A

H. Kim et al. / Physica B 302–303 (2001) 88–100 93

While we temporarily adopt this view as a workinghypothesis reflected in the level ordering in Fig. 3,the experimental results for B jj ½1 1 1� usingcircularly polarized light unambiguously establishthe correctness of this choice.The Zeeman effect with B jj ½0 0 1� provides a

suitable starting point for comparing theory andexperiment. Table 2 shows that, in the scatteringconfiguration yðzxþ zyÞz [36] with Bjj½0 0 1�jjz,only the 10; 20; 30, and 40 Zeeman components aswell as the Raman-EPR transitions E10 and E20

are allowed. In Fig. 4, at B ¼ 4 T, close to oL, thefrequency of the laser line, four lines are clearlyresolved both in the Stokes and the anti-Stokesparts of the spectrum; two of these, labeled ‘‘TA’’and ‘‘LA’’, are identified as the Brillouin compo-nents associated with transverse and longitudinalacoustic phonons propagating along ½0 %1 1�. E10

and E20, the Raman-EPR lines, appear distinctly

resolved. Lines 10; 20; 30, and 40, the Zeemancomponents of D0, are also displayed in the figure.The results in Fig. 5 for yðxxþ xyÞz are recordedat 6 T. In this scattering geometry, only lines 1–4,E1 and E2 are allowed. The higher magnetic fieldemployed in the measurement prevented the TAand LA Brillouin components from obscuring E1and E2. The appearance of lines 1 and 4 and theabsence of 2 and 3 follow from Table 2 withg25g3.In Fig. 6 we exhibit the results with B jj ½1 1 1� in

backscattering, the incident light along B beingcircularly polarized #r� and the scattered analyzedfor #r0

�. Denoting by#n; #g, and #f unit vectors

parallel to ½1 1 %2�, ½%1 1 0�, and ½1 1 1�, respectively,we have #r� ¼ ð#g � i#nÞ=

ffiffiffi2

pand #r0

� ¼ ð#n � i#gÞ=ffiffiffi2

p: The experimental results for the four config-

urations %fð #r�; #r0�Þf are displayed in Figs. 6(a)–(d).

In Fig. 6(a), ð #r�; #r0�Þ, transitions 1, 3

0, and 40

appear, whereas in Fig. 6(b), one sees 10; 20; and 4.Using the Raman tensors appropriate forB jj ½1 1 1� [31], the Raman amplitudes calculatedfor the four polarization configurations are dis-played in Table 3. The change in the angularmomentum of the radiation upon scattering, DS, is0 for ð #rþ; #r

0�Þ and ð #r�; #r

0þÞ, 2 for ð #rþ; #r

0þÞ and �2

for ð #r�; #r0�Þ. The second column in the table lists

Fig. 5. The Zeeman spectrum of the D0 transition recorded as in

Fig. 4 except for B ¼ 6 T, the scattering geometry being

yðxxþ xyÞz. Lines 1, 4, E1, and E2 are labeled according toFig. 3.

Fig. 4. The Zeeman spectrum of the D0 line of a boron acceptor

in a man-made diamond recorded with a piezoelectrically

scanned ð5þ 4Þ tandem Fabry–P!erot interferometer. The

spectrum was recorded at 4 T with B jj ½0 0 1� in the right anglescattering geometry, the incident and the scattered directions

being along y jj ½0 1 0� and z jj ½0 0 1�. The measurements weremade at 11 K as estimated from the intensity ratio of the

transverse acoustic (TA) and longitudinal acoustic (LA) Stokes

Brillouin components with respect to their anti-Stokes counter-

parts. oL is the unshifted frequency of the exciting laserradiation at lL ¼ 5145 (A and G identifies its ‘‘ghost’’ due to the

leakage in the tandem operation. D0 labels the zero field position

of the G8 ! G7 transition; E10 and E20, the Raman-EPR

transitions; and 10; 20; 30, and 40 the Zeeman components of D0

labeled according to Fig. 3.

H. Kim et al. / Physica B 302–303 (2001) 88–10094

the change in the pseudo-angular momentum ofthe hole, d, for each of the eight Zeemancomponents. An inspection of the table revealsthat for lines 1–4, DS þ d ¼ 0 consistent withconservation of ‘‘angular momentum’’. In con-trast, lines 10 and 20 in ð #rþ; #r

0þÞ and 3

0 and 40 inð #r�; #r

0�Þ appear as a consequence of the departure

from spherical symmetry ðassociated with g2 6¼ g3Þ.The transitions 2 and 3 allowed in both ð #rþ; #r

0�Þ

and ð #r�; #r0þÞ conforming to DS ¼ d ¼ 0 are shown

in Figs. 6(c) and (d). Had we employed linearlypolarized light and analyzed the scattered lightlinearly, the primed lines and their unprimedcounterparts would appear together in the samescattering configuration preventing their clearresolution.The appearance of the eight Zeeman transitions

in the four polarization configurations conformsto the choice of the level ordering in Fig. 3. Wenote that the interval between 40 and 4 is equal tothat between 1 and 10 as can be seen from theseparation between the vertical lines in the figure.And since the initial states of 40 and 4 and of 1 and10 are þ32 :G8 and �32 :G8, respectively, clearly theirspacings represent the splitting of the G7 Zeemanlevels. Since 40ð1Þ occurs at a higher energy than

Table 3

Theoretical Raman amplitudes of the Zeeman components of D0 with B jj ½1 1 1�, observed in the backscattering geometry along B

employing circularly polarized incident #s� and scattered #s0� radiation defined in the text. Amplitudes identified in boxes satisfy theconservation of ‘‘angular momentum’’, i.e., sum of the change in the photon angular momentum ðDSÞ and d ¼ m�M. The allowedprimed transitions, which do not conform to the conservation of ‘‘angular momentum’’, arise from the departure from spherical

symmetry of the valence band ðg2 6¼ g3Þ

Line ð #sþ; #s0�Þ ð #sþ; #s0þÞ ð #s�; #s0þÞ ð #s�; #s0�Þ

DS ! 0 2 0 �2d#

1 2 0 0 0 2ifþð2g3 þ g2Þffiffiffiffiffiffiffiffiffiffiffið2=3Þ

p2 0 �i

ffiffiffi2

pg3g� 0 i

ffiffiffi2

pg3g� 0

3 0 iffiffiffi2

pg3gþ 0 �i

ffiffiffi2

pg3gþ 0

4 �2 0 2if�ð2g3 þ g2Þffiffiffiffiffiffiffiffiffiffiffið2=3Þ

p0 0

10 1 0 ð2ih�=ffiffiffi3

pÞðg3 � g2Þ 0 0

20 1 0 �2ikðg3 � g2Þ 0 0

30 �1 0 0 0 2ikðg3 � g2Þ

40 �1 0 0 0 �ð2ihþ=ffiffiffi3

pÞðg3 � g2Þ

Fig. 6. The Zeeman components of D0, observed in back-

scattering along B jj ½1 1 1�, at 7 T and 5 K, the incident

(scattered) light polarized (analyzed) using circular polariza-

tions defined in the text. Spectra excited with the 6471 (A Krþ

laser line and recorded on a CCD-based triple grating spectro-

meter. The lines are labeled according to Fig. 3.

H. Kim et al. / Physica B 302–303 (2001) 88–100 95

4ð10Þ, the ordering of the G7 Zeeman levels is thatadopted in Fig. 3. However, had the ordering ofthe G8 sublevels been reversed, the lines in Fig. 6(a)would be relabeled 40, 30, and 1 in order ofincreasing energy while in Fig. 6(b) they would be4, 20, and 10, and again 1ð40Þ occurs at a higherenergy than 10ð4Þ demonstrating that the levelordering of �12 :G7 and þ

12 :G7 remains the same as

dictated by the experiment. Hence g1=2 is positive.[Eqs. (8) and (10)]. We note that the labeling of 30

in Fig. 6(a) and that of 20 in Fig. 6(b) are fixed bythe selection rules in Table 3, irrespective of thelevel ordering of G8. The occurrence of 30 at ahigher energy compared to that of 20 thendemonstrates that the level ordering in Fig. 3 isthe only one consistent with experiment. It is alsonoteworthy that the separation between the pairsð3; 30Þ and ð20; 2Þ must be equal and correspond tothe G7 splitting; thus a relabeling of lines 2 and 3 inFigs. 6(c) and (d) as 3 and 2 would result in anunacceptable contradiction. In conclusion, theexperimentally observed selection rules and linepositions establish unambiguously the level order-ing in Fig. 3.

3.2. Magnetic field dependence of the Zeemancomponents of D0

In Figs. 7 and 8 we display the experimentallyobserved positions of the Zeeman components ofD0 for B jj ½0 0 1� and ½1 1 1�, respectively, asfunctions of B. The transition energies arecalculated by solving the secular equations govern-ing the mixing of c1=2 with f1=2 and of c�1=2 withf�1=2. In Figs. 7 and 8 the solid lines represent fitsof the calculated expressions thus obtained tothose experimentally observed. In this manner, weobtain values for g1, g2, and Q2 þQ3 with the datafor B jj ½0 0 1� and g1, g2, and Q2 for B jj ½1 1 1�. Theenergy spacing between lines 30 and 20, on the onehand, and that between 40 and 1, on the other, arepredicted to be linear in B and given by �2mBBg1and �mBBð2g1 þ g2Þ, for B jj ½0 0 1� and B jj ½1 1 1�,respectively. The experimental results for thesespacings plotted as a function of B are displayed inFig. 9. The magnetic field dependence of thesespacings is clearly linear with B and the linear leastsquares fits yield g1 ¼ �0:373� 0:007 and

Fig. 7. The energies of lines 1; 10; 20; 30; 4, and 40 as functionsof magnetic field, with B jj ½0 0 1�; lines 2 and 3, though allowed,are too weak to be observed (see text). The solid lines are

theoretical, calculated using g1 ¼ �0:373; g2 ¼ �2:112;Q2 þQ3 ¼ �0:159 cm, energy being expressed in cm�1. The

inset displays the magnetic field dependence of the four Raman-

EPR lines. The curves in the inset passing through the

experimental points, drawn as guides to the eye, are consistent

with an intrinsic Jahn–Teller splitting.

Fig. 8. The energies of lines 10; 1; 2; 20; 30; 3; 4, and 40 asfunctions of B with B jj ½1 1 1�. The solid lines are drawn with g1and g2 as in Fig. 7 and Q2 ¼ �0:013 cm.

H. Kim et al. / Physica B 302–303 (2001) 88–10096

g2 ¼ �2:112� 0:021. The data shown in Figs. 7and 8, using g1 and g2 thus deduced, lead to Q2 ¼�0:013� 0:008 cm and Q3 ¼ �0:146� 0:016 cm.As we have demonstrated [37], the D0 line at B ¼ 0exhibits a spontaneous splitting of ð0:80� 0:04Þcm�1 attributed to a static Jahn–Teller effect. Theextrapolation of the Zeeman lines to a single zero-field position at D0 does not reflect this separation,since the minimum magnetic field at which theZeeman levels could be fully resolved is 1 T. Asshown in Ref. [37], the Jahn–Teller splitting resultsfrom a tetragonal distortion along h1 0 0i whichimplies that at zero field the states �3

2 :G8 and�12 :G8 have different energies. This feature is

brought out distinctly in the inset in Fig. 7, wherethe positions of the Raman-EPR lines E1, E2, E10,and E20 are shown. The extrapolations of the highfield values of the positions of E2 and E20 to B ¼ 0converge to a value different from that of thesimilarly obtained extrapolations of E1 and E10.The Hamiltonian matrix for the G8 þ G7 mani-

fold in the states cM and fm when B is appliedalong h1 1 0i, factors into two 3� 3 submatricesbecause, on the one hand, c�3=2; c1=2, and f1=2belong to the G3 irreducible representation of %Cs,while, on the other hand c3=2; c�1=2, and f�1=2belong to G4. With parameters determined fromFigs. 7 and 8, numerical diagonalization allows

one to obtain the eigenvectors and eigenvalues ofH 0 as functions of B.Fig. 10 shows the Zeeman components of D0 for

four different polarization configurations accessi-ble with Bjjx0jj½1 1 0�, y0 jj ½%1 1 0�, and z jj ½0 0 1�, at6 T and 5 K. The intensities based on thecalculated Raman amplitudes displayed in Table 4are fully verified by the experimental results. Wenote that, in the ðx0zÞ polarization, all the fourprimed lines (10–40) appear, whereas in ðy0y0Þ thefour unprimed lines (1–4) are observed. In ðx0y0Þ allthe primed lines are allowed, but their intensities,being proportional to g22, are well below the level ofdetection. Similarly, in ðy0zÞ lines 1 and 4 appearwith considerable strength, in agreement with Table4. The table predicts that lines 2 and 3 should beabsent in ðy0zÞ; however, the small admixture ofc�3=2 makes them allowed but with negligibleintensity as verified by the experimental results.

Fig. 9. The magnetic field dependence of the spacing between

lines 30 and 20 as well as that between 40 and 1. The straight lines

passing through data for B jj ½0 0 1� and ½1 1 1� permit adetermination of the g-factors g1 and g2.

Fig. 10. Raman spectra of the Zeeman transitions of D0 with

B jj ½1 1 0� recorded in the right angle scattering geometryallowing the four polarization configurations zðy0zÞx0; zðy0y0Þx0;zðx0zÞx0, and zðx0y0Þx0. B ¼ 6 T and a nominal temperature of5:0 K. The spectral analysis was performed with a CCD-basedtriple grating spectrometer; lL ¼ 6471 (A line of a Krþ laser.

The Zeeman components are labeled 1; 10; 2; 20; 3; 30 and 4;40 following the energy level scheme shown in Fig. 3. The

feature labeled with a ‘‘þ’’ mark is of unknown origin.

H. Kim et al. / Physica B 302–303 (2001) 88–100 97

3.3. Extreme mass anisotropy of the valence bandof diamond

In the theory of the electronic states of theacceptor-bound holes in the elemental semicon-ductors, the departure from the spherical approx-imation needs to be established as a prerequisitefor the calculation of eigenvalues and eigenvectorsof the bound states. From Eq. (6) we note that thespherical model holds when g2 ¼ g3. By the sametoken if g2 is negligible compared to g3, as appearsto be the case for diamond [31], the contribution ofthe cubic part of the acceptor Hamiltonianbecomes significant (See, for example, Eq. ð3:65Þin Ref. [2]). Correspondingly, the polarizabilitytensor governing D0 (Eq. (6)) exhibits a similarseparation. The experimental manifestation of thelack of isotropy in the Raman spectrum of D0 is theappearance of lines that would otherwise be absentunder strict pseudo-angular momentum conserva-tion. From Table 3 it is immediately clear thatlines 10; 20; 30; and 40 would be absent in Fig. 6had g2 equalled g3.A quantitative measure of ðg2=g3Þ is accessible

from the results in Table 4 and Fig. 10 for B jj½1 1 0� at 6 T. The intensity ratios ðI3=I4Þ : ðI30=I40 Þand ðI2=I1Þ : ðI20=I10 Þ are temperature independentbecause the primed and the corresponding un-primed line have the same initial state and hencethe thermal factors cancel.The calculated ratios are 0.6 and 1.2 for B ¼ 6 T

and g2 0:1g3. The procedure of line fitting in thiscontext yielded 0.65 for the first and 1.23 for thesecond. From the experimental ratios one can

directly deduce ðg2=g3Þ using the theoreticallycalculated Raman amplitudes as linear combina-tions of g2 and g3 given in Table 4. The theoreticalvalue of ðI3=I4Þ : ðI30=I40 Þ deduced from the table is

3g3 � g23ðg3 þ g2Þ

gþhþf�k

� �2: ð11Þ

From the values of the mixing parametersgþ; hþ; f�, and k calculated [35] for B ¼ 6 T,ðgþhþ=f�kÞ ¼ 0:885. Equating the ratio (11) aboveratio to its experimental value of 0.65, we deducethat

g2g3

ffi 0:07:

Similarly, the ratio ðI2=I1Þ : ðI20=I10 Þ is

3g3 � g23ðg3 þ g2Þ

g�h�fþk

� �2: ð12Þ

We find ðg�h�=fþkÞ ¼ 1:243 at 6 T and, equatingthe ratio (12) to the experimental value 1.23, wefind

g2g3

¼ 0:09:

We remark that use of Table 4 implies theneglect of mixing of c�3=2 into the pairsðc�1=2;f�1=2Þ. When this additional mixing isincluded, the values of ðg2=g3Þ turn out to be 0.13and 0.10, respectively. We conclude that ourearlier, approximate estimate of ðg2=g3Þ to be 0:1 is validated by the more detailed analysis.Furthermore, one can assert that g2 and g3 have thesame sign.

Table 4

Raman amplitudes for B jj ½1 1 0�; #x0 jj ½1 1 0�; #y0 jj ½%1 1 0�; z jj ½0 0 1�

ðy0zÞ ðy0y0Þ ðx0zÞ ðx0y0Þ

1 iffiffiffi6

pg3 fþ

ffiffiffiffiffiffiffiffiffiffiffið3=2Þ

pðg3 þ g2Þfþ 0 0

2 0 �ð3g3 � g2Þg�=ffiffiffi2

p0 0

3 0 ð3g3 � g2Þgþ=ffiffiffi2

p0 0

4 iffiffiffi6

pg3 f� �

ffiffiffiffiffiffiffiffiffiffiffið3=2Þ

pðg3 þ g2Þf� 0 0

10 0 0ffiffiffiffiffiffiffiffiffiffiffið3=2Þ

pg3h� �i

ffiffiffiffiffiffiffiffiffiffiffið3=2Þ

pg2h�

20 0 0 �ð3=ffiffiffi2

pÞg3k ð3i=

ffiffiffi2

pÞg2k

30 0 0 �ð3=ffiffiffi2

pÞg3k �ð3i=

ffiffiffi2

pÞg2k

40 0 0ffiffiffiffiffiffiffiffiffiffiffið3=2Þ

pg3hþ i

ffiffiffiffiffiffiffiffiffiffiffið3=2Þ

pg2hþ

H. Kim et al. / Physica B 302–303 (2001) 88–10098

4. Concluding remarks

The unique optical properties of diamond andthe present state of Raman spectroscopy in thevisible in comparison with that in other spectralranges have enabled a comprehensive and precisestudy of D0, an important electronic transitionassociated with acceptors in elemental semicon-ductors. This transition between the spin–orbitsplit 1s ground states of the acceptor-bound hole isdescribed within the framework of the effectivemass theory formulated in terms of the Luttingerparameters, g2 and g3, of the valence bandmaximum. Group theory in combination withthe theory of electronic Raman scattering enabledus [14] to deduce the extreme mass anisotropy ofthe valence band maximum of diamond. Thisemerges in an unambiguous fashion in the absenceof Zeeman components in scattering configura-tions, allowed by symmetry but of negligibleintensity due to the significantly smaller magnitudeof g2 in comparison to that of g3. In the presentinvestigation, we have observed all the eightZeeman components of D0 and, in addition,discovered the four Raman transitions within theG8 ground state multiplet. This has been experi-mentally possible thanks to the high resolution,the discrimination against parasitic radiation aswell as the significantly enlarged free spectralrange of a multi-passed, multi-scanned, tandemFabry–P!erot spectrometer. The theoretically de-duced selection rules in terms of appropriatepolarizability tensors have permitted the selectionof optimum scattering configurations which yieldunambiguous assignments of quantum numbers.The magnitude and sign of the g-factors revealedby the level ordering of the Zeeman sublevels; thecontributions to the energy eigenvalues quadraticin B making the G8 sublevels unequally spaced andthereby generating the four distinct Raman-EPRlines; the comparison of the relative intensities ofthe experimentally observed Zeeman componentswith those theoretically predicted allowing thedetermination of the magnitude and sign of(g2=g3Þ; . . . these are the definitive results ensuingfrom the present study.A fascinating feature of the selection rules

obeyed by the Zeeman–Raman transitions is the

intimate relationship between time-reversalsymmetry, magnetic field direction and thecircular polarization of the incident and=or thescattered radiation, and the mutual relationshipbetween the Stokes and anti-Stokes spectra(A previous observation of a closely related effectin II–VI diluted magnetic semiconductors has beenreported by Petrou et al. [38]). This emerges in aparticularly striking manner in the spectrarecorded with B jj z, incident light propagatingalong y and scattered along z and analyzed as #r0

�.In ðz #r0

�Þ only the anti-Stokes components of30; 40; E10, and E20 appear, whereas 10 and 20

are observed only in the Stokes spectrum.In contrast, ðz #r0

þÞ allows 30; 40; E10 and E20 as

Stokes components and 10 and 20 as anti-Stokes.(See Fig. 4 in Ref. [39].) The microscopic selectionrules predict this mutual exclusion. In ðz #r0

�Þ; E10

and E20, absent in the Stokes, appear in the anti-Stokes spectrum with intensities proportional tog23 f

2� and g23 f

2þ, respectively, f�=fþ being 0:9 at

B ¼ 4 T; exactly the opposite is the case in ðz #r0þÞ;

E10 and E20 being forbidden in the anti-Stokesbut allowed in the Stokes, also with the samerelative intensities.Finally, we draw attention to the spontaneous

lowering of the symmetry of G8 due to Jahn–Tellersplitting reported by us recently [37]; in the inset ofFig. 7, the convergence of E2 and E20 and of E1and E10 as B tends to zero, is an additionalmanifestation of the Jahn–Teller effect.

Acknowledgements

The authors acknowledge support fromthe National Science Foundation GrantNo. DMR 98-00858 at Purdue Universityand from the US Department of Energy, BESMaterial Sciences Grant No. W-31-109-ENG-38at Argonne National Laboratory.

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