Electronic Supplementary Material (ESI) for Nanoscale. This journal is © The Royal Society of Chemistry 2015 Electronic Supplementary Information: Low Ensemble Disorder in Quantum Well Tube Nanowires Christopher L. Davies,∗a Patrick Parkinson,b Nian Jiang,c Jessica L. Boland,a Sonia Conesa-Boj,a H. Hoe Tan,c Chennupati Jagadish,c Laura M. Herz,a and Michael B. Johnston.a‡ (a) 3.950 nm (b) GaAs-QW 2.026 nm 2.181 nm 4.0 nm 3.962 nm 50 nm 20 nm GaAs-core Fig. S1 TEM image of top of B50 sample Fig. S2 TEM image of bottom of B50 sample S1 TEM Figure S1(a) corresponds to a low magnification bright field TEM image of a representative cross-section of the sample B50. The thickness of the GaAs QW has been measured in different regions. S2 1D Finite Square Well Model The variations in thickness are found to be around 4 nm and 2 For a semiconductor the Fermi-Dirac distribution for electrons in nm in the edges and in the facets, respectively, confirming the the conduction band and holes in the valence band is given by, disorder in the GaAs QW. In the HR-TEM image performed in 1 one of the edges of the nanowire cross section, figure S1(b), the f = ; (1) e,h exp((E − Ec,v) ) + 1 variation in the QW thickness (marked by white dashed lines) f b between the edge and the facets is clearly visible. c,v where b = 1=kBT, T is the electron temperature and Ef is the Figures S2 and S3 are additional TEM images of sample B50 Fermi energies of the electrons and holes. In the high temperature taken from the top and bottom of a nanowire. These images limit we can ignore the +1 in the denominator, and the electrons demonstrate the inhomogenous well width around the nanowire. and hole occupancies will follow Boltzmann statistics, fe,h ∝ exp(−Eb) : (2) a Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Ox- The rate of emission is proportional to the occupancy of the elec- ford, OX1 3PU, United Kingdom. b School of Physics and Astronomy and the Photon Science Institute, University of trons multiplied by the occupancy of the holes. That is, Manchester, Manchester M13 9PL, United Kingdom. c Department of Electronic Materials Engineering, Research School of Physics and Engi- f (E) ∝ fe fh ∝ exp(−(Ee + Eh)b) ; (3) neering, The Australian National University, Canberra, ACT 0200, Australia. z [email protected] f (E) ∝ exp(−(E − Eg)b) ; (4) 1 let us write, N Y(x) = ∏y(xi) ; (10) i=1 equation (9) then reduces to, 2 ¶ y(xi) 2mE 2 = − 2 y(xi) : (11) ¶xi h¯ By considering the boundary equations (6 and 7) the normalised wave function for the ith component must be given by, 1 y(xi) = 1 sin(kixi) ; (12) L 2 with k = n p=L, and so the overall wave–function is given by, Fig. S3 TEM image of top of B50 sample i i 1 N y(x) = 1 ∏sin(kixi) : (13) V 2 i=1 where E is the energy of the photon emitted on recombination The volume occupied by each k-state in k-space is, and Eg is the band gap of the semiconductor material. For a semi- p N V = : (14) conductor the energy dependence of the emission spectrum in the k L classical limit is proportional to the Boltzmann distribution, with The number of allowed states with jkj 2 [k;k + dk] is described by an energy-offset equal to the band gap energy, multiplied by the the function g(k)dk where g(k) is the density of states, joint density of states for the electrons and holes: V in k-space of a shell of an N-D sphere g(k)dk = ; (15) I(E) ∝ g(E) f (E) ; (5) V in k-space occupied per allowed state where f (E) is as above and g(E) is the joint density of states for for the case of N=3 and N=2 we have, the electrons and holes. The core will have a 3–dimensional den- 4pk2dk sity of states but the QWT layer is confined in one direction so we g3D(k)dk = ; (16) p 3 need to use the 2–dimensional density of states for this case. L 2pkdk g2D(k)dk = : (17) p 2 S2.1 Density of States L Consider an N–dimensional (N-D) box of side L (Volume= LN). A Now rewriting in terms of energy using, E =h¯2k2=2m∗ and dE = particle inside this box experiences a potential V given by, h¯2dk=m∗ we have, 3 ∗ V(xi) = 0 for 0 < xi < L ; (6) L 2m 1 g3D(E)dE = E 2 dE ; (18) p2 h¯2 V(xi) = ¥ for xi < 0 or L < xi ; (7) L2 2m∗ where x are the orthogonal coordinates of the system, for exam- g2D(E)dE = dE; (19) i p h¯2 ple for a 3–dimensional system xi = x;y or z. In position repre- Dropping the numerical factors gives, sentation the time–independent Schrodinger¨ equation for a non- relativistic particle for the ith coordinate is given by, 1 g3D(E)dE ∝ E 2 dE ; (20) h¯2 ¶ 2Y(x) g (E)dE ∝ dE : (21) − 2 +V(x)Y(x) = EY(x) : (8) 2D 2m ¶xi Equation (7) implies that Y(x) ! 0 at xi = 0 and xi = L. For 0 < xi < L we have that, ¶ 2Y(x) 2mE 2 = − 2 Y(x) ; (9) ¶xi h¯ 2 S2.2 Model 100 Using the density of states g3D(E) for the core and g2D(E) for the QWT we have, 10-1 p E − ECore BCore(E) ∝ E − ECore exp − ; (22) kBT e1!hh1 ! 10-2 E − EQWT BQWT(E) ∝ exp − ; (23) kBT PL intensity (arb. units) -3 e1!hh1 10 where T is the temperature, and ECore and EQWT are the band gap energies in the core and the QWT. Equation (22) can be recognised as the Maxwell-Boltzmann distribution, and equation 10-4 (23) can be considered as a two-dimensional version of the Maxwell- 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 Energy (eV) Boltzmann distribution. The nanowires, however, do not have a perfect structure and so the spectral peaks will also be broadened Fig. S4 PL spectrum of a nanowire from sample B100 shown on a by defects and disorder, which was modelled using a Gaussian, logarithmic scale. The blue dots are the measured data and the red G(E). To combine the two effects we take the convolution of the dashed line is the model. two, Cap Shell QWT Shell Core ICore(E) ∝ GCore(E) ⊗ BCore(E − ECore) ; (24) ΔE IQWT(E) ∝ GQWT(E) ⊗ BQWT(E − EQWT) ; (25) ΔEe CB where, E E ! AlGaAs E Core E2 QWT GCore(E) = exp − 2 ; (26) 2s ΔEh Core ΔEVB and, ! E2 Fig. S5 A schematic diagram showing the band gap of a GQWT(E) = exp − ; (27) 2s 2 GaAs/Al0.4Ga0.6As quantum well. The horizontal red lines represent the QWT energy level of the ground state for a bound particle in the potential well The model for the measured photoluminescence is then a linear where DECB and DEVB are the barrier heights. CB and VB are the combination of these two peaks, conduction and valence band respectively. IPL(E) = aCoreICore(E) + aQWTIQWT(E) ; (28) layers. As stated previously this confinement will increase the where aCore and aQWT are constants dependent on the intensity of energy of the ground state transition. The increase in energy due emission from the core and QWT. A least squares fit was used to to confinement is needed to be able to deduce the width of the determine the following parameters: T;ECore;sCore;EQWT;sQWT. well. Consider a finite well of width d and depth V centred on Using the values for ECore and EQWT and a simple model for the x = 0, and a particle whose effective mass differs in the barrier, band gap, as illustrated in Figure S5, it is possible to calculate an ∗ ∗ mb (Al0.4Ga0.6As), and the QWT, mq (GaAs) (b = QWT; q = estimate for the well width. Core). The wavefunction of the particle is given by, Figure S4 shows a typical spectrum of a nanowire from sample y (x) = C exp(±kx) for jxj < d=2 ; (29) B100 displayed on a logarithmic scale. The straight line seen q in the high energy tails is indicative of a Boltzmann distribution 0 yb(x) = C exp(±kx) for jxj > d=2 ; (30) (exp(−(E − E0)=kBT)) and is further confirmation of our models validity. where, s ∗ s ∗ 2mqE 2m k = and k = b (V − E) : (31) h¯2 h¯2 S2.3 Finite Well Model for the Band Structure Continuity conditions (y and its derivative with respect to x are The wavefunction of excited electrons and holes in the QWT layer continuous) restrict the allowed k-states to solutions of, will undergo a perturbation due to quantum confinement effects from the finite potential well created from the difference in band kd m∗k tan = q ; (32) gap between the Al Ga As barrier layers and the GaAs QWT ∗ 0.4 0.6 2 mbk 3 rearranging this equation to be in terms of energy gives, As such, the ratio of the surface area to volume ratio of the QWT and the core is R=d = 20 to 100.