Crystal Structure of Graphite, Graphene and Silicon Dodd Gray, Adam McCaughan, Bhaskar Mookerji∗ 6.730—Physics for Solid State Applications (Dated: March 13, 2009) We analyze graphene and some of the carbon allotropes for which graphene sheets form the basis. The real-space and reciprocal crystalline structures are analyzed. Theoretical X-ray diffrac- tion (XRD) patterns are obtained from this analysis and compared with experimental results. We show that staggered two-dimensional hexagonal lattices of graphite have XRD patterns that differ significantly from silicon standards. The wide-variety of carbon allotropes and their associ- spacing between sheets c = 6.71A.˚ The sheets align such ated physical properties are largely due to the flexibility that their two-dimensional hexagonal lattices are stag- of carbon’s valence electrons and resulting dimensionality gered, either in an ABAB pattern or an ABCABC pat- of its bonding structures. Amongst carbon-only systems, tern. The ABAB alignment is shown in Figure 1, which two-dimensional hexagonal sheets—graphene—forms of indicates four atoms per unit cell labeled A, B, A’, and the basis of other important carbon structures such as B’, respectively. The primed atoms A–B on one graphene graphite and carbon nanotubes. (:: Say something about layer are separated by half the orthogonal lattice spacing interesting band structure here) from the A’–B’ layer; BB’ atomic pairs differ from their In the following, we will examine the planar lat- corresponding AA’ pairs in their absence of neighboring tice structure of graphene and its extension to higher- atoms in adjacent layering planes. The coordinates of dimensional lattice structures, such as hexagonal these atoms forming the basis are given by: graphite. We first analyze the lattice and reciprocal- c space structures of two-dimensional hexagonal lattices of ρA = (0, 0, 0) ρA′ = 0, 0, carbon, and use the resulting structure factors to esti- 2 mate the x-ray diffraction (XRD) intensities of graphite. a 1 a a c ρB = , 1, 0 ρB′ = − , − , (1.1) We conclude by comparing its calculated XRD spectra to 2 √3 2√3 2 2 experimental spectra of graphene and crystalline silicon. With respect to an orthonormal basis, the primitive lattice vectors are given by, 1. PRELIMINARY QUESTIONS a = a √3/2, 1/2, 0 a = a = 2.46A˚ 1 − | 1| 1.1. Lattice Structure a2 = a √3/2, 1/2, 0 a2 = a = 2.46A˚ | | Our discussion of the crystal structure of graphite fol- a3 = c (0, 0, 1) a3 = c = 6.71A˚. (1.2) lows partially from D.D.L. Chung’s review of graphite [1]. | | When multiple graphene sheets are layered on top of each The magnitudes of the primitive lattice vectors corre- other, van der Walls bonding occurs and the three di- spond to the lattice constants parallel and perpendicu- mensional structure of graphite is formed with a lattice lar to the graphene sheet. The corresponding ABCABC layer forms a rhombohedral structure with identical lat- tice spacing parallel and orthogonal to the layer. 1.2. Reciprocal Lattice Structure Recall that the reciprocal lattice vectors bi are defined as a function of the primitive lattice vectors ai such that a a b 2 3 1 = 2π a a× a FIG. 1: In-plane structure of graphite and reciprocal lattice 1 3 3 a· ×a vectors [1]. 3 1 b2 = 2π × a2 a3 a1 a· ×a b 1 2 3 = 2π a a× a (1.3) 3 · 1 × 2 ∗Electronic address: [email protected], [email protected], mook- [email protected], The reciprocal lattice vectors for graphite are then 2 of points in space using the specified lattice vectors and atom bases. Normals generated from lattice vector sums were used to extract planes and display them. Varying colors were used to depict vertical spacing between adja- cent planes. 2.2. Structure Factors and X-Ray Diffraction Intensities Calculate the structure factor for all the 2 reciprocal lattice vectors Kl < 16 (2π/a) . The structure factor is calculated as n −iKi·ρi FIG. 2: Reciprocal lattice planes [001], [110], and [111]. Mp (Ki) = fc (Ki) e Xj=1 given by, where fc is the structure factor of Carbon and ρi are the b 2π 1 b 2π 2 basis vectors of our lattice. We find that only four unique, 1 = , 1, 0 1 = K a √3 − | | a √3 non-zero values of Mp ( i) occur in the reciprocal lattice. 2π 1 2π 2 Each of these corresponds to the height of a diffraction b2 = , 1, 0 b2 = intensity peak and their relative values are referenced in a √3 | | a √3 Table I. 2π 2π b = (0, 0, 1) b = . (1.4) 3 c | 3| c Calculate the ratio of the intensities ex- pected for the following lines of the diffraction The reciprocal lattice plane generated by the b and b 1 2 pattern with respect to the [111] line: [100], vectors forms the outline of the first Brillouin zone, as [200], [220], [311], and [400]. depicted in Figure 1. The intersection of the the planes kz = 2π/c with the plane forms a hexagonal prism of ± Including the structure factor, there are other several height 4π/c. factors contributing to the intensities of the diffraction peaks [3]: 1.3. Atomic form factors 1. The Lorenz correction is a geometric relation al- tering the intensity of an x-ray beam for different As carbon is the only element present in graphene and scattering angles θ. graphite, the atomic form factor is uniform across the entire crystal, and thus can be factored out when calcu- 2. The multiplicity factor p is defined as the num- lating the structure factor. Thus the atomic form factor ber of different planes having the same spacing has no effect on the relative intensities of x-ray diffrac- through the unit cell. Systems with high symme- tion occuring in different planes of graphite. According tries will have different planes contributing to the to the NIST Physics Laboratory, the atomic form factor same diffraction, thereby increasing the measured of carbon varies from 6.00 to 6.15 e/atom with incident intensity. radiation ranging from 2 to 433 KeV [2]. 3. Temperature, absorption, polarization each con- tribute higher-order corrections ultimately ignored 2. X-RAY DIFFRACTION in our calculation. These include Doppler broad- ening from thermal vibrations in the material, ab- sorption of x-rays, and the polarization of initially 2.1. Planes in the Reciprocal Lattice unpolarized x-rays upon elastic scattering. Provide pictures of the crystal and of the These factors all contribute to the relative intensity of a reciprocal lattice in the [100], [110], and [111] [hkl] diffraction peak given by planes. Indicate the vertical positions of 2 atoms with respect to the plane. 2 1 + cos 2θ I[hkl] (θ)= p Ma (Km) . (2.1) | | sin2 θ cos θ Pictures of the crystal and of the reciprocal lattice in the [100], [110], and [111] planes are included in Fig- Using the formula from the previous question to calcu- ure 2. In MATLAB, the crystal was represented as a set late the ratios of the structure factors in the given planes, 3 diffraction? The crystal structure of another common semiconduc- tor material, silicon (Si) is featured in Figure 3. Silicon forms a diamond cubic crystal structure with a lattice spacing of 5.42A.˚ This crystal structure corresponds to a face-centered cubic Bravais lattice whose unit-cell basis contains 8 atoms located at vector positions, a d = ~0 d = (1, 3, 3) 0 4 4 a a d = (1, 1, 1) d = (2, 2, 0) 1 4 5 4 a a d = (3, 3, 1) d = (2, 0, 2) 2 4 6 4 a a d = (3, 1, 3) d = (0, 2, 2) . (2.2) 3 4 7 4 FIG. 3: Crystal structure of silicon. The structure factor contributing to its X-ray diffrac- tion pattern is given by TABLE I: X-Ray diffraction intensities for graphite and sili- con [4, 5]. Structure factors are included in parentheses. n (j) −iKm·dj Ma (Km) = f (Km) e Si C (Graphite) a ◦ ◦ Xj=1 2θ ( ) Exp. 2θ ( ) Calc. Exp. = f(1+( 1)h+k +( 1)k+l +( 1)h+l [111] 28.47 100.00 — (3) — − − − h+k+l 3h+k+l 3h+3k+1 [100] — — 42.77 4(3) 3.45 +( 1) +( i) +( i) − − − [200] — — — (1) — +( i)h+3k+1) [002] — — 26.74 106(16) 100 − = f 1+( 1)h+k +( 1)k+l +( 1)h+l [220] 47.35 64.31 — (1) — − − − [311] 37.31 37.31 — (3) — 1+( i)h+k+l . (2.3) [400] 69.21 9.58 — (1) — · − This term undergoes a number of simplifications based we obtain the following results presented in Table I. Cal- on the parity of its Miller indices. If [hkl] are all even culations show that planes [100], [200], [220] and [400] ex- and are divisible by 4, then Ma (Km) = 8f. If they are hibit relative diffraction intensities 1/3 that of the [111] not divisible by 4 or have mixed even and odd values, and [311] planes. The [002] plane exhibits the highest in- then Ma (Km) = 8f. Lastly if [hkl] are all odd, then tensity diffraction, 16/3 that of the [111] and [311] planes. Ma (Km) = 4f (1 i). ± We also found several non-zero structure factors that are The experimental X-ray diffraction intensities from not present in the experimental data. these contributions are listed in Table I. The intensity values for silicon were measured with respect to a ref- erence value I/I0 = 4.7, which is a direct ratio of the 2.3.