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Chapter 14 Graphene-based Transistor 14.1 Band structure of graphene 14.2 Doped graphene 14.3 Processing of graphene-based transistor 14.4 Electrical properties of graphene-based transistor 1 14.1 Band structure of graphene c a b 2ab cos(120 ) 2 a0 a1 y a2 2 2 2a02 2a02 (0.5) 3a02 c 3a0 o 3a0 3 3 a1 ( a0 , a0 ) ( 3 , 1) 2 2 2 3a0 3 3 a2 ( a0 , a0 ) ( 3 , -1) 2 2 2 x 2dsinθ=nλ We study crystal structure through the diffraction of photons, neutrons, and electrons. The diffraction depends on the crystal structure and on the wavelength. According to the Bragg law, reflection can occur only for wavelength λ≤2d(d is interspacing). Although the reflection from each plane is specular, for only certain values of θ will the reflection from all parallel planes add up in phase to give a strong reflected beam. Each plane in crystal reflects 10-3 to 10-5 of the incident radiation, so that 103 to 105 planes may contribute to the formation of the Bragg-reflected beam in a perfect crystal. 2 The electron number density n(r) is a periodic function of r, with periods a1, a2, a3 in the directions of the three crystal axes. Thus n(r+T) n(r) (1’) T is a translation function of the form T=u1a1 +u 2a 2 +u3a 2 where u1,u 2 , and u3 are constants A lattice translation operation is defined as the displacement of a crystal by a crystal translation vector described in Eq. (1’), any two lattice points are connected by a vector of this form. r T r ' =r+T r’ Such periodicity creates an ideal situation for Fourier analysis. The most interesting properties of crystals are directly related to the Fourier components of the electron density. 3 Expanding n(x) in a Fourier series of sines and cosines, we have n( x a) n0 [C p cos(2 px / a) S p (2 px / a)] n( x) or n( x) n p exp(i 2 px / a) (1) (2) p We say that 2πp/a is a point in the reciprocal lattice or Fourier space of the crystal. The reciprocal lattice points tell us the allowed terms in the Fourier series (1) and (2). A term is allowed if it is consistent with the periodicity of the crystal, as shown in Fig. 2. Other points in the reciprocal space are not allowed in the Fourier expansion of a periodic function. n(x) a a a ‧ ‧ ‧ ‧2 ‧4 0 a a 4 2 a a a Fig. 2 a G 4 A periodic function n(x) of period a, and the terms 2p/a that may appear in the Fourier transform . n( x) np exp(i 2 px / a) p The extension of the Fourier analysis to periodic functions n(r) in three dimensions is straightforward. We must find a set of vector G such that n(r) nG exp(iG r) (3) G is invariant under all crystal translations T that leave the crystal invariant. Constructing the axis vectors b1, b2, and b3 of the reciprocal lattice: b1 2 a 2 a3 a 3 a1 a a , b2 2 , b3 2 1 2 a1 a 2 a 3 a1 a 2 a 3 a1 a 2 a 3 (4) 5 If a1, a2, and a3 are primitive vectors of the crystal lattice, then b1, b2, and b3 are the primitive vectors of the reciprocal lattice. Each vector defined by (1) is orthogonal to two axis vectors of the crystal lattice. Thus b1, b2, and b3 have the property bi a j 2ij where ij = 1 if i=j and ij = 0 if i ≠ j (5) Points in the reciprocal lattice are mapped by the set of vectors G v1b1 v2b2 v3b3 (6) where v1 , v2 , v3 are integers. A vector G of the form is a reciprocal lattice vector. Every crystal structure has two lattices associated with it, the reciprocal lattice and the reciprocal lattice. A diffraction pattern of a crystal is a map of the reciprocal lattice of the crystal. A microscope image is a map of the crystal structure in real space. 6 Vectors in the direct lattice have the dimensions of [length]; vectors in the reciprocal lattice have the dimensions of [1/length]. The reciprocal lattice is a lattice in the Fourier space associated with the crystal. Wavevectors are always drawn in Fourier space, so that every position in Fourier space may have a meaning as a description of a wave, but there is a special significance to the points defined by the set of G’s associated with a crystal structure. The vectors G in the Fourier series (3) are just the reciprocal lattice vectors (6), for then the Fourier series representation of the electron density has the desired invariance under any crystal translation T= u1a1+u2a2+u3a3. From (3) n(r+T) nG exp(iG r) exp(iG T) G (7) After derivation, we have n(r+T) n(r) Diffraction condition: Let k being an incident wave and G is a reciprocal lattice Vector, we have (8) 2k G G 2 This particular expression is often used as the condition for diffraction. 7 Brillouin zones A Brillouin zone is defined as a Wigner-Seitz primitive cell in the reciprocal lattice. The value of the Brillouin zone is that it gives a vivid geometrical interpretation of the diffraction condition 2k G G 2 (8) we work in reciprocal space, the space of the k’s and G’s . Select a vector G from the origin to a reciprocal lattice point. Construct a plane normal to this vector G at its midpoint. This plane forms a part of the zone boundary, as shown in Fig. 3. Any vector from the origin o the plane 1, such as k1, will satisfy the diffraction condition. 1 1 k1 ( G c )=( G c ) 2 2 2 Fig. 3 a Fig. 3 b 8 Construction of the first Brillouin zone for an oblique lattice in two dimension. Construction of the first Brilluin zone for an oblique lattice in 2-D. We first draw a number of vectors from O to nearby points in the reciprocal lattice. Next we construct lines perpendicular to these vectors at their midpoints. The smallest enclosed area is the first Brillouin zone (Fig. . Fig. 4 Fig. 5 a Fig. 5 b Introduction to Solid Physics by C Kittel 9 Fig. 6(a) Primitive lattice of graphene, (b) Reciprocal lattice of graphene and its Brillouin zone E (k x , k y ) 0 kya ka 3k x a 2 y 1 4cos cos 4cos 2 2 2 Fig. 7 Enengy band of graphene derived from the Tight-binding approximation Physics Bimonthly 2011.4, p.191 10