Creator:John E. Hove Date Created:January 10, 1955 Place Created:Downey, California Keywords:graphite,electronic density Context:article reprinted from The Physical Review ************************************************** Electronic Density of States of Graphite John E. Hove Reprinted from The Physical Review, Vol. 97, No. 6, pp. 1717-1718, March 15, 1955 Reprinted from The Physical Review, Vol. 97, No. 6, 1717-1718, March IS, 19SS Printed in U. S. A. Electronic Density of States of Graphite* John E. Hove Nuclear Engineering and Manufacturing, North American Aviation, Inc., Downey, California (Received January 10, 1955) IT has been shown by Carter and Krumhansl1 that the electronic density of states of graphite near the top of the filled band is asymmetric about the energy where this band touches the next, unfilled, band. This conclusion is based on a modification of the Wallace2 band structure. This modification consists of recognizing the difference in the number of neighbors between the four atoms in the unit cell and thus assigning two of Wallace's diagonal matrix elements a different value than the other two. The discontinuity of the density of state at the band edge is proportional to the difference (/7ii — 7/22) and if this is large enough (greater than 271) the bands need not touch at all. It should be mentioned that asymmetry in the energy contours is a necessary feature to explain the sizeable negative Hall coefficient.3 The purpose of the present note is to point out that the effect of including next-nearest neighbors (in the basal plane) is to introduce an asymmetry in the density of states in the same qualitative fashion as above. Next-nearest neighbors in the plane are easily taken into consideration and, in fact, Wallace2 has already done so, although he neglects such terms when calculating the density of states. From reference 2, the energy of a state k, including nearest and next-nearest neighbors in the plane and nearest neighbors out of the plane, is: e= —71 COS5C£*±[7I2 cos2|cfeI+l7o2a2Kx„2]i -bo'ah*S. (1) Here ki„2 = ki2+k„2, where k= k — k(corner), € is measured from the band edge, and 70, 70', and 71 are the resonance integrals involving coplanar nearest and next-nearest and interplanar nearest neighbors, respectively. The only effect of including 70' is a term in kx„2. In the two-dimensional approximation (71 = 0) the 70' term does not affect the density of states to a first order. However, when 71 is retained, the terms in 70' and in 70 are of the same order for small kxv and the density of states curve is altered markedly. The calculation for this case can be readily performed with the assumption that 7o'M1+—(ir+12i)07?i) vStt27o2L 2 271 + (2) For -27i<€<0, 271 V271/ +67ri7oT)i) (2) 271 r * |«| ,V(e)~- 1--7J017H--(tt— 12tjot?i V5t27o2L 2 27i V271/ f-6jnj0i?i) (3) o Fig. 1. The graphite density of states (the solid line is for -yo' = 0). The effect of 70' is seen in Fig. 1. There is a discontinuity in N(e) at zero and both its value and slope are less for the lower band. This is the same qualitative feature found by Carter and Krumhansl. The next-nearest neighbor effect may be of equal or greater importance than the effect of coordination number, although estimates of relative magnitude are difficult to make. It is, however, significant that these two modifications (dissimilar in nature) have similar effects on the density of states curve. This is probably a general property of the graphite lattice symmetry in that if the correct matrix elements were expanded in a Fourier series the above asymmetry would generally appear. The coefficients in the series would be disposable constants and not directly identifiable with any particular overlap integral by itself. The effects considered above and by Carter and Krumhansl would, however, be a major part of the first order coefficients. This change in the density of states is in the correct direction to explain the negative Hall coefficient. The author would like to acknowledge discussions of this topic with J. A. Krumhansl. * This note is based on studies conducted for the U. S. Atomic Energy Commission. 1 J. L. Carter and J. A. Krumhansl, J. Chem. Phys. 21, 2238 (1953). 2 P. R. Wallace, Phys. Rev. 71, 622 (1947). 3 G. Hennig, J. Chem. Phys. 20, 1438 (1952).