Quantum of optical absorption in two-dimensional ...

1 Quantum of optical absorption intwo- dimensional semiconductorsHui Fanga,b,c, Hans A. Bechteld, Elena Plise, Michael C. Martind, Sanjay Krishnae, Eli Yablonovitcha,b,1, and Ali Javeya,b,c,1aDepartment of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720;bMaterials Sciences Division, Lawrence BerkeleyNational Laboratory, Berkeley, CA 94720;cBerkeley Sensor and Actuator Center, University of California, Berkeley, CA 94720;dAdvanced Light Source Division,Lawrence Berkeley National Laboratory, Berkeley, CA 94720; andeCenter for High Technology Materials, University of New Mexico, Albuquerque, NM 87106 Contributed by Eli Yablonovitch, May 26, 2013 (sent for review February 11, 2013)The optical absorption properties of free-standing InAs nanomem-branes of thicknesses ranging from 3 nm to 19 nm are investigatedby Fourier transform infrared spectroscopy.

2 Stepwise absorption atroom temperature is observed, arising from the interband transi-tions between the subbands of 2D InAs nanomembranes. Interest-ingly, the absorptance associated with each step is measured to be , independent of thickness of the membranes. The exper-imental results are consistent with the theoretically predictedabsorptance Quantum ,AQ= /ncfor each set of interband tran-sitions in a 2D semiconductor, where is thefine structure con-stant andncis an optical localfield correction factor. Absorptancequantization appears to be universal in 2D systems including III Vquantum wells and optical properties of heterostructure Quantum wells (QWs)have been extensively studied since the 1970s, in GaAs/AlGaAs (1), GaInAs/AlInAs (2, 3), InGaAs/InP (4), and HgCdTe/CdTe (5). Here we do a careful quantitative examination of theintrinsic absorption properties of free-standing 2D semiconductorthinfilms, which has previously been done only for layered struc-tures, such as MoS2(6).

3 (The criterion of real two-dimensionalityis that the material thickness be smaller than the electronBohr radius.)Previous work has shown that graphene, a 2D semimetal, hasa universal value of light absorption , namely , where is thefine structure constant (7). Here, we use free-standing InAsmembranes with exceptionally small thickness as a model ma-terial system to accurately probe the absorption properties of 2 Dsemiconductors as a function of thickness. We demonstrate thatthe magnitude of the light absorption is an integer product ofa Quantum of absorptance. Specifically, each set of interbandtransitions between the 2D subbands results in a Quantum unit ofabsorptance ofAQ /nc, wherencis the optical localfieldcorrection factor. The total absorptance for thefirst several setsof interband transitions is simply given asA=MAQ, whereMisthe integer number of allowed transitions for a given photonenergy.

4 The result here appears to be universal, except for smallcorrection factors associated with higher , there has been a high level of interest in exploringthe fundamental science (6 10) and associated devices (11 20)of free-standing ( , attached to a substrate by van der Waals orother weak forces) 2D semiconductors . Two- dimensional layeredsemiconductors [ , MoS2(11), WSe2(15), GaSe (16), Bi2Sr2-CaCu2Ox(17), Bi2Se3(18), and Bi2Te3(19) or diamond/zinc-blend structures InAs (12) and InGaSb (20)] can be readilymounted on virtually any support substrate, thereby enabling awide range of novel device concepts and practical one example system, InAs Quantum membranes (QMs) withadjustable thicknesses down to a few atomic layers have beenrealized by a layer transfer process onto a user-defined substrate(12). The approach enables the direct optical absorption studiesof fully relaxed ( , unstrained) (21) 2D III V semiconductorsby using transparent substrates, without the constraints of theoriginal growth substrate (10).

5 Here, we use InAs membranes ofthicknessLz 3 19nm on CaF2support substrates as a modelmaterial system for examining the absorption properties of 2 Dsemiconductors. Given the large Bohr radius of 34 nm (22) inbulk InAs, strong Quantum confinement of carriers is readilyobtained for sub-20-nm Quantum membrane thicknesses. Notethat there are only 5 unit cells in a 3-nm-thick InAs QM, givenits lattice constant of nm (23). In such a quantummechanically confined 2D system, there is electronic band dis-persion in the two unconfined directions, with discrete energysubband edges at values determined by Quantum confinement(24) in the third direction. In our case, the InAs QMs can beeffectively treated as infinitely deep potential wells, because theyare confined by air on one side and by a wide band-gap (25) CaF2substrate on the other side. Fig. 1 Ashows the optical microscopeimage of a periodic array of 5- m-wide InAs strips (of 3nmthickness), on a CaF2crystal substrate produced by the layertransfer process (SI Text).

6 CaF2was chosen as the support sub-strate as it is optically transparent for the wavelength range ofinterest. From visual appearance (Fig. 1A), it is evident that ul-trathin InAs QMs exhibit strong optical contrast, thereby allow-ing for the detailed optical characterization even for thefilmsconsisting of a few atomic layers in 1 Bshows the calculated energy levels of a 9-nm InAs QMwith CaF2and air boundaries. The molecular beam epitaxy-grownInAs samples are not intentionally doped, eliminating bandfilling(the Fermi level was calculated from an estimated bulk electrondensity of 2 1016/cm3) (26). Here, the effects of surface accu-mulation layers due to surface defects are ignored (seeFig. S1for surface accumulation layer effects). Fig. 1 Cdepicts thequalitative band structure of a 2D InAs QM, with vertical arrowsindicating the interband transitions from thenth heavy hole (hh)/light hole (lh) subband to thenth electron (e) subband.

7 Here wedefineen hhnanden lhnas thenth set of interband to spatial matching of electron/hole wave functions in dif-ferent energy subbands, interband transitions in Quantum wellsare favored when n=n n=0, wheren andnare thesubband indexes for electrons and holes, respectively (27).Consequently, there are two allowed transitions to each electronsubband: one from the correspondinghhsubband and the use Fourier transform infrared (FTIR) microspectroscopy(Fig. 2) to probe optical interband transitions (in the range <h < eV) in InAs QMs at room temperature (SI Text).Transmittance and reflectance spectra were collected in therange of 2,414 9,656 cm 1( eV) over an aperture sizeof 50 50 m2, with 8 cm 1(1 meV) resolution and 512 Author contributions: and designed research; , , , , and research; , , , , , and contributed new reagents/analytic tools; , , and analyzed data; and , , and wrote the authors declare no conflict of available online through the PNAS open access whom correspondence may be addressed.

8 E-mail: or article contains supporting information online 11691|PNAS|July 16, 2013|vol. 110|no. for all samples with different thicknesses. AtmosphericH2O and CO2effects were removed using OMNIC software(Thermo Scientific). The absorption spectra of InAs QMs foreach thickness were obtained by subtracting the transmittanceand reflectance from a normalized 100% spectrum, yielding Thefinal spectra were generated by dividing the measuredspectra by the fractional areafill factor of the InAs strips. Fig. 4shows the overlaid absorptance (A) spectra of InAs QMs withLz 3 nm, 6 nm, 9 nm, 14 nm, and 19 nm. Clear step-like featuresare observed in the absorptance spectra arising from the quan-tized interband transitions between the 2D subbands (10). Thespacings between the measured absorptance steps are in quan-titative agreement with the calculated interband energy spacings(seeFig.)

9 S2for the detailed analysis). Note that intersubbandtransitions are negligible due to our transverse electric (TE)polarization. The experimentalfinding here is that the individualabsorptance steps plateau at (SD) for all samples,regardless of the QM thickness. The result was reproducible formultiple samples and shed light on this observed absorption behavior, the elec-tron photon interaction in a semiconductor material is theoret-ically evaluated from Fermi s golden rule. If a light wave withelectricfield~E, polarization vector^e, and frequency is incidentperpendicular to a direct band-gap semiconductor membranewith a thicknessLz,inaninfinitely deep potential well model(28), the optical absorption coefficient is =e2nrc om2o LzZ211=m*en+1=m*hn ^e ~pcv 2;[1]which is a step function for each interband transition, whereeisthe electron charge,nris the real part of surrounding refractiveindex,cis the speed of light, ois the vacuum permittivity,moisthe free electron mass,Zis Planck s constant,men* andmhn* +(eV)z (nm)CaF29nmInAsAire2e1 EFhh2hh1lh2lh1 Eke1hh1e2hh2lh1lh2 Distance ( m)020406010 m~3 nm10 mABCFig.

10 1.(A) optical microscope image of InAs 2D strips ( 3 nm thick) ona CaF2substrate. (Inset) The atomic force microscope image of a single InAsstrip. (B) CaF2/InAs (9 nm)/air Quantum well band diagram with energy re-ferring to E(z=0+), abbreviated as E0+.EFshows the quasi-Fermi level,wherease1,e2,hh1,hh2,lh1, andlh2are the indexes for the conduction andvalence subbands. Note that only thefirst two conduction/valence subbandedges are shown. (C) Qualitative band structure schematic of a 2D InAs QM,with arrows indicating the allowed optical interband transitions betweenvalence and conduction interferometerIR detectorCaF2substrateMirrorMicroscope objectiveIncident lightReflected lightTransmitted lightFig. illustration of the FTIR microspectroscopy setup used forthe absorption measurements. The absorption spectrum is obtained fromthe measured transmission and reflection spectra. The incident light anglewas actually perpendicular to the (%)E (eV) (%)E (eV) (%)E (eV) (%)E (eV) (%)E (eV)hh1lh1hh1lh1hh1lh1hh2lh2hh1lh1hh2lh2 hh3lh3hh4hh1lh1hh2hh3hh4hh5lh2lh3lh4lh56 nm9 nm14 nm19 nm~3 nmABCDEFig.