Direct measurement of quantum phases in graphene via photoemission spectroscopy
Choongyu Hwang, Cheol-Hwan Park, David A. Siegel, Alexei V. Fedorov, Steven G. Louie, Alessandra Lanzara
DDirect measurement of quantum phases in graphene via photoemission spectroscopy
Choongyu Hwang, , , Cheol-Hwan Park, , , David A. Siegel, , ,Alexei V. Fedorov, , Steven G. Louie , , ∗ and Alessandra Lanzara , † Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA. Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA. Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA. and These authors contributed equally to this work. (Dated: August 30, 2018)Quantum phases provide us with important information for understanding the fundamental prop-erties of a system. However, the observation of quantum phases, such as Berry’s phase and thesign of the matrix element of the Hamiltonian between two non-equivalent localized orbitals in atight-binding formalism, has been challenged by the presence of other factors, e. g. , dynamic phasesand spin/valley degeneracy, and the absence of methodology. Here, we report a new way to di-rectly access these quantum phases, through polarization-dependent angle-resolved photoemissionspectroscopy (ARPES), using graphene as a prototypical two-dimensional material. We show thatthe momentum- and polarization-dependent spectral intensity provides direct measurements of (i)the phase of the band wavefunction and (ii) the sign of matrix elements for non-equivalent orbitals.Upon rotating light polarization by π/
2, we found that graphene with a Berry’s phase of n π ( n = 1for single- and n = 2 for double-layer graphene for Bloch wavefunction in the commonly used form)exhibits the rotation of ARPES intensity by π/n , and that ARPES signals reveal the signs of thematrix elements in both single- and double-layer graphene. The method provides a new techniqueto directly extract fundamental quantum electronic information on a variety of materials. PACS numbers: 79.60.Jv, 03.65.Vf, 31.15.aq, 81.05.ue, 73.22.Pr
I. INTRODUCTION
Quantum phases are the most beautiful example ofquantum physics and essential to understand physics inany material. For example, Berry’s phase, the accumu-lated phase in the eigenfunction acquired by evolving thequantum system adiabatically around a closed loop in theparameter space of the Hamiltonian [1], has been shownto be responsible for the Aharonov-Bohm effect [2], thehalf-integer quantum Hall effect [3–5], etc. Another im-portant example is the sign of the hopping matrix ele-ment (or hopping integral) (cid:104) φ | H | φ (cid:105) of the Hamiltonianbetween two non-equivalent localized orbitals φ and φ in a tight-binding formalism. This phase, a fundamen-tal quantity in determining the electronic structure of asystem, is dictated by the characteristics of the atomisticinteraction, e. g. , whether it is attractive or repulsive.Both of these are important to directly extract funda-mental quantum electronic information on a variety ofmaterials [6–8].In graphene, the Berry’s phase is theoretically ex-tracted from the spinor eigenstate, which are π for single-and 2 π for double-layer graphene [3]. From recent stud-ies, the nπ Berry’s phase in the commonly used form ofthe spinor states is, in fact, related to the pseudospinwinding number n of a particle as it travels in a loopin k -space which encloses the Dirac point [32]. Thesevalues have been measured through magneto-transport ∗ Electronic address: [email protected] † Electronic address: [email protected] experiments [4, 5] that are typically neither capable ofmeasuring Berry’s phase of a specific electron bandstruc-ture nor free from spin/valley degeneracy of the elec-tron bandstructure of a system under study. Addition-ally, this method requires a strong magnetic field, whichbreaks time-reversal symmetry in graphene. Meanwhile,the signs of hopping integrals between non-equivalent or-bitals for graphene (graphite) have only been determinedby ab initio calculations [9], e. g. , using maximally local-ized Wannier functions [10]. Since the sign of hoppingintegral depends on the characteristics of the localizedorbitals and the interaction between them, it is crucial indetermining the electron bandstructure within a tight-binding formalism. However, the absence of methodol-ogy has led to use the signs following the well-knownSlonczewski-Weiss-McClure model [11, 12] without ex-perimental verification for the past few decades. Ad-ditionally, the sign of hopping integral between non-equivalent orbitals has never been determined experimen-tally for any material.Given the high momentum-resolving power of angle-resolved photoemission spectroscopy (ARPES), ARPESis an ideal candidate to solve the above issues on thedetermination of quantum phases. For example, thephase difference between the matrix elements describ-ing two different optical transitions at the (110) sur-face of platinum was extracted from a combined studyof a spin-resolved ARPES experiment and a theoreticalmodel [13]. Also, ARPES has been employed to study thecharacteristics of the spinor eigenstates in graphite [14]and graphene [15], which revealed an interference ef-fect between photo-excited electrons [14, 15]. However,these theoretical studies, within a tight-binding formal- a r X i v : . [ c ond - m a t . m t r l - s c i ] S e p ism, have incorrectly treated the interaction Hamilto-nian, which is the key part in the photoemission processas it describes the interaction between photons and elec-trons. Moreover, it has not been clear how Berry’s phaseenters in ARPES intensity and the sign of hopping in-tegral has only been speculated without any comparisonwith experiments [15], which naturally leads to incorrectvalues.Here we report that ARPES can indeed provide in-formation on these quantum phases, e. g. , the Berry’sphase and the sign of hopping integral between non-equivalent orbitals. The phase factor in the spinor eigen-state of graphene [3] gives rise to strong intensity vari-ation around a constant energy contour. Upon rotat-ing light polarization by π/
2, we found that graphenewith a Berry’s phase of n π ( n = 1 for single- and n = 2 for double-layer graphene) exhibits the rotationof ARPES intensity maxima by π/n , which gives impor-tant advantages compared to the conventional magneto-transport method [4, 5]. Additionally, we found thatfull polarization-dependence of ARPES signal reveals thesign of hopping integrals in both single- and double-layergraphene (graphite can also be understood), e. g. , γ (cid:48) > γ (cid:48) >
0, which is the first experimental determinationof the sign of hopping integral between non-equivalent or-bitals for any material by any method.
II. SAMPLE PREPARATION
Single- and double-layer graphene samples weregrown epitaxially on n -doped 6 H -SiC(0001) surfaces byelectron-beam heating, as detailed elsewhere [16]. An SiCsample was mounted in a prep-chamber with a base pres-sure of 5 × − Torr to remove a thick oxide layer fromthe sample by heating at 600 ◦ C for a few hours. Theclean sample was then transferred to a custom-designedchamber equipped with low-energy-electron microscopy(LEEM) with a base pressure of 2 × − Torr and heatedto 1000 ◦ C under Si flux to improve the surface condi-tions for graphene growth. The sample temperature wasraised to 1400 ◦ C or 1600 ◦ C (determined by an opti-cal pyrometer) to make single- or double-layer graphene,respectively.The surface morphology was monitored in situ dur-ing the sample growth by LEEM at the National Cen-ter for Electron Microscopy at Lawrence Berkeley Na-tional Laboratory. The thickness of fabricated graphenesamples was determined by LEEM measurements per-formed at room temperature following the standard pro-cedure [17, 18]. In particular, the electron reflectivityversus kinetic energy curve varies significantly with thenumber of graphene layers providing position-dependentmeasurements on the number of graphene layers. A typ-ical bright field image for double layer graphene is shownin Fig. 1(a) over 4 µ m × µ m range, recorded with elec-tron beam of kinetic energy 3.5 eV denoted as the dashedline in Fig. 1(b). In order to determine the number of graphene layers at each position, the electron reflectiv-ity is plotted as a function of electron kinetic energy, asshown in Fig. 1(b), where the number of dips is the sameas the number of graphene layers. Regions 1, 2, and 3 inFig. 1(a) show 1, 2, and 3 dips, respectively, correspond-ing to single-, double-, or triple-layer graphene, respec-tively. These regions are painted in black, white, andgrey, respectively, in Fig. 1(c). The fractions of regionsin the sample covered by different numbers of graphenelayers were determined from the areal fractions of differ-ently colored regions in Fig. 1(c). In particular, we findthat the double-layer graphene sample contains ∼
74 %of double-layer and ∼
22 % of single-layer graphene.
III. EXPERIMENT
We have performed polarization-dependent ARPESexperiments on single- and double-layer graphene at 10K using a photon energy of 50 eV at beam-lines 10.0.1and 12.0.1 of Advanced Light Source at Lawrence Berke-ley National Laboratory. In Figs. 2(a) and 2(b), we showthe typical geometry of ARPES experiments: a beam ofmonochromatized light with energy (cid:126) ω and polarizationvector (cid:126)ε is incident on a sample, resulting in the emissionof photoelectrons in all directions. The polarization vec-tor of light is referenced with respect to the sample nor- FIG. 1: (a) A LEEM image using an electron energy of 3.5 eVover 4 µ m × µ m range. (b) Reflectivity spectra for the threeregions (1, 2, and 3) specified in (a). (c) Post-processed imageof (a) showing the regions covered by single-, double-, andtriple-layer graphene. (d) A histogram showing the fractionsof single-, double-, and triple-layer graphene in our sampleused for double-layer graphene measurements. FIG. 2: (a) X-polarization geometry. (b) Y-polarization geometry. A beam of monochromatic lights with energy (cid:126) ω =50 eVand polarization vector (cid:126)ε is incident on a sample. The light polarizations in X- and Y-polarization geometries are almostparallel to the x and y axes, respectively. (c, d) Measured intensity maps of single-layer graphene at energy E = E F withX- and Y-polarized lights, respectively. Intensity maxima are denoted by white arrows and the electronic band structure ofsingle-layer graphene is drawn in the cartoon. (e) Constant-energy ARPES intensity maps for single-layer graphene at E F withX- and Y-polarized light. (f) The angle-dependent intensity profiles of single-layer graphene are obtained by integrating theconstant-energy intensity map along the radial direction from the Dirac point, in which the solid and dashed lines denote theexperimental data for X- and Y-polarized lights, respectively. The angle θ is measured from the + k x direction. The plottedquantities are with respect to the intensity minimum. mal. In the experiment presented here, two different ge-ometries were employed as shown in Figs. 2(a) and 2(b).In one geometry shown in Fig. 2(a), the polarization oflight is almost parallel to the x axis, while in the othershown in Fig. 2(b) to the y axis; hence, we define thesetwo geometries as X- and Y-polarization, respectively.These geometries have the advantage with respect to theconventional s - and p -polarizations used in previous stud-ies [19, 20], to measure the whole two-dimensional vari-ation of the intensity maps around a singular (Dirac)point and not just the intensity distributions along twocharacteristic lines in momentum space. This aspect be-comes particularly relevant for some experimental condi-tions, e. g. , photon energy and sample orientation (i. e. ,the mixture of light polarizations), when the intensitymaps (or initial electronic states) are neither symmetricnor anti-symmetric with respect to the reflection plane.Under this condition in fact, the conventional notationswould not give appropriate information on the symmetryof the initial states.Figures 2(c) and 2(d) show the experimental photo-electron intensity maps at the Fermi level, E F , versus thetwo-dimensional wavevector k for single-layer graphene,for the two polarization geometries. Here, E F is 0.4 eVabove the Dirac point energy, E D [17, 21, 22]. Themain feature in the intensity maps of both geometriesis an almost circular Fermi surface centered at the K point as shown in Figs. 2(c) and 2(d), as expected fora conical dispersion. This is in good agreement with arecent polarization-dependent ARPES study on single-layer graphene when using photons with energy lowerthan 52 eV [23]. Surprisingly we find that the the angularintensity distribution is quite different for the two polar-izations: for the X-polarization geometry, the minimumintensity position is in the first Brillouin zone, whereasfor the Y-polarization geometry, the maximum intensityposition is in the first Brillouin zone, suggesting a π rota-tion of the maximum intensity in the k x - k y plane aroundthe K point upon rotating the light polarization by π/ K point in Fig. 2(f). Thereis an overall shift of the intensity maxima (minima) by ∼ π upon changing the light polarization from X (blacksolid line) to Y (black dashed line), although the latterappears to be slightly shifted by ∼ π/
10 with respect to π . As we will show later, this is due to the presence ofa finite polarization component along the k x direction inour experimental geometry.We note that not only the angular position of theintensity maximum, but also the absolute value of it FIG. 3: (a, b) Measured intensity maps of double-layer graphene at energies E = E D + 0 .
25 eV (= E F ) and E = E D − .
95 eV.Intensity maxima are denoted by white arrows and the electronic band structure of double-layer graphene is drawn in thecartoon. (c) Measured intensity maps of double-layer graphene at energy E = E F . (d) The angle-dependent intensity profilesof double-layer graphene at energy E = E F , in which the solid and dashed lines denote the experimental data for X- andY-polarized lights, respectively. The plotted quantities are with respect to the intensity minimum. changes upon changing light polarization. The maxi-mum intensity ratio from experiments is X-polarized/Y-polarized=21.4. However, this number itself is not verymeaningful, because the measured ARPES intensity isaffected by the difference in the experimental geometriesfor X- and Y-polarized lights (the difference in the out-of-plane component of light polarization, photon flux perunit area, etc., which are the factors that cannot becontrolled to be the same in different experimental ge-ometries). On the other hand, the ratio from our the-ory that will be discussed later provides X-polarized/Y-polarized=0.83, assuming that the experimental param-eters for two geometries are the same except for the in-plane light polarization.A similar study on Bernal stacked double-layergraphene reveals a strong and complicated momentum-, band-, and polarization-dependence as shown inFigs. 3(a) and 3(b), that is qualitatively different fromthat of single-layer graphene. Like single-layer graphene,the double-layer sample is slightly n doped [22], there-fore, only three of the four π bands are occupied andhence detectable with ARPES as shown in the cartoon ofFig. 3(a). The most prominent feature is that, when thelight polarization is changed from X to Y, the maximumintensity positions around the K point in the k x - k y planeare rotated by ∼ π/ E F ). This isin striking contrast with the single-layer case where therotation is ∼ π as seen from the raw MDCs in Fig. 3(c)and the photoelectron intensity maps integrated over theradial direction around the K point shown in Fig. 3(d).Due to trigonal warping effects [24], however, the rota-tion for higher-energy states is not exactly π/ IV. THEORETICAL ANALYSIS
To the best of our knowledge, the only models in theliterature describing the polarization-dependence of theARPES intensity in graphite [14] and graphene [15] aresubstantially different from our results. Previous studies,in fact, predict a small polarization-dependence [14] andno polarization-dependence [15] of the photoelectron in-tensity maps, respectively. Therefore, to be able to repro-duce our experimental findings and understand what liesbehind this nontrivial polarization dependence, we havedeveloped a new model. In particular, we first considerthe Hamiltonian using the tight-binding model based onthe p z orbital of each carbon atom using two parame-ters: t and t for the in-plane nearest-neighbor (A-B orA (cid:48) -B (cid:48) ) and the inter-layer vertical (B-A (cid:48) ) hopping inte-grals, respectively, as schematically drawn in Fig. 4(a).The parameters t and t correspond to − γ (cid:48) and γ (cid:48) , re-spectively, in the well-known Slonczewski-Weiss-McClure(SWMc) model [11, 12]. In our calculation, we have used | t | = 3 .
16 eV and | t | = 0 .
39 eV, which are the values inTable II of Gr¨uneis et al . [9], but we do not fix the signs ofthem. Note that all four possible choices of the signs giveexactly the same electron energy band structure withinthis two-parameter tight-binding model.With this setup, the tight-binding Hamiltonian of adouble-layer graphene for two-dimensional wave vector k = ( k x , k y ) using a basis set composed of Bloch sums oflocalized orbitals on each of the four sublattices (A, B, FIG. 4: (a) Schematic of single- and double-layer graphene.(b) The Brillouin zone. Here, b = b (0 , b = b (cid:16) − √ , − (cid:17) and b = b (cid:16) √ , − (cid:17) are the three vectorsconnecting the in-plane nearest neighbor atoms where theinter-carbon distance b = 1 .
42 ˚A, and the lattice constantis a = √ b . The positions of the K and K (cid:48) points are (cid:0) π a , (cid:1) and (cid:0) − π a , (cid:1) , respectively. A (cid:48) , and B (cid:48) ) reads H ( k ) = t g ( k ) 0 0 t g ∗ ( k ) 0 t e − i k z d t e i k z d t g ( k )0 0 t g ∗ ( k ) 0 . (1)Here, g ( k ) = (cid:88) i =1 exp( i k · b i ) (2) with b i ’s defined as in Fig. 4(a), and k = 1 √ N (cid:88) R A e i k · R A φ ( r − R A ) , (3) k = 1 √ N (cid:88) R B e i k · R B φ ( r − R B ) , (4)etc. We note that often the k dependence of the ba-sis function is suppressed in the spinor notation for sim-plicity. In Eq. (1), we have considered a phase differ-ence e ± i k z d arising from the finite inter-layer distance d and the perpendicular component of electron wave vec-tor k z . Here, k z is not part of the Bloch momentum,but the z component of the photoelectron wave vector,which is determined by the photon energy. This quantityplays a crucial role in determining the ARPES intensityof double-layer graphene as will be discussed later andalso of multi-layer graphene as previously reported [25].The additional interaction Hamiltonian coupling toelectromagnetic waves of wavevector Q for a double-layergraphene ˆ H intdouble can be obtained by using the velocityoperator ˆ v = (cid:104) ˆ r , ˆ H (cid:105) /i (cid:126) , where ˆ r = i (cid:126) ( ∇ k , ∂ k z ) inthe k -representation and (cid:126) is the Planck’s constant, as − ec ˆ A · ˆ v [26] [ e is the charge of an electron, c is the speedof light, and the external vector potential is given by A ( r , t ) = A Q e i ( Q · r − ωt ) ] [28] , i. e. , H intdouble ( k , Q ) = − e (cid:126) c A Q · t ∇ k g ( k ) 0 0 t ∇ k g ∗ ( k ) 0 − i d t e − ik z d ˆ z i d t e ik z d ˆ z t ∇ k g ( k )0 0 t ∇ k g ∗ ( k ) 0 . (5)The transition matrix elements in Eq. (5) are those takenbetween basis functions of Bloch sums of p z orbitalsof wavevectors k + Q and k . Equation (5) is validwhen 1 / | Q | is much larger than the distance betweenthe nearest-neighbor atoms b , i. e. , when the variationin A ( r , t ) over a length scale of b is much smaller than A ( r , t ) itself. We shall eventually take the Q → H ( k ) = t g ( k ) t g ∗ ( k ) 0 , (6)and H intsingle ( k , Q ) = − e (cid:126) c A Q · t ∇ k g ( k ) t ∇ k g ∗ ( k ) 0 . (7) H int is critical to explain the polarization depen-dence of I k , because it describes the interaction be-tween electrons and photons. The lack of polarizationdependence in previous studies [14, 15] is indeed dueto the way in which H int is incorrectly treated. Inone case [15], the light interaction is completely ne-glected by setting H int = 1, while in the earlier study ongraphite [14], the velocity operator v is replaced by themomentum p /m = − i (cid:126) ∇ /m , where (cid:126) is the Planck’sconstant and m the free-electron mass. This replace-ment works [27, 28] only when the Hamiltonian is lo-cal, whereas a tight-binding Hamiltonian, which has beenused in the previous studies [14, 15] as well as our study,is intrinsically non local. The experimental finding ofa strong polarization dependence of I k in Figs. 2 and 3clearly shows the need for a more complete theoreticaltreatment. We have developed a theory using the widelyadopted tight binding model with one p z -like localizedorbital per carbon atom, but employing the appropriateinteraction Hamiltonian with the velocity operator. Avery good agreement between our model and the exper-imental results is obtained for all polarizations and forboth single- and double-layer graphene, when we com-pare Fig. 2 with Fig. 5 and Fig. 3 with Fig. 6 as will bediscussed later.In order to understand what lies behind the observednon-trivial and unexpected wavevector-dependent pho-toelectron intensity I k , we need to calculate the abso-lute square of the transition matrix element M s k = (cid:104) f k + Q | H int ( k , Q ) | ψ s k (cid:105) , where | ψ s k (cid:105) is a single- ordouble-layer graphene eigenstate with s = ± | f k + Q (cid:105) is the plane-wave final state projected ontothe p z orbitals of graphene [both | ψ s k (cid:105) and | f k + Q (cid:105) areexpressed using the basis set of Bloch sums of local-ized p z orbitals at sublattices A and B in Fig. 5(a)] and H int = − ec A · v [26], which should not be neglected inphotoemission process [15]. The use of a projection ofthe final plane-wave state onto the Bloch sum, which –when using plane-waves basis – is effectively composedof multiple plane-waves [29], allow to explain the non-trivial polarization dependence of the ARPES intensitydistribution in Figs. 2(c) and 2(d). Since the polariza-tion of A is in the x-y plane, the projection of | f k (cid:105) ontothe σ -states of graphene will result in zero contribution tothe transition matrix elements and hence are neglected inthis analysis. For simplicity of notation, and without anyloss of generality, in the rest of this section, we shall takethe limit of Q → H int ( k ) = H int ( k , Q ) and | f k (cid:105) = | f k + Q (cid:105) . For single-layer graphene, we may use | f k (cid:105) = 1 √ (cid:18) (cid:19) k (8)and for double-layer graphene | f k (cid:105) = 12 k . (9) For photons (with energy ≈
50 eV) used in the exper-iment, k z of the planewave final state is much largerthan the variation of two-dimensional Bloch wavevector k around a single Dirac point, leading to only a small vari-ation in k z with any change in k around a single Diracpoint. For light with a nonzero polarization componentalong the z direction, it will only give rise to an addi-tive isotropic term to the photoelectron intensity that isindependent of the in-plane polarization of the light.Now, let us consider the case where k is very closeto the Dirac point K as shown in Fig. 4(b), and define q = k − K ( | q | (cid:28) | K | ). According to Eq. (2), g ( q + K ) ≈ − √ a ( q x − iq y ) , (10)where a is the lattice parameter. For single-layergraphene, therefore, H ( q + K ) ≈ − √ at ( q x σ x + q y σ y ) , (11)and H intsingle ( q + K ) ≈ √ e (cid:126) c at ( A x σ x + A y σ y ) , (12)where σ x and σ y are the Pauli matrices. The energyeigenvalue and wavefunction of Eq. (11) are given by E s k = √ a | t | s | q | and | ψ s k (cid:105) = 1 √ (cid:18) e − iθ q / − sgn( t ) s e iθ q / (cid:19) , (13)respectively, when θ q is the angle between q and the + k x direction. Using Eqs. (8), (12), and (13), the transitionmatrix element is given for light polarized along the x direction by M x − pol s k ∼ exp( − iθ q / − sgn( t ) s exp( iθ q / . (14)It follows that for s = +1 (states above the Dirac pointenergy), | M x − pol+1 k | ∼ sin ( θ q /
2) (15)and | M x − pol+1 k | ∼ cos ( θ q /
2) (16)with t > t <
0, respectively.Similarly, for light polarized along the y direction, thetransition matrix element is given by M y − pol s k ∼ exp( − iθ q /
2) + sgn( t ) s exp( iθ q / . (17)It follows that for s = +1 (states above the Dirac pointenergy), | M y − pol+1 k | ∼ cos ( θ q /
2) (18)
FIG. 5: (a, b) Calculated intensity maps of single-layer graphene for X- and Y-polarized lights, respectively. The insets arethe results of calculations [14] using the simplified momentum operator instead of the correct velocity operator. An arbitraryenergy broadening of 0.10 eV has been used. Intensity maxima are denoted by white arrows. (c, d) The angle-dependentintensity map of single-layer graphene for X- and Y-polarized lights, respectively, in which the solid black, solid red, and dashedblue lines denote the experimental data, the calculated results obtained by assuming the actual light polarization used in theexperiment, and the calculated results obtained by assuming perfectly Y-polarized light, respectively. The theory results shownin (a) and (b) have adopted the light polarization used in the actual experiment shown in Figs. 2(a) and 2(b)). The plottedquantities are with respect to the intensity minimum. and | M y − pol+1 k | ∼ sin ( θ q /
2) (19)with t > t <
0, respectively.In both cases, we can explain the rotation of the inten-sity maximum in the photoelectron intensity map aroundthe K point by π upon the change from X- to Y-polarizedlight. Comparing Eqs. (15) and (16) with Eqs. (18)and (19), irrespective of the sign of t , the maxima ofthe photoemission intensity map of single-layer grapheneis rotated by π when the light polarization is rotatedby π/
2, in agreement with experiment shown in Fig. 2.Moreover, the theoretical results with t < | t | = 3 .
16 eV (fitted to previous experi-ments [9]) reproduces quite well the salient features inthe experimentally measured intensity maps.This is even more clear from the angular dependenceof theoretical photoelectron intensity drawn with the redsolid lines compared to experimental results drawn withthe black solid lines in Figs. 5(c) and 5(d) . Note thatexperimental intensity maximum for Y-polarized lightshows additional shift by ∼ π/
10 in Fig. 2(f). This ad-ditional shift is well understood by a finite polarizationcomponent along the k x direction. When we assume theactual light polarization used in the experiment shown inFig. 2(b), the theoretical intensity exactly matches withthe experimental result. On the other hand, when weassume the ideal Y-polarization, the intensity maximumappears at π as shown with the blue dashed line in inFig. 5(d). Therefore, we determine through experimentthat the inter-orbital hopping matrix element t betweentwo in-plane nearest-neighbor carbon p z orbitals is nega-tive; we will come back to this point later.Recent theoretical study on the matrix element insingle-layer graphene [23] has found that, in order to de- scribe the matrix element for Y-polarized light from first-principles calculations using plane-wave basis, one needsmultiple plane-wave components for the final photo-emitted electron state. Since we consider a projection ofthe final state onto the Bloch sum, which – when usingthe plane-waves basis – is effectively composed of multi-ple plane-waves [29], our approach can explain the non-trivial polarization dependence of the ARPES intensitydistribution in Figs. 2(c) and 2(d). Additionally, our re-sult obtained by using photons with energy 50 eV is ingood agreement with the recent study [23] based on first-principles calculations using photons with energy lowerthan 52 eV. This suggests that, at photon energy below52 eV and only when the correct interaction Hamilto-nian is employed, the projection of the final state ontothe tight-binding Bloch sum may describe the true finalstate qualitatively. However, since our theoretical frame-work is within tight-binding formulation, and not fromfirst-principles calculations with plane-wave basis, con-vergence tests with respect to the number of plane-wavesand the character of the true final state are beyond thescope of this paper and, in fact, has been done in a recentstudy [23].In the case of double-layer graphene, for simplicityof the analysis, we confine our discussion to the innerparabolic bands (bands 1 and 2 in the cartoon in Fig. 3),although we considered all the four bands in our the-oretical calculations shown in Figs. 6(a) and 6(b). Indouble-layer graphene, for electronic states with energy | E | (cid:28) | t | of the Hamiltonian in Eq. (1), the energyand wavefunction are given by E s k ≈ s (cid:126) q / m ∗ where m ∗ = (cid:126) | t | / t and | ψ s k (cid:105) ≈ √ e − iθ q − sgn( t ) s e iθ q + ik z d , (20) FIG. 6: (a, b) Calculated intensity maps of double-layer graphene for X- and Y-polarized lights, respectively. An arbitraryenergy broadening of 0.10, 0.10, and 0.15 eV have been used for bands 1, 2, and 3, respectively. Panels denoted by ‘Ideal DG’show results obtained by considering a double-layer graphene alone and those denoted by ’SG+DG’ show results consideringthe contribution from some single-layer fraction of the sample as well (see text). Intensity maxima are denoted by whitearrows. (c, d) The angle-dependent intensity map of single-layer graphene for X- and Y-polarized lights, respectively, in whichthe solid black, solid red, and dashed green lines denote the experimental data, the calculated results obtained by consideringthe contribution from single-layer graphene portion of the sample, and the calculated results for ideal double-layer graphene,respectively. The theory results shown in (a) and (b) have adopted the light polarization used in the actual experiment asshown in Figs. 2(a) and 2(b). The plotted quantities are with respect to the intensity minimum. respectively [30]. The phase difference e ik z d between thetwo graphene layers in Eq. (1) appears here as well. UsingEqs. (5), (9), and (20), the transition matrix element isgiven for light polarized along the x direction by M x − pol s k ∼ exp( − iθ q ) − sgn( t ) s exp( iθ q + ik z d ) . (21)It follows that for s = +1 (states above the Dirac pointenergy), | M x − pol+1 k | ∼ sin ( θ q + k z d/
2) (22)and | M x − pol+1 k | ∼ cos ( θ q + k z d/
2) (23)with t > t <
0, respectively. The perpendicularcomponent of the wavevector k z reads [31], k z = (cid:112) m e ( E KE + V inner ) / (cid:126) − k (24)where E KE is the kinetic energy of the photo-electronand V inner the inner potential. Note here that k is atwo-dimensional Bloch wavevector, i, e. , it has no out-of-plane component. The inner potential has been measuredfor graphite by analyzing the energy dispersion at normalemission (i. e. , k = 0) [31]. According to Eq. (24), k z d ≈ . π .Similarly, for light polarized in the y direction, thetransition matrix element for a double-layer graphene is given by M y − pol s k ∼ exp( − iθ q ) + sgn( t ) s exp( iθ q + ik z d ) . (25)It follows that for s = +1 (states above the Dirac pointenergy), | M y − pol+1 k | ∼ cos ( θ q + k z d/
2) (26)and | M y − pol+1 k | ∼ sin ( θ q + k z d/
2) (27)with t > t <
0, respectively.In both cases, we can explain the rotation of the in-tensity maximum in the photoelectron intensity maparound the K point by π/ t , the max-ima of the photoemission intensity map of a double-layergraphene is rotated by π/ π/
2, in agreement with experiment shownin Fig. 3. If we assume that t >
0, which is qualita-tively in agreement with experiment shown in Figs. 3(a)and 3(b), I x − pol+1 k ∝ | M x − pol+1 k | ∼ sin ( θ q + k z d/
2) and I y − pol+1 k ∝ | M y − pol+1 k | ∼ cos ( θ q + k z d/
2) for the upperband ( s = +1). Therefore, we have determined that thevertical inter-layer hopping matrix element t betweentwo carbon p z orbitals sitting on top of each other is pos-itive; we will come back to this point later.Although this model can overall account for the ex-perimental data of double-layer graphene, there is adiscrepancy: the measured photoemission intensity at E = E D + 0 .
25 eV along the + k x direction for the X-polarization geometry is finite, whereas the theory pre-dicts this value to vanish as shown in the E D + 0 .
25 eVmap of the Ideal DG in Fig. 6(a). We believe that this dis-crepancy arises from the finite size of the light beam spot( ∼ × µ m ) which covers not only the double-layergraphene portion but also some single-layer graphene,as discussed in Fig. 1. In fact, double-layer graphenesamples inevitably contain a finite amount of single-layergraphene [17, 18]. The fraction of single-layer cover-age can be obtained by LEEM measurements [17, 18]:from this analysis shown in Fig. 1, we find that thedouble-layer graphene sample used here contains ∼ ∼
22% single-layer graphene. When thetheoretical photoelectron intensity maps of single- anddouble-layer graphene are correspondingly weighted andaveraged, the results denoted by SG+DG in Figs. 6(a)and 6(b) are in excellent agreement with the experimen-tal data shown in Figs. 3(a) and 3(b). This is even moreclear from the angular dependence of the photoelectronintensity maps shown in the red solid lines in Fig. 6(c)and 6(d). Note that the presence of single-layer graphenedoes not affect the results for the Y-polarization, which isobvious when we compare the red solid and green dashedlines in Fig. 6(d), because both the intensity maxima ofsingle- and double-layer graphene occur near θ = π . V. DISCUSSIONA. Berry’s phase
We have shown that, when the light polarization isrotated by π/
2, the maximum intensity position in I k in the k x - k y plane of single- and double-layer grapheneis rotated by π/n , where n = 1 and n = 2 for single-and double-layer graphene, respectively. The physicalmeaning of these rotations, whose origins rests on thephase factor exp ( ± i n θ q /
2) of the sublattice amplitudeof the wavefunctions [24], becomes clear upon a completecirculation of q around the Dirac point K ( θ q → θ q +2 π ), which directly gives a Berry’s phase β = n ∆ θ q / n π [24]. Recently the Berry’s phase interpretation of n [3–5] has been given by a different interpretation interms of pseudospin winding number [32].The fact that n θ q / I k in the formof either sin ( n θ q / ϕ ) or cos ( n θ q / ϕ ) withsome constant ϕ , demonstrates that the matrix el-ements directly contain information on the Berry’sphase. The rotation of light polarization givesan additional phase π/ ± i n θ q / → exp ( ± i n θ q / i π/ I k is modified accord-ingly from sin ( n θ q / ϕ ) → cos ( n θ q / ϕ ) orcos ( n θ q / ϕ ) → sin ( n θ q / ϕ ) upon the rotation of light polarization by π/
2, i. e. , the π/n rotation of I k .This prediction is exactly realized in our experimentalresults.The power of our method is that it can be extendedin a straightforward way to other materials with Berry’sphase β = n π (not necessarily π or 2 π ). In this case,the photoemission intensities for X- and Y-polarizationgeometries are given by I x − pol k ∝ sin ( n θ q / ϕ )and I y − pol k ∝ cos ( n θ q / ϕ ), where ϕ is a system-dependent constant. The important feature is that therotation of light polarization by π/ π/n for a system with β = n π re-gardless of the constant ϕ . Thus, we have demonstratedhere that the Berry’s phase can directly be measuredfrom polarization-dependent ARPES.Unlike methods based on magneto-transport experi-ments [4, 5], our new method has three important advan-tages. (i) The Berry’s phase of a specific electronic bandcan be measured, because ARPES has the angle-resolvingpower and also because one can set up a tight-bindingHamiltonian focussing on only the electronic states of in-terest: those two results can directly be compared witheach other. (ii) Due to the angle-resolving power, themeasured result is free from valley degeneracy for thecase of graphene. (iii) Our method does not need electricgating, which is essential for the transport measurements. B. The sign of inter-orbital hopping integrals
Another important finding of our study is that we candirectly extract, for the first time, the sign and the ab-solute magnitude of the inter-orbital hopping integrals(IOHIs) between non-equivalent localized orbitals of atight-binding Hamiltonian from experiment. Until now,in fact, the sign determination of an IOHI has resortedno to any experimental method, but to ab initio calcu-lations, e. g. , using maximally localized Wannier func-tions [10]. In order to understand an ambiguity relatedwith the experimental sign determination, we take thesimplest one-dimensional example, and extend the dis-cussion to a more complicated tight-binding model ofgraphitic systems than the one described previously. Weconsider simple tight-binding models having s -like local-ized states, the values of which are all positive in realspace (we can arbitrarily set this gauge without losinggenerality.) If there is only one localized orbital per unitcell in a one-dimensional tight-binding model as drawnin Fig. 7(a), the energy bandstructure varies with thesign of the IOHI t (cid:48) as shown in Fig. 7(b). Hence, thesign of IOHIs between “equivalent” orbitals can alwaysbe trivially determined [33].However, if we consider the case where there are twonon-equivalent localized orbitals φ s and φ (cid:48) s whose valuein real space is positive and if we denote the IOHI be-tween the nearest neighboring orbitals by t (cid:48)(cid:48) as drawn inFig. 7(c), the actual band structure is invariant when wechange the sign of t (cid:48)(cid:48) as shown in Fig. 7(d). Therefore,0 FIG. 7: (a) Schematic of a one-dimensional crystal having s -type orbital per unit cell. The nearest neighbor hoppingintegral is t (cid:48) . (b) Calculated electron energy band structureof the system depicted in (a) with different choices for thesign of t (cid:48) . (c) Schematic of a one-dimensional crystal havingtwo nonequivalent s -type orbitals per unit cell (i. e. , havingdifferent on-site energies). We assume that all the distancesbetween the centers of the nearest neighbor orbitals are thesame and hence the corresponding hopping integrals, denotedby t (cid:48)(cid:48) . (d) Calculated electronic band structure of the systemdepicted in c with different choices for the sign of t (cid:48)(cid:48) . even when the actual electronic band structure is empiri-cally determined, the sign of t (cid:48)(cid:48) cannot be determined. Ingeneral, an empirical tight-binding model with more thanone orbital per unit cell has this sign ambiguity problemfor IOHIs between “non-equivalent” orbitals, thus pre-venting an experimental measurement of IOHI from justthe energy band dispersions.We could understand this degeneracy as follows. If wedenote the Bloch sums of the two localized orbitals by φ ( k ) and φ ( k ), the tight-binding Hamiltonian H usingthis basis set reads H ( k ) = (cid:18) (cid:104) φ ( k ) | H | φ ( k ) (cid:105) (cid:104) φ ( k ) | H | φ ( k ) (cid:105)(cid:104) φ ( k ) | H | φ ( k ) (cid:105) (cid:104) φ ( k ) | H | φ ( k ) (cid:105) (cid:19) . (28)Note here that the matrix elements (cid:104) φ i ( k ) | H | φ j ( k ) (cid:105) , inwhich i, j ∈ { , } , involve not only the on-site or nearest-neighbor hopping processes but also all the other possiblehopping processes. Now, it is obvious that the followingHamiltonian H (cid:48) ( k ) has exactly the same eigenvalues as H ( k ): H (cid:48) ( k ) = (cid:18) (cid:104) φ ( k ) | H | φ ( k ) (cid:105) − (cid:104) φ ( k ) | H | φ ( k ) (cid:105)− (cid:104) φ ( k ) | H | φ ( k ) (cid:105) (cid:104) φ ( k ) | H | φ ( k ) (cid:105) (cid:19) . (29)What we have done by going from H ( k ) to H (cid:48) ( k ) isto change the sign of the IOHI between the two non-equivalent localized orbitals. In fact, the two matri-ces H ( k ) and H (cid:48) ( k ) are related by a unitary trans-form, which does not change the eigenvalues of a matrix, H (cid:48) ( k ) = U † H ( k ) U with U = (cid:18) − (cid:19) . (30) On the other hand, if we change the signs of the diagonalmatrix elements (cid:104) φ i ( k ) | H | φ i ( k ) (cid:105) , in which i ∈ { , } ,we get a different eigenvalue spectrum. Thus, there isno ambiguity in the sign of the IOHI between equivalentorbitals. This simple example illustrates that one cannotdetermine the signs of the IOHIs between non-equivalentorbitals just by looking at the measured electron energybandstructure.The tight-binding model for double-layer graphenethat we used in our calculations is based on the p z or-bitals of carbon atoms with two parameters: t and t forthe nearest-neighbor in-plane and the vertical inter-layerhopping integrals, respectively. The parameters t and t correspond to − γ (cid:48) and γ (cid:48) in the well-known Slonczewski-Weiss-McClure model, respectively [11, 12], and we haveused | t | = 3 .
16 eV and | t | = 0 .
39 eV [9]. The photoe-mission intensity map in the k x - k y plane is strongly de-pendent on the signs of both t and t as shown in Fig. 8.Because the four different choices of the signs produce ex-actly the same electronic band structure, it has not beenknown from experiments which choice of the signs of theIOHIs is physically correct, although the absolute valueshave been experimentally estimated [34, 35].This sign-ambiguity problem still exists even whenwe include more complicated hopping processes in themodel, especially the non-vertical inter-layer hopping in-tegrals γ (cid:48) and γ (cid:48) , there still exist unitary transformsthat leave the energy eigenvalues unchanged. In a tight-binding model having four hopping integrals ( γ (cid:48) , γ (cid:48) , γ (cid:48) ,and γ (cid:48) ), the following four different sets of parametersgive exactly the same electron energy bandstructure: ( γ (cid:48) , γ (cid:48) , γ (cid:48) , γ (cid:48) ), ( γ (cid:48) , − γ (cid:48) , − γ (cid:48) , − γ (cid:48) ), ( − γ (cid:48) , γ (cid:48) , γ (cid:48) , − γ (cid:48) ), and( − γ (cid:48) , − γ (cid:48) , − γ (cid:48) , γ (cid:48) ), assuming that the first set is com-posed of the values currently accepted and used when theSWMc model is considered. In principle, there shouldbe eight different sets of parameters giving the same en-ergy bandstructure; however, from our knowledge thatthe nearest-neighbor intralayer hopping integrals in dif-ferent graphene layers are the same, we have reduced thenumber of candidates to four. This is also the reason whywe considered only the four cases in Fig. 8.We have shown that the choice of t < t > γ (cid:48) > γ (cid:48) > FIG. 8: (a-d) Calculated intensity maps for four different choices of the signs of nearest-neighbor in-plane and vertical inter-layerhopping integrals, t and t , respectively. Case (c) agrees with the experimental results. VI. SUMMARY
We have shown that ARPES can be used as a power-ful tool to directly measure quantum phases such as theBerry’s phase of a specific electronic band in graphenewith advantages compared to the interference type ofmeasurements [36–38] which do not give any informationon the band-specific Berry’s phase, and the sign of thehopping integral between non-equivalent orbitals, nevermeasured for any material before. The experimental andtheoretical procedures developed here can be applied instudying the electronic, transport, and quantum interfer- ence properties of a huge variety of materials.
Acknowledgments
We gratefully acknowledge D.-H. Lee, J. Graf, C. M.Jozwiak, S. Y. Zhou and H. Zhai for helpful discussions.This work was supported by the Director, Office of Sci-ence, Office of Basic Energy Sciences, Materials Sciencesand Engineering Division, of the U.S. Department of En-ergy under Contract No. DE-AC02-05CH11231. [1] A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, andJ. Zwanziger
The Geometric Phase in Quantum Systems (Springer-Verlag, Berlin Heidelberg, 2003).[2] Y. Aharonov and D. Bohm,
Phys. Rev , 485 (1959).[3] E. McCann and V. I. Fal’ko,
Phys. Rev. Lett. , 086805(2006).[4] Y. Zhang, Y. -W. Tan, H. L. Stormer, and P. Kim, Nature , 201 (2005).[5] K. S. Novoselov, E. McCann, S. V. Morozov, V. I. Fal’ko,M. I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin, andA. K. Geim,
Nature Phys. , 177 (2006).[6] Y. Taguchi, Y. Oohara, H. Yoshizawa, N. Nagaosa, andY. Tokura, Science , 2573 (2001), and referencestherein.[7] S. Murakami and N. Nagaosa,
Phys. Rev. Lett. ,057002 (2003).[8] D. Hsieh et al. , Science , 919 (2009).[9] A. Gr¨uneis et al. , Phys. Rev. B , 205425 (2008).[10] N. Marzari and D. Vanderbilt, Phys. Rev. B , 12847 (1997).[11] J. C. Slonczewski and P. R. Weiss, Phys. Rev. , 272(1958).[12] J. W. McClure,
Phys. Rev. , 612 (1957).[13] S. -W. Yu et al. , Surf. Sci. , 396 (1998).[14] E. L. Shirley, L. J. Terminello, A. Santoni, and F. J.Himpsel,
Phys. Rev. B , 13614 (1995).[15] M. Mucha-Kruczy´nski et al. , Phys. Rev. B , 195403(2008).[16] E. Rollings et al. , J. Phys. Chem. Solids , 2172 (2006).[17] D. A. Siegel et al. , Appl. Phys. Lett. , 243119 (2008).[18] T. Ohta et al. , New J. Phys. , 023034 (2008).[19] D. Pescia, A. R. Law, M. T. Johnson, and H. P. Hughes, Solid State Commun. , 809 (1985).[20] K. C. Prince, M. Surman, Th. Lindner, and A. M. Brad-shaw, Solid State Commun. , 71 (1986).[21] S. Y. Zhou et al. , Nature Mater. , 770 (2007).[22] Th. Seyller, K. V. Emtsev, F. Speck, K. -Y. Gao, and L.Ley, Appl. Phys. Lett. , 242103 (2006). [23] I. Gierz, J. Henk, H. H¨ochst, C. R. Ast, and K. Kern, Phys. Rev. B , 121408(R) (2011).[24] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.Novoselov, and A. K. Geim, Rev. Mod. Phys. , 109(2009).[25] T. Ohta et al. , Phys. Rev. Lett. , 206802 (2007).[26] P. Y. Yu and M. Cardona, Fundamentals of Semiconduc-tors: Physics and Materials Properties ; Springer-Verlag:Berlin Heidelberg, 1999.[27] A. F. Starace
Phys. Rev. A , 1242 (1971).[28] S. Ismail-Beigi, E. K. Chang, and S. G. Louie, Phys. Rev.Lett. , 087402 (2001).[29] C. -H. Park and S. G. Louie, Nano Lett. , 1793 (2009).[30] Ando, T. J. Phys. Soc. Jpn. , , 104711.[31] Zhou, S. Y.; Gweon, G. -H.; Lanzara, A. Ann. Phys. , , 1730-1746.[32] C. -H. Park and N. Marzari, arXiv:1105.1159v4.[33] M. Z. Hasan et al. , Phys. Rev. Lett. , 246402 (2004).[34] W. W. Toy, M. S. Dresselhaus, and G. Dresselhaus, Phys.Rev. B , 4077 (1977).[35] A. Misu, E. E. Mendez, and M. S. Dresselhaus, J. Phys.Soc. Jap. , 199 (1979).[36] S. A. Werner, R. Colella, A. W. Overhauser, and C. F.Eagen, Phys. Rev. Lett. , 1053 (1975).[37] A. Tomita and R. Y. Chiao, Phys. Rev. Lett. , 937(1986).[38] T. Bitter and D. Dubbers, Phys. Rev. Lett.59