Magnetoabsorption experiments on the A exciton in InSe have been performed in search of a unified correspondence relation between low and high magnetic field levels of a hydrogenlike state. Results show that the nodal-surface conservation rule of Shinada et al. is a more appropriate representation of this relationship than the so-called noncrossing rule.磁場中の水素原子の固有状態は解析的に解くことが不可能であるために、物理学上の永年の問題となってきた。磁場の強さが無限大の極限では無磁場のときとは量子数の全く異なる漸近解を持つことが知られているが、この強磁場極限準位に無磁場状態がどのように接続するかという疑問が問題の核心である。本研究では、InSeの励起子が物理的に理想に近い水素原子類似体であり、かつ固体効果によって水素原子に対するよりも数桁低い磁場で強磁場極限に到達しうることに着目して磁気光吸収を測定した結果、準位の接続に関して波動関数の節面数保存仮説を強く支持するデータが得られた

Date, Muneyuki

Hori, Hidenobu

Kuroda, Noritaka

Nishina, Yuichiro

黒田, 規敬

仁科, 雄一郎

堀, 秀信

伊達, 宗行

Kumamoto University Repository

熊本大学学術リポジトリValidity of the conservation rule of nodalsurface numbers in the wave function ofhydrogenic exciton in magnetic fieldjournal orpublication titlePhysical Review Lettersvolume 48number 17page range 1215-1219year 1982-04-26その他の言語のタイトル磁場中の水素原子様励起子の波動関数についての節面数保存則の妥当性URL http://hdl.handle.net/2298/9610doi: 10.1103/PhysRevLett.48.1215VALIDITY OF THE CONSERVATION RULE OF NODAL SURFACE NUMBERS INTHE WAVE FUNCTION OF HYDROGENIC EXCITON IN MAGNETIC FIELDNoritaka Kuroda, Yuichiro Nishina, Hidenobu Hori, and Muneyuki DateReprinted from PHYSICAL REVIEW LETTERS Vol. 48, No. 17, 26 April 19821215Volume 48, Number 17 PHYSICAL REVIEW LETTERS 26 April 1982Validity of the Conservation Rule of Nodal Surface Numbers in the Wave Functionof Hydrogenic Exciton in Magnetic FieldNoritaka Kuroda and Yuichiro NishinaThe Research Institute for Iron, Steel and Other Metals, Tohoku University, Sendai 980, JapanandHidenobu Hori and Muneyuki DateFaculty of Science, Osaka University, Toyonaka, Osaka 560, Japan(Received 11 December 1981)Magnetoabsorption experiments on the A exciton in InSe have been performed in searchof a unified correspondence relation between low and high magnetic field levels of a hydro-genlike state. Results show that the nodal-surface conservation rule of Shinada et at. isa more appropriate representation of this relationship than the so-called noncrossing rule.PACS numbers: 78.20.Ls, 71.35.+ ZIt has been recognized as an everlasting problem of both theoretical and experimental studiesto solve for the energy eigenvalues of Wannierexciton in the intermediate magnetic field range,y = #o)c/2(ft*=l, where Hu)c is the cyclotron energy of the exciton and <R* the effective Rydbergconstant. For tracing the transition in characters of hydrogenic exciton levels with respect tothe field, Kleiner1 has proposed a topologicalconcept that for any strength of the field the number of nodal surfaces of the wave function for agiven level has to be kept constant. Elliott andLoudon2 were the first to refer to this rule intheir theoretical analysis of absorption spectrumin a high magnetic field of y > 2. Subsequently,Shinada, Akimoto, Hasegawa, and Tanaka (SAHT]Phave modified Kleiner's concept in such a waythat the excitonic states with a common magneticquantum number m and a parity would interactwith each other in the intermediate field range iftheir total (polar and radial) nodal-surface numbers are equal, and derived a modification ofKleiner's correspondence relations between low-and high-field levels. Boyle and Howard (BH),4on the other hand, have proposed that the correspondence should be established only on the© 1982 The American Physical Society 1215Volume 48, Number 17 PHYSICAL REVIEW LETTERS 26 April 1982basis of the noncrossing rule for such levels,thus proposing a new correspondence scheme.There have been a number of works, especiallyof Wagner and McCombe,5 on the magnetospec-troscopy of shallow donors in various semiconductors in which they claim quantitative consistency with BH's correspondence relations. Itshould be pointed out, however, that BH*s relations are identical with SAHT's for Is, 2pOt±1,%Po9±i> 4pOt±lf etc. Thus no optical transitioninvolving any of these levels can discriminateSAHTs relations from BH's. The magnetospec-troscopy of a donor is inherently not an appropriate probe in search of the level correspondence between low- and high-field levels.An essential spectroscopic difference betweenSAHT's relations and BH's is .found in the behavior of nd0 levels. According to SAHT, a levelwhich originates from a d0 should cross over theN = 0 Landau edge to track with the N = 1 edge ina high-field limit: The 3d0 level, for example,forms a hybridization with the 2s level so thatthe upper branch connects to the (100+) state,where the notation of the high-field state implies(Nmv*) of SAHT. According to BH's relations,on the other hand, all the m * 0 bound statesshould lie below the N = 0 Landau edge at anyfield strength; the (100+) state develops out ofcontinuum states of the exciton as the field increases. It is clear that a magnetospectroscopyof an exciton is much more appropriate than thatof an impurity for tracing such levels experimentally. It is for this reason that we report onthe magnetoabsorption of the A exciton in InSe infields up to 39.5 T. The results show that themost reasonable correspondence is given bySAHT.One cannot be too careful on the choice of thesubstance with its electronic band structure assimple as possible. According to Lipari andAltarelli,6 the diamagnetic shift of the groundstate in the group IV, III-V, and II-VI compoundscannot be described in terms of a simple hydro-genie exciton model, but exhibits a complicatedbehavior due to the cubic anisotropy, the k -linearterm, and the intervalence band couplings. Tl-halide compounds have been examined by Koba-yashi7 up to the field of 15 T (y = 0.25). Only theIs and 2s lines in T1C1 show the field dependencesatisfying the noncrossing rule. But there areseveral levels which obviously cross over theLandau edges (see Fig. 5 of Ref. 7). The pres-cence of a strong exciton-phonon coupling as wellas a comparatively large exciton reduced massmakes further detailed investigation difficult. Atypical layer semiconductor, GaSe, has been amaterial of great interest for both theoretical3and experimental8'9 analyses. Our recent magneto-optical absorption experiments8 of the Bexciton in GaSe up to a field of 38 T {y = 0.7)clearly shows that the line originating from the2s level crosses over the N = 0 Landau edge at y= 0.65. Furthermore, Miura et al.g have measured the absorption spectra in the extremely high-field range between 50 and 170 T. In fields above- 60 T, they have found an unidentified line belowthe N= 1 Landau edge which seems to extrapolateto our 2s-like line toward lower field. Underthese circumstances, the experiment on the details of level connections around y = 1 is highlydesirable as has been pointed out by Miura et al.To the best of our knowledge, the only casewhich barely satisfies all of the physical andtechnical requirements as mentioned above isthe A exciton in 3#-InSe. The second valence-band edge, A, is well decoupled from the B andC band edges since the crystalline field anisotropy is as large as - 1.33 eV in the quasicubicmodel and the spin-orbit coupling constant of theSe atom is 0.47 eV, both of which are much larger than (ft* = 15 meV.10 Although the reduced-massparameter is somewhat anisotropic, the anisotropy parameter for the exciton, €±IA±/€\\H\\, isvery close to unity.11 The above value of (ft* together with ]LtJL=0.12m0 (Ref. 11) gives y = l fora field of -30 T which may be reached by ourpulsed method without considerable difficulty.A single crystal of 3£-InSe has been grown bythe Bridgman method. The sample is cleaved toa thickness of 1-2 /im. Measurements have beenperformed by the same method as described inRef. 8.Two typical sets of transmission spectra measured at 4.2 K in the Faraday configuration withmagnetic field parallel to the c axis are shownin Figs. l(a) and l(b). The Is line which lies at2.5475 ±0.0005 eV at zero field exhibits a Zeemansplitting as recognized by the difference betweenright-circular polarization and left-circular polarization curves. The center of the splittingshows a diamagnetic shift. The second weak absorption, though unobservable in the zero field,appears already in the field of 7.3 T at 2.564±0.001 eV as shown in Fig. l(b), and becomesstronger toward higher field. The third absorption at higher photon energy becomes observableat the field of ~ 12 T near 2.59 eV, A significantZeeman splitting is also observable for the sec-1216Volume 48, Number 17 PHYSICAL REVIEW LETTERS 26 April 1982(Q)2.60 2.58 2-56Photon Energy (eV)2.54 2.57 2.56 255 254Photon Energy (eV)FIG. 1. Typical transmission spectra of InSe at 4.2 K in the Faraday geometry with external magnetic field parallel to the c axis of the hexagonal unit cell. The third absorption bands quoted in the text are shown by arrows. Themagnetic field in teslas for right-circular polarization (rep: solid line) is given by the numbers, and that for left-circular polarization (lep: dashed line) by those in parentheses. The sample thickness is (a) (No. 3) 1.25 nm and(b) (No. 4). 1.70 /im. The ordinate is the ratio of the transmitted to the incident light intensity in arbitrary units.ond line. The photon energies for*, these absorption peaks are plotted as a function of magneticfield in Fig. 2.The ground state of the A exciton consists ofparallel and antiparallel configurations of electron and hole spins. In the Faraday geometry,only the exciton with antiparallel spins may beoptically created.12 Their energy eigenvalues,Elf2, in a magnetic field, H, parallel to the caxis are expressed13 as follows:(^-^)VB2i/2]l/2, CDwhere Eo is the ground-state energy at zero field,D the diamagnetic energy, Aex the exchange energy, ALT the longitudinal to transverse splittingof the exciton, gA the effective g factor of the Avalence band in the direction of the c axis, gethat of the conduction band, and jll b the Bohrmagneton. The Lyddane-Sachs-Teller relationapplied to the A exciton of InSe gives ALT =0.3meV. It follows from Eq. (1) that the Zeemansplitting of 3.8 ±0.4 meV at 30 T deduced fromFig. 2 for the Is line gives ^-#.= 2.2±0.2.In order to estimate the value of D, we make afit of the theoretical results of Cabib, Fabri, andFiorio (CFF)14 to our experimental points withSa~Se =2.2 obtained above. The best fit is obtained with jLtx = 0.102m0 as shown in Fig. 2 bydash-dotted lines. Then the deduced value of ju±allows us to scale the magnetic field in terms ofy, and thereby to compare the observed energiesof the second and third absorption lines with theoretical values. The magneto-optical dispersioncurves derived by Tanaka and Shinada (TS)15 andby CFF are shown by solid and dash-dotted lines,respectively, in Fig. 2.We notice that our second and third absorptionlines are in excellent agreement with the (100+)and (200+) curves, respectively, in magneticfields above 12 T. According to TS, the adiabaticpotential employed in their calculation gives inaccurate solutions in the low-field range. Theyargue that the (100+) line should extrapolate tothe 2s level at zero field and (001+) to 3d0, if thepotential is adequately modified and no interaction between these levels takes place. Becauseof the presence, however, of the interaction asproposed by SAHT and supported by the exactcalculation of CFF, these two levels admix together and repel each other. Consequently, thefirst excited state, (001+), which correspondsto the lower hybrid branch would change itscharacter from 2s-like to 3<io-like as the field1217Volume 48, Number 17 PHYSICAL REVIEW LETTERS 26 April 19822.602.58co"oQ_2.562.5 4-1.00 20 30 40Magnetic Field (T)50FIG. 2. Magnetic field dependences of energies ofabsorption peaks measured with rep (circles) and lep(squares) light. Theoretical curves of the lowest threeLandau edges and excitonic levels calculated by Tanakaand Shinada and those by Cabib et al. are shown bydashed, solid, and dash-dotted lines, ^respectively.The reduced energy in the ordinate is scaled in unitsof Oi* with its origin at the direct energy gap in thezero field.increases from y = 0 through y =0.3. Hence itsoscillator strength remains as weak as 11%to 18% of the Is-like (000+) line at any fieldstrength,14 and thereby its observation is extremely difficult. On the other hand, the upperbranch, (100+), has the Zd0 character at zero field,but would be transformed to the 2s-like as a result of the interaction, so that its oscillatorstrength increases almost linearly with magneticfield to exceed that of the (001+) line above y= 0.5.15 Such a field dependence as expected forthe 2s-like state is in good agreement with ourexperimental result on the second absorptionband. Furthermore experimental points alongthe (100+) line do not exhibit any evidence of repulsion by the N = 0 Landau edge, but rather intersect it near the field of 10 T, thus disprovingBH's correspondence relation. In the light ofthis result, the unidentified line observed byMiura et al.9 below the N = 1 Landau edge inGaSe may be attributed to the (100+) level whichconnects to the 2s-like line observed by thepresent authors.8 If one applies the same argument to the third absorption band, it is assignedto the 3s-like (200+) state after experiencing theinteraction with 4<2O and 5gQ levels.3From the above results, one can conclude thatthe level connection of a hydrogenic exciton uponthe Rydberg-to-Landau transition satisfies thenodal-surface-number conservation rule, although the detail of interactions between excitonic levels is not clear yet. Besides the works ofTS and CFF, numerical calculations of energylevels have been reported by a number ofgroups.4'16"18 Apart from Ref. 16, their resultsshow obvious violations of the noncrossing rule[see the (210) level in Ref. 4, the 3d0 level inRef. 17, and the (210) and (110) levels in Ref. 18],but no contradiction with SAHT's rule. Furthermore SAHT's rule can explain all of the experimental results cited4*5l7i8f9t18 in this paper.The authors would like to acknowledge the valuable technical assistance by Mr. Y. Mori in thecourse of measurements. This work was supported in part by a Grant-in-Aid for ScientificResearch from the Ministry of Education, Science, and Culture in Japan.lW. H. Kleiner, Massachusetts Institute of TechnologyLincoln Laboratory Progress Report, February, 1958(unpublished), p. 36.2R. J. Elliott and R. Loudon, J. Phys. Chem. Solids15, 196 (1960).3M. Shinada, O. Akimoto, H. Hasegawa, and K. Tanaka, J. Phys. Soc. Jpn. 28, 975 (1970).4W. S. Boyle and R. E. Howard, J. Phys. Chem. Solids1.9, 181 (1961).5R. J. Wagner and B. D. McCombe, Phys. Status Solidi(b) 614, 205 (1974); see also the references therein.6N. O. Lipari and M. Altarelli, in Physics in High Magnetic Fields, edited by S. Chikazumi and N. Miura(Springer-Verlag, Berlin, 1981), p. 180.7K. Kobayashi, in Solids and Plasmas in High Magnetic Fields, edited by R. L. Aggarwal, A. J. Freeman,and B. B. Schwartz (North-Holland, Amsterdam, 1979),p. 84.8Y. Sasaki, N. Kuroda, Y. Nishina, H. Hori, M. Shi-noda, and M. Date, in Physics in High Magnetic Fields,edited by S. Chikazumi and N. Miura (Springer-Verlag,Berlin, 1981), p. 195.9N. Miura, G. Kido, H. Katayama, and S. Chikazumi,J. Phys. Soc. Jpn., Suppl. A 49, 409 (1980).10N. Kuroda, I. Munakata, and Y. Nishina, Solid StateCommun. 33, 687 (1980).1218Volume 48, Number 17 PHYSICAL REVIEW LETTERS 26 April 1982UN. Kuroda and Y. Nishina, Solid State Commun. 34,481 (1980).12H. Venghaus, S. Suga, and K. Cho, Phys. Rev. B16, 108 (1973).4419 (1977).13I. Broser and M. Rosenzweig, Phys. Rev. B 22,2000 (1980).(1972).15K. Tanaka and M. Shinada, J. Phys. Soc. Jpn. 34,16N. Lee, D. M. Larsen, and B. Lax, J. Phys. Chem.Solids 34, 1059 (1973).17H. C. Praddaude, Phys. Rev. A 6t 1321 (1972).14D. Cabib, E. Fabri, and G. Fiorio, Solid State Com- 18S. Narita and M. Miyao, Solid State Commun. 9mun. 9>, 1517 (1971), and Nuovo Cimento B ^0, 185 2161 (1971).1219

ジバチュウ ノ スイソ ゲンシヨウ レイキシ ノ ハドウ カンスウ ニ ツイテ ノ セツメンスウ ホゾンソク ノ ダトウセイ

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ジバチュウノスイソゲンシヨウレイキシノハドウカンスウニツイテノセツメンスウホゾンソクノダトウセイ

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ジバチュウ ノ スイソ ゲンシヨウ レイキシ ノ ハドウ カンスウ ニ ツイテ ノ セツメンスウ ホゾンソク ノ ダトウセイ

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ジバチュウノスイソゲンシヨウレイキシノハドウカンスウニツイテノセツメンスウホゾンソクノダトウセイ