Calculate the charge carrier density for intrinsic diamond

Question

Question 1: Surface Transfer Doping of Diamond

Diamond has a large band gap of 5.47 eV, which makes diamond extremely insulating.

(i) Calculate the charge carrier density for intrinsic diamond at room temperature (300 K). What is the volume of a diamond sample for it to contain one mobile electron carrier?

Note: effective density of states in the valence band of diamond at RT Nv = 3×10¹⁹ cm^-3 effective density of states in the conduction band of diamond at RT Nc = 1×10¹⁹ cm^-3

Figure 1. Schematic diagram showing the energy level alignment before and after surface transfer doping.

(ii) Comment on the doping efficiency of boron for diamond at RT.

(iii) Plot the areal hole density p (in unit of cm^-2) as a function of the surface valence band position u_s in the range of -0.5 eV to 1 eV. Use log scale for the areal hole density.

(iv) Show that ELUMO - EF in Eqn. (1.3) can be rewritten as ELUMO - EF = Δ₀ + ΔΦ(p) + us(p).2

(v) Plot the transfer doping efficiency η = N_A-/N_A as a function of areal hole density p in the range of 109 cm^-2 to 1014 cm^-2 for initial activation energy Δ₀ of 0.2 eV, 0 eV, -0.2 eV, -0.5 eV, and -1.0 eV respectively (all in one graph) at RT. Use log scale for the areal hole density. Why does the doping efficiency drop to zero above a certain acceptor coverage?

(vi) Plot areal hole density p as a function of C60F48 coverage NA in the range of 10¹¹ molecules/cm² to 10¹⁴ molecules/cm2 for initial activation energy Δ₀ of -0.5 eV, -0.6 eV and -0.7 eV at RT. Use double log scale.

(vii) Suppose that we can independently determine the areal hole density on diamond as a function of molecular coverage using Hall effect measurements, and obtain data as follows:

Hole Density (cm-2)	1.5×1011	4.0×1012	9.0×1012	1.0×1013	1.2×1013
Molecular Coverage (cm-2)	1.5×1011	4.0×1012	1.0×1013	2.5×1013	1.0×1014

Using the data and combine with your plot in (vi), determine the initial activation energy Δ0 for C60F48 and thus determine the electron affinity of C60F48. The ionisation energy of diamond is 4.4 eV.

(viii) What determines the maximum achievable areal hole density on diamond? What is your strategy in choosing the best-performing surface acceptors to enable p-type surface transfer doping of diamond?

Question 2: Surface Transfer Doping of Graphene

Surface transfer doping also proves to be an effective and non-destructive doping method for two-dimensional materials like graphene.

(i) Follow Figure 1 in Question 1, draw a similar energy level alignment diagram, before and after surface transfer doping, for a surface acceptor with a given acceptor energy Δ₀ on graphene. In your graph, label major energy levels and indicate graphene bands using the Dirac cone.

(ii) Show that the density of states of graphene near Dirac points can be derived as:

N (E) = 2|E - E_D|/ Π (ηvF )  (1.5)

in which ED is the energy of Dirac points, and vF = 1×10⁶ m/s is the Fermi velocity of graphene. Note that the degeneracy in graphene is 4 due to the two inequivalent carbon atoms in the unit- cell (on top of the regular spin degree of freedom).

(iii) From the density of states, derive that the electron or hole carrier density in graphene can be related to the energy distance between the Dirac point and the Fermi energy (in low T limit):

n[or p] = ((E - E )²/DF)/ Π (ηvF )

(iv) Suppose again we are using C60F48 as surface acceptors for p-type transfer doping of graphene, and the graphene sample is intrinsic at the beginning (i.e. Fermi energy lies at the Dirac point). Follow the procedures described in Question 1 for diamond and with the help of the energy level diagram you have sketched in (i), establish the correlation between the areal hole density in graphene and the areal electron density in the acceptor layer. You should use Eqn. 1.6 so that you can write areal electron density in the acceptor layer as a function of areal hole density p. Use the same C as the one used for diamond/C60F48.

(v) Plot the transfer doping efficiency η = N_A- / NA as a function of areal hole density p in the range of 109 cm-2 to 1014 cm-2 for initial activation energy Δ0 of 0.2 eV, 0 eV, -0.2 eV, -0.5 eV, and -1.0 eV respectively (all in one graph) at RT. Use log scale for the areal hole density.

(vi) Plot areal hole density p as a function of C60F48 coverage NA in the range of 1011 molecules/cm2 to 1015 molecules/cm2 for initial activation energy Δ0 of 0.2 eV, 0 eV, -0.2 eV, -0.5 eV, and -1 eV at RT. Use double log scale.

(vii) If you are going to achieve n-type surface transfer doping of graphene in order to build planar graphene p-n junction devices. What will be your strategy in choosing the surface donors? List one or more potential candidates for n-type surface transfer doping of graphene and state your rationale.