Diameter-driven crossover in resistive behaviour of heavily doped self-seeded germanium nanowires

The dependence of the resistivity with changing diameter of heavily-doped self-seeded germanium nanowires was studied for the diameter range 40 to 11 nm. The experimental data reveal an initial strong reduction of the resistivity with diameter decrease. At about 20 nm a region of slowly varying resistivity emerges with a peak feature around 14 nm. For diameters above 20 nm, nanowires were found to be describable by classical means. For smaller diameters a quantum-based approach was required where we employed the 1D Kubo–Greenwood framework and also revealed the dominant charge carriers to be heavy holes. For both regimes the theoretical results and experimental data agree qualitatively well assuming a spatial spreading of the free holes towards the nanowire centre upon diameter reduction.

where is the electrostatic potential at the surface, is vacuum permittivity, is the dielectric constant of germanium, and is the charge of the electron. The carrier concentration will fall off gradually away from the surface, and there will not be a sharp distinction between the occupied space charge region and the empty centre of the nanowire. For this reason Equation S2 is only valid when .
The volume of the space-charge region is the volume of the whole nanowire minus the inner cylindrical region with no charge: and thus we have for the carrier concentration in the space charge region: where we have again assumed that there is a constant carrier concentration over some distance . Finally, by multiplying by the charge of the electron and the mobility, , we can find the dependence of the resistivity, , on the diameter: . (S5)

Wavefuction of the Charge Carriers
To calculate the mobility with the Kubo-Greenwood formula we first need to calculate the momentum relaxation times for different scattering processes. For this we require the wavefunction of the carriers in the wire. We model the nanowire as an infinite cylindrical well, assuming that the mobile holes are present throughout the entire wire volume. The advantage of this approach is that it allows us to solve for the wavefunction analytically. In cylindrical coordinates ( ( )) the wavefuction, ( ) can be decomposed into its radial, ( ), angular, ( ), and longitudinal, ( ), parts [8]: For our calculations we only require the radial part due to cylindrical symmetry of the system. The radial part of the wavefunction is given by [8]: where , , is the m-th Bessel function of the first kind and is the n-th positive zero of the m-th Bessel function. Each possible pair of values for and corresponds to a different subband. We will label each subband with a single index, , instead of with and for clarity in later equations. We note that the assumption of an infinite potential well leads to parabolic bands. However, in low dimensional structures the bands are no longer well described as parabolic which can be accounted for by a nonparabolic correction term, . We took the value of to be 0.7 eV -1 for heavy holes and 0.2 eV -1 for light holes, as has previously been calculated for germanium nanowires [10].
The number of charge carriers in a nanowire can be calculated by integrating the density of states, ( ), over energy up to the Fermi level, : This expression is only strictly valid at 0 K, but can be used if the Fermi level does not change by a significant amount as the sample is heated to room temperature, which should be the case for our quasi metallic nanowires [11]. Inserting Equation   S4 for and ( ) (see below) into Equation S8 allows us to find the Fermi level as a function of diameter and the density of surface states.

Density of States and Expectation Value of Potential Energy
We calculate the density of states in one dimension starting from the nonparabolic bands. In this the following expressions hold 1 where ( ) is the group velocity of the carrier in subband at energy and is the valley degeneracy. It should be noted that these expressions are valid for either the conduction band or the valence band. In the case of the valence band there are no valleys and so =1. Furthermore, the effective mass will no longer be constant as the bands are no longer parabolic. The effective mass that appears in the expressions above is the effective mass for the parabolic band.
In these equations also the term appears which is the expectation value of the potential energy for a given subband [4,9], ∫ ( ) ( ) Due to the fact that we are setting the potential inside the wire to be at all points the term vanishes everywhere, simplifying the calculation.

Hole-Phonon Scattering
Charge carriers can be scattered by either acoustic or optical phonons. Acoustic phonons have little or no momentum at the Brillouin zone centre, in contrast to optical phonons [11]. The momentum relaxation time for a hole in the subband due to scattering from an acoustic phonon is given by [5]: where is the average acoustic deformation potential which, is the density of germanium, is the speed of sound in germanium, ( ) is the density of states in the sub-band , and is the so called form factor for a one dimensional wire, given by [5]: The momentum relaxation time associated with scattering from optical phonons is given by [5]: where is the average optical deformation potential, is the energy of an optical phonon, ( ) is the Fermi-Dirac distribution, is the phonon number, determined by ( ( ) ) [12], and the signs account for forward and backward scattering, both of which are included in the calculation. We use the angular frequency of bulk phonons in our model which has been found to be a good approximation for germanium nanowires with diameters above 10 nm [8].
Finally we come back to the effective deformation potentials, in Equation S10 and in Equation S12 . describes the local energy shift of the valence band that is caused by the presence of a phonon distorting the crystal structure [13,14]. The acoustic deformation potential is a second rank tensor quantity [13,14], related to the strain tensor. For the valence band there are three deformation potentials [15] due to the fact that the valence band is derived from a p-orbital, that is, deformation potential related to (i) isotropic deformations, (ii) deformations along the [100] direction and (iii) deformations along the [111] direction. Despite this complexity it has been found that a single, scalar, effective deformation potential can be used which results in an error of less than 5% in the final calculation [16,17]. Similarly the effective optical deformation potential should be a vector quantity that is related to the displacement of the atoms by the phonon [13,14]. This too can be approximated by an effective scalar which has been shown to introduce a negligible error [16,17].

Coulomb Scattering
The momentum relaxation time for a carrier in a subband to scatter due to a Coulomb scattering centre located at is given by the following set of equations [18]: assumed that all of the holes were located in the germanium core. In this case the Green's function is given by the following set of equations [4]:  To demonstrate that the value of the Green's function is not strongly affected by the parameters thickness and dielectric constant of the shell, we examined the effect of changing them to a range of values. We assumed the value of the dielectric constant to be that of silicon, 11.9 [19], and that of germanium oxide, 7.4. The value of the Green's function was approximately 10 % lower of the case of the silicon compared to germanium oxide. A change of less than 1 % was seen when we changed the thickness of the shell from 20 nm to 1 nm. Therefore the choice of parameters here does not have a large bearing on the final result. For these reasons we took the dielectric constant to be the same as for germanium oxide and assumed a constant thickness of 1 nm.
In Figure S1 we show the Green's function for scattering from the first ( ) to the second ( ) subband for a nanowire with a 12 nm diameter and a wire with a

S8
wire. The Green's function increases strongly approaching the surface of the wire, where the trapped charge is located, especially for the larger diameter wires. In this case the value of the Green's function at the surface ( ) is 360 times larger than the value at the centre ( ) of the nanowire for the 12 nm wire, and 3958 times larger in the case of the 17 nm wire. For this reason the Coulomb scattering will be strongly affected by the distribution of the carriers within the wire, especially for larger diameter wires. We expect that at large diameters the charge carries will be located closer to the surface, and so scattering from electrons in the interface states will play a larger role. We do not take into account this carrier distribution and we are therefore likely to underestimate quantitatively the Coulomb scattering especially for large diameter wires.
In Table S1 the values of the parameters used in the calculation are shown. The calculation of the mobility involves solving the Kubo-Greenwood formula [6,7], which contains an integral over the energy from zero to infinity. In principle we should therefore include infinitely many subbands. In practice however we solve this integral numerically and include only a limited number of subbands. Only holes near to the Fermi level contribute to the mobility as quantified (following Refs. [4] and [5]) by the term ( ( )).  shows the position of the Fermi level relative to the subband minima.

S9
In Figure S2a we plot the position of the bottom of the first 20 heavy-hole bands, in blue, and the first 3 light hole bands, in orange. It becomes apparent that the energy separation between the bottoms of the bands is increased as the diameter is decreased. Although we have plotted the bottom of the subband ( ) here, the same behaviour is seen for all values of . Figure S2b shows the same evolution of the band bottoms as Figure S2a, but with a smaller energy scale to better show the change of the Fermi level at different diameters. It should be noted that in Figure S2b the scale on the y-axis is in steps of 25 meV, which is the thermal energy at room temperature, and therefore provides insight into the diameter at which single subband effects may be occurring.
The mobility of light and heavy holes, and respectively, will be different due to their dissimilar effective masses. We therefore calculate the mobility of light and heavy holes separately and then find the total mobility : (S15) S11 Figure  Scattering into light-hole and heavy-hole bands is taken into account in both cases.
In general the light holes will have a higher mobility due to their lower effective mass, however the mobility of the material will be mainly determined by the heavy holes due to their greater density of states [10]. This is shown in Figure S3a where we plot the density of states for light and heavy holes in a nanowire with a 22 nm diameter. It can be seen that over the entire energy range the density of states of the heavy holes is 1 to 2 orders of magnitude larger than the light holes. This is due to both the fact that the density of states in a single band is higher, as well as from the fact that at any given energy there will be more heavy-hole bands contributing to the density of states due to the smaller energy separation between heavy-hole bands. The position of the Fermi level at each diameter is indicated in Figure S3a by the position of the black dots, and it is clear that there are a greater number of heavy hole bands beneath the Fermi level at every diameter. It has been found that in germanium at down to 40 K. We therefore expect that there should not be a substantial change at room temperature.
The average subband spacing for different wire diameters was found using the expression: where ( ) is the energy of the bottom of the subband for a wire of radius and ( ) is the number subbands used at that radius. It is plotted in Figure S3b. It is evident that the average spacing is comparable to the thermal energy at room temperature, 25 meV. The subband spacing close to the Fermi level exceeds 25 meV for the smallest diameter wires that we consider, and there are only a few subbands within a of the Fermi level at all diameters. We note that the one dimensional Kubo-Greenwood formula has previously been used when the average subband spacing was found to be 10 meV [5]. Therefore the use of the one dimensional formula is reasonable. Figure S4: Form factor (Equation S11) for scattering from the first to the second subband.The fact that the form factor increases as the radius of the nanowire is decreased is clearly seen. This stems from the normalisation prefactor, √ ( ) in Equation S7. This impacts on the diameter dependence of the resistivity, as it results in the decrease in the phonon-limited mobility as the wire radius is decreased.

Inserting Equation S4 into Equation S2
we can find the width of the space charge region for a given nanowire radius , electrostatic surface-potential and . We assume that the relative permittivity is the same as in bulk, 16 for the case of germanium [19]. has been found to be 0.3 eV for p-type germanium nanowires with a native oxide [3]. In Figure S5 we plot the ratio of to as a function of nanowire diameter. We have assumed that cm -2 . We note that for a wire with a diameter of about 18 nm.