Letter pubs.acs.org/NanoLett
Luminous Efficiency of Axial InxGa1−xN/GaN Nanowire Heterostructures: Interplay of Polarization and Surface Potentials Oliver Marquardt,* Christian Hauswald, Martin Wölz, Lutz Geelhaar, and Oliver Brandt Paul-Drude-Institut für Festkörperelektronik, Hausvogteiplatz 5−7, 10117 Berlin, Germany S Supporting Information *
ABSTRACT: Using continuum elasticity theory and an eightband k·p formalism, we study the electronic properties of GaN nanowires with axial InxGa1−xN insertions. The three-dimensional strain distribution in these insertions and the resulting distribution of the polarization fields are fully taken into account. In addition, we consider the presence of a surface potential originating from Fermi level pinning at the sidewall surfaces of the nanowires. Our simulations reveal an in-plane spatial separation of electrons and holes in the case of weak piezoelectric potentials, which correspond to an In content and layer thickness required for emission in the blue and violet spectral range. These results explain the quenching of the photoluminescence intensity experimentally observed for short emission wavelengths. We devise and discuss strategies to overcome this problem. KEYWORDS: Nanowires, surface potentials, electronic properties, InxGa1−xN, eight-band k·p formalism
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improved electron−hole overlap and thus to a higher internal quantum efficiency.13 Moreover, GaN-based nanowires can be synthesized on virtually arbitrary substrates without degradation in their crystal quality.14,15 Compared to the present state-of-the-art in RGB lighting technology (GaN-based on sapphire substrates for blue and green, GaP-based on GaAs substrate for red), GaN-based nanowires thus promise to enable an integral solution for solidstate lighting applications on a single platform such as Si. Besides the obvious economical benefits of this potential scenario, the group III nitrides are nontoxic materials,16 thus reducing the environmental impact of solid-state lighting17 compared to the present status. Consequently, GaN nanowires with axial InxGa 1−xN insertions have attracted much interest, and using molecular beam epitaxy (MBE) as well as metal−organic chemical vapor deposition, such structures have been fabricated by several groups.18−26 Conspicuously, the emission wavelengths reported for these nanowire heterostructures cluster at the green, amber, and red parts of the visible spectrum.18,19,21−25 This fact may appear natural given that long-wavelength emission is part of the motivation to investigate these structures. However, we have recently reported that the photoluminescence (PL) intensity of such structures monotonically decreases with decreasing In content. Due to this phenomenon, we found it difficult to obtain emission at wavelengths corresponding to the blue spectral range or even shorter ones.27
roup-III nitride semiconductors constitute the materials system of choice for general lighting applications. Commercially available white light-emitting diodes (LEDs) are based on a blue InxGa1−xN/GaN LED pumping a phosphor which in turn emits in the yellow to amber range. The luminous efficacy of these devices can be higher than 150 lm/W but decreases significantly when lower color temperatures and higher color rendering indices are demanded.1 Since the band gap of InxGa1−xN spans a range from 380 to 1800 nm, thus covering the entire visible spectrum,2−4 the development of phosphor-free white LEDs having both high efficiency and high color quality should be possible.5 However, it has been proven difficult to synthesize InxGa1−xN/GaN heterostructures with the In content required for red emission while still retaining a high internal quantum efficiency. Both the tendency of phase separation6 and the large lattice mismatch between InN and GaN (10%) impede the growth of InxGa1−xN layers on GaN with a high In content and the desired structural perfection. To overcome this limitation of planar heterostructures, GaN nanowires with axial InxGa1−xN insertions are currently discussed as a promising alternative with a high potential for the realization of red−green−blue (RGB) LEDs.7−9 In this context, the main advantage of nanowires compared to planar structures lies in their high aspect ratio (typically larger than 10). In contrast to planar structures, where strain has to be eventually relieved plastically above a critical compositionthickness product, the free sidewalls of nanowires permit the elastic relaxation of strained axial insertions.10−12 Simultaneously, this elastic strain release is expected to result in reduced piezoelectric fields, which in turn should lead to an © XXXX American Chemical Society
Received: April 26, 2013 Revised: June 14, 2013
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In this paper, we show that this finding also holds when varying the thickness of the insertion. Regardless of the growth conditions and the actual In content and thickness of the insertion, we have been unable to obtain any emission at all at wavelengths shorter than 430 nm. To clarify this phenomenon, we systematically examine the electronic structure of axial InxGa1−xN/GaN nanowire heterostructures within the framework of an eight-band k·p formalism, taking into account deformation, polarization, and surface potentials. We identify the interplay of polarization and surface potentials to be the decisive factor controlling the luminous efficiency of these InxGa1−xN/GaN nanostructures. We discuss possible remedies of this problem to achieve a high luminous efficiency throughout the whole visible spectrum. Figure 1 displays the integrated PL intensity vs the peak wavelength measured at room temperature for three series of
facilitates the elastic relaxation of a strained axial insertion. Consequently, both strain and polarization in such an axial heterostructure exhibit complex three-dimensional (3D) spatial distributions. The spatial confinement of the quantized electron and hole states depends on these potentials in a highly nontrivial way, making a full 3D simulation of the elastic and electronic properties of the structure imperative. In fact, such a simulation has been recently carried out for a selected axial InxGa1−xN/GaN nanowire heterostructure, taking into account strain and polarization fields, but neglecting surface potentials.31 For the specific structure considered in ref 31, a lateral separation of electrons and holes was predicted to occur, an effect which could very well be responsible for our experimental findings summarized in Figure 1. We theoretically investigate a hexagonal GaN nanowire containing an InxGa1−xN disk of thickness t and In content x. The composition of the alloy is assumed to be homogeneous. Deviations from a perfect random alloy could obviously be important in the present context, as potential fluctuations of sufficient magnitude may localize electrons and holes independent of the spatial potential distribution of the homogeneous matrix. However, the existence of these alloy fluctuations in InxGa1−xN is controversially discussed, particularly with regard to their potential magnitude and scale. In recent studies, evidence is accumulating that these deviations from a random alloy are in fact much more subtle than believed previously and only occur on an atomic scale.32−38 Such atomic-scale potential fluctuations are beyond the capabilities of the continuum model used in our work and will require the use of a computationally expensive atomistic approach. Within the present study, we hence focus on the understanding of the influence of surface and polarization potentials within InxGa1−xN disks of homogeneous In contents. The diameter of the wires under investigation in Figure 1 exhibits a broad distribution with a mean of about 80 nm which we have chosen for our model system. We use a linear continuum elasticity model and minimize the strain energy in the entire nanowire to obtain the strain distribution in the InxGa1−xN disk, which in turn determines the polarization fields arising from piezoelectricity. Together with the contribution from the spontaneous polarization of the wurtzite crystal, we refer to these potentials simply as polarization potentials. Having setup these basic properties, we employ a linear continuum elasticity model to compute strain and piezoelectric potentials, which then enter an eight-band k·p model to calculate the electron and hole single particle states and binding energies. Both formalisms are implemented in a plane-wave framework within the S/Phi/nX software package39,40 (for further details see refs 41 and 42). The material parameters required for the calculation of the elastic and piezoelectric properties as well as the deformation potentials of InN and GaN are taken from ref 43, except for the fact that we used a negative value e15 = e31, following a recent study.44 However, a different choice of this parameter, as well as the choice of different sets of deformation potentials (e.g., from ref 45) do not qualitatively alter the result of our study. The bulk electronic parameters, that is, the effective masses or band edges, were taken from ref 4. Again, deviations from the values of these parameters do not notably alter the result of our study. Using established parameters for the polarization of GaN and InN, a charge carrier separation solely due to deformation potentials as reported in ref 31 could not be reproduced. Instead, we found that electrons and holes are located in the
Figure 1. Integrated PL intensity vs emission wavelength at room temperature for InxGa1−xN/GaN nanowire heterostructures with different In contents (red squares and triangles) and disk thicknesses (blue circles). The shaded area provides a guide to the eye, and the arrows indicate the directions along which the In content (red) and the disk thickness (blue) increase.
samples where we have varied either the In content (triangles and squares) or the disk thickness (circles). For further experimental details concerning these samples, see the Supporting Information. Other series of samples obtained with different growth conditions show the same monotonic decline of the integrated intensity with decreasing wavelength, which is the exact opposite of what is known from planar InxGa1−xN/GaN quantum wells. For these structures, the difficulty to obtain a high luminous efficiency for longer wavelengths is well-documented (“green gap”)28−30 and is easily understood from the fact that thicker wells and higher In contents inevitably result in longer radiative lifetimes and thus reduced internal quantum efficiencies due to the increasing role of internal electrostatic fields. The experimentally observed behavior of InxGa1−xN/GaN nanowire heterostructures is thus equally unexpected and puzzling. The fundamental differences between planar quantum well and nanowire quantum disk structures are their geometry and the absence/presence of free surfaces. The latter may result in nonradiative surface recombination, and it is not inconceivable that this effect becomes more important for weak carrier confinement as obtained for low In contents and thin InxGa1−xN disks. The former, that is, the high aspect ratio of the nanowires, is an intrinsic property of their shape and B
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single-particle charge densities with increasing In content (top panel) and with increasing disk thickness (bottom panel) for a surface potential of 40 meV, simulating a doping level of 5 × 1016 cm−3. In all cases, the polarization potential arising from strain and the spontaneous polarization leads to a spatial separation of electron and hole states along the growth direction. The surface potential resulting from surface Fermi level pinning and the finite carrier density in the nanowire acts as an attractive potential for either the hole or the electron states, giving rise to an additional in-plane spatial separation. For the positive surface potential chosen in this figure, the electrons are always localized in the center of the nanowire, but the holes are pulled toward the nanowire sidewalls due to the surface potential. The influence of the surface potential is of particular importance for low In content and thin disks, as in these cases the polarization potentials are weak. For higher In contents (Figure 3, top right) and thicker disks (Figure 3, bottom right), the polarization dominates the confining potential of the hole state such that the holes are forced toward the center of the bottom interface of the disk. For very thick disks, the polarization potential induces a complete spatial separation of electrons and holes along the growth direction, an effect well-known from planar quantum wells. As a measure for the change in recombination efficiency with varying In content and disk thickness, we use the threedimensional overlap 6 of the electron−hole ground state:
center of the nanowire regardless of the In content and disk thickness as expected from the type-I bulk band offset and the hexagonal symmetry of the nanowire cross section. In recent work on axial GaN/AlxGa1−xN nanowires, surface potentials VS arising from Fermi level pinning at the free sidewalls of the nanowire were explicitly taken into account.46 For the nanowires under consideration in the present work, the sidewalls are made up of the M-planes of GaN, for which a Fermi level pinning with a value of 0.6 eV has been determined experimentally.47 Little is known about the Fermi level pinning at the corresponding surfaces of InxGa1−xN, except that we encounter carrier depletion for GaN and carrier accumulation for InN. We have thus considered both positive and negative surface potentials with a magnitude that is maximum at the surface and drops to its minimum at the center of the wire. Solving Poisson’s equation for a cylinder, a typical doping level of 1017 cm−3 corresponds to a parabolic potential with a maximum potential drop of eVS = ±80 meV that enters the k·p model as an additional, external potential and is not treated selfconsistently. The Coulomb potential arising from a single charge e in the center of the wire can be estimated analytically when a point charge is assumed. In GaN, this potential amounts in fact to 3.7 meV in a distance of 40 nm, which is 1 order of magnitude smaller than the surface potential at the surface. Hence, we restricted our investigation to a minimum doping density of 2.5 × 1016 cm−3. For doping densities below 1 × 1016 cm−3 (which will be difficult to achieve experimentally), surface potentials and potentials arising from localized charge carriers will be of the same order of magnitude, thus demanding a selfconsistent treatment of the surface potentials. Qualitatively, we expect that charge carrier confinement depends on the interplay between these surface potentials and the polarization potentials. Figure 2a shows the influence of the
6=
∑ ∑ ∑ ρe (r1, r2 , r3)ρh (r1, r2 , r3) r1
r2
r3
(1)
with ρe(r1, r2, r3) and ρh(r1, r2, r3) being the electron and hole ground state charge densities and r1, r2, and r3 denoting the spatial discretization of the super cell. Since we are interested in an order-of-magnitude effect (cf. Figure 1), the overlap 6 is a suitable quantity to reveal trends of the recombination efficiency depending on thickness and composition, analogously to an earlier study of GaN quantum dots.48 Furthermore, we focus here exclusively on the electron and hole ground states. For situations where the magnitudes of surface and polarization potentials are similar, however, the participation of excited states may become important. In this case, the computation of dipole matrix elements is imperative, since for a transition between excited states, the selection rules for the transition may determine its strength rather than the overlap. To guide the optimization of axial nanowire heterostructures for emission at specific wavelengths, future work will thus demand the calculation of the whole sequence of excited states and the dipole matrix elements of the transitions between them. Figure 4a and b show the electron−hole ground state overlap as a function of the In content x for different magnitudes of surface potentials. In the absence of a surface potential (VS = 0: black solid line), the overlap decreases monotonically with increasing polarization potential, that is, In content, since electrons and holes are pulled apart by the electrostatic field within the QW. The increase of the overlap beyond In contents of 30% occurs as the lateral charge carrier separation is maximized here, but an in-plane compression of the carriers is achieved by the stronger polarization potentials. In Figure 4c and d, the overlap is shown as a function of the disk thickness for a constant In-content of 10%. As expected, the overlap decreases monotonically with larger disk thickness due to stronger polarization potentials.
Figure 2. Schematic profile of the valence band energy Evb across the center of the nanowire at the bottom interface of the InxGa1−xN disk. The bulk valence band edge is indicated as a dash−dotted line. Energies are not to scale. (a) Taking into account strain and the resulting piezoelectric polarization, the profile is determined by the polarization potential with a magnitude VP. (b) Influence of an additional surface potential with a magnitude VS.
polarization potential on the valence band energy Evb across the center of the nanowire at the bottom of the InxGa1−xN disk. Evidently, the hole ground state will be confined in the center of the disk regardless of the magnitude of the polarization potential. The situation changes entirely, if a surface potential is additionally taken into account as depicted in Figure 2b. In this case, the location of the quantized hole state depends on the strengths of both the polarization and the surface potential. A similar argument holds for the localization of electrons at the top surface of the disk for surface potentials of opposite sign. Next, we investigate this interplay of the polarization and surface potentials for InxGa1−xN/GaN disks for different In contents and thicknesses. Figure 3 shows the evolution of the C
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Figure 3. Electron (red) and hole (blue) charge densities in an InxGa1−xN disk in a GaN nanowire for different In contents (top) and different disk thicknesses (bottom) and a surface potential of 40 meV. The wire is indicated in white; the surrounding vacuum is depicted in yellow. Note that the wire diameter and disk thickness are not to scale.
Figure 4. (a and b) Electron−hole 3D charge density overlap 6 as a function of the In content of a 5 nm thick InxGa1−xN/GaN disk. (c and d) 6 as a function of the thickness of an In0.1Ga0.9N disk in a GaN nanowire. The surface potentials are assumed to be positive in a and c and negative in b and d. In all figures, the solid black line indicates the limiting case for a vanishing surface potential.
With a finite surface potential, however, the situation changes entirely for low In content or thin disks. In both cases, even a surface potential as small as 20 meV can induce a dramatic reduction of the electron−hole overlap. The reason for this result is the fact that surface potentials tend to separate opposite charges and pull holes toward the surface for VS > 0 (cf. Figure 4a and c), or electrons for VS < 0 (cf. Figure 4b and d). The overlap 6 as defined in eq 1 is a three-dimensional quantity and includes the vertical spatial separation of electrons and holes along the growth direction as in planar quantum wells. A quantity more suitable to represent the lateral
separation of electrons and holes due to the surface potential is obtained by normalizing 6 with the one-dimensional (1D) electron−hole overlap 61D =
∑ ρ′e (r3)ρ′h (r3) (2)
r3
where ρ′e (r3) =
∑ ∑ ρe (r1, r2 , r3) r1
r2
and D
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counteracting the decreasing surface potential as pointed out above. In addition, for the weak surface potentials in thin nanowires, a self-consistent treatment of the potentials and the associated electronic states becomes imperative. The broad diameter distribution obtained by the self-induced growth of nanowires impedes a systematic experimental investigation of the luminous efficiency of InxGa1−xN/GaN nanowire heterostructures and its dependence on the diameter. Well-defined diameters have been obtained for nanowire arrays synthesized by selective-area growth,49−52 making such samples highly attractive for this purpose. To maximize the electron−hole overlap in these axial heterostructures, the piezoelectric potential should actually be as large as possible without decreasing the vertical overlap 6 1D. This aim can be achieved by synthesizing thin wells with a large In content. The actual optimum design parameters, however, are determined by the desired emission wavelength and cannot be generalized. Their computation requires the calculation of two-dimensional maps of the overlap 6 , or better the dipole matrix elements, as a function of In content and thickness for a constant ground-state energy. Since the piezoelectric potentials depend on the elastic relaxation of strain at the nanowire sidewalls, these maps have to be computed for various nanowire diameters. For thin nanowires, surface potentials should be considered self-consistently to include the interaction of potential and charge. All in all, the exploration of the optimal design parameters of InxGa1−xN/GaN nanowires heterostructures is feasible, but will require substantial computational resources and time. Finally, while these systematic studies would focus on the properties of the active region of the nanowire heterostructure, it will also be important to incorporate and evaluate the influence of surface potentials in full device simulations for the investigation of nanowire-based LEDs. Alternatively, we may try experimentally to minimize the surface potential in these structures. This potential arises from two factors: (i) the pinning of the Fermi level at the nanowire sidewalls, and (ii) the finite concentration of carriers in the active region of the device. The former may be reduced by either growing a thick GaN shell around the nanowires,53,54 or by a passivation of the free surfaces by using a suitable chemical agent.55 The latter, that is, a reduction of the charge carrier concentration in the active region, is more difficult to achieve. The incorporation of the shallow donor O into the axial InxGa1−xN insertions is high due to both the N polarity of the nanowire and the low growth temperature (which is necessary to incorporate significant amounts of In).56,57 The resulting high carrier concentration could be compensated by the intentional incorporation of deep acceptors, such as C or Fe,58,59 but these impurities are often the cause of nonradiative recombination and are to be avoided in the active region of a light-emitting device at all costs. An approach with shallow impurities such as Mg does not bear this problem60 but leads to other difficulties: these impurities participate in radiative transitions, inducing a strong redshift of the electroluminescence for lower operating current. A counterdoping in excess of what is needed for compensation will even turn the active region p-type, thus shifting the position of the p−n-junction of the nanowire LED. We finally note that surface potentials created by Fermi-level pinning are a property common to all semiconductors. Most of these semiconductors, however, exhibit significantly smaller polarization potentials since they are not as piezoelectric as the
∑ ∑ ρh (r1, r2 , r3) r1
r2
are the projections of the electron and hole charge densities onto the growth axis r3. This 1D overlap is affected only by a vertical separation of electrons and holes and is similar to (but not identical with) the conventional definition of the electron and hole overlap integral for planar quantum wells. The normalized overlap 6 /6 1D allows us to quantify the effect of the lateral separation of electrons and holes. We can even combine the data shown in Figure 4 in a single graph by plotting the normalized overlap via the ratio of the surface and the polarization potentials |VS/VP|. The normalization thus allows us to separate the lateral spatial separation of electrons and holes from the one along the growth direction whose effects are well-known and understood for planar group IIInitride quantum wells. Figure 5 shows that the data obtained
Figure 5. Combined data from Figure 4a and b (red +) and Figure 4c and d (blue ×) normalized by the 1D overlap 6 1D plotted as a function of the ratio between the surface potential and the polarization potential. The red solid arrow indicates the direction of increasing In content, and the blue dashed arrow indicates the direction of increasing disk thickness. The thick gray line is a guide to the eye.
for different In contents and thicknesses indeed follow the same, universal trend when represented in this way. This plot thus serves as a useful guideline for the design of axial InxGa1−xN/GaN nanowire heterostructures. In particular, it is obvious from Figure 5 that, once the ratio between the extrema of the surface and polarization potential exceeds a critical value (|VS/VP| > 0.2), the normalized electron−hole overlap 6 /6 1D is reduced by orders of magnitude, correspondingly reducing the internal quantum efficiency of these systems significantly below the one of a planar quantum well of similar layer thickness and In content. The attempt to minimize the polarization potentials by designing the heterostructure such as to encourage elastic relaxation at the free nanowire surfaces is thus rather counterproductive, since a vanishing piezoelectric field inevitably results in a complete in-plane spatial separation of electrons and holes due to surface potentials. In this context, it is also of interest to consider the impact of the nanowire diameter (which we have kept constant in the present work) on the electron−hole overlap. Qualitatively, it is expected that the surface potential becomes dominant in thicker nanowires since their magnitude increases quadratically with the wire diameter. However, with decreasing diameter, the strain within the InxGa1−xN disk relaxes progressively,12 thus E
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group-III nitrides, and they may not even possess a spontaneous polarization. The interplay between surface and polarization potentials, investigated here for InxGa1−xN/GaN nanowires, is likely to be an important phenomenon for other axial semiconductor nanowire heterostructures as well. Since this interplay depends in a nontrivial way on a number of material and system properties, its impact on charge carrier recombination needs to be elucidated for each given system individually by employing a full 3D simulation of the nanowires’ electronic properties.
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ASSOCIATED CONTENT
S Supporting Information *
Additional experimental data: PL spectra and sample description. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank Raffaela Calarco for fruitful discussions and Lutz Schrottke for a critical reading of the manuscript. The sample synthesis has been funded by the German government BMBF project MONALISA (Contract no. 01BL0810).
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REFERENCES
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