Band-Edge Diagrams of Core−Shell ... - ACS Publications

ARTICLE pubs.acs.org/JPCC

Band-Edge Diagrams of Core
Shell Semiconductor Dots M.-E. Pistol*,† and C. E. Pryor*,‡ † ‡

Solid State Physics/The Nanometer Structure Consortium, Lund University, P. O. Box 118, SE-221 00 Lund, Sweden Department of Physics and Astronomy and Optical Science and Technology Center, University of Iowa, Iowa City, Iowa 52242, United States

bS Supporting Information ABSTRACT: We have calculated band-edge diagrams for spherical core
shell nanocrystal quantum dots for all combinations of AlN, GaN, and InN, as well as all combinations of AlP, GaP, AlAs, GaAs, InP, InAs, AlSb, GaSb, and InSb, as a function of core radius, with the outer radius of the shell held fixed. We have calculated the Γ- and the X-conduction band minima, the valence band maximum, and the effective masses using strain-dependent eight-band kp theory, with a linear continuum model for the strain. We have found all the band alignments that may occur, and identified all combinations where one material becomes metallic due to a negative gap. Structures which are suitable for biological applications have been identified. We provide a figure which allows easy calculation of the confinement energy using a single band model.

’ INTRODUCTION Semiconductor nanocrystal quantum dots have attracted a great deal of interest due to the wide range of electrooptic properties that can be obtained. Beyond purely scientific interest, there has been promising work on applications for LEDs,1
3 lasers,4
6 photovoltaics,7,8 biosensors,9,10 and biological markers.11,12 In most applications the primary goal is to obtain strong fluorescence at a particular wavelength. For example, in many biological applications it is advantageous to use dots emitting in the infrared since tissue is more transparent in the infrared than in the visible.13 For nanocrystals composed of a single material, the choice of material, size, and shape can all be varied to influence the electronic properties. Additional control may be obtained using a core
shell structure in which a shell of a different material surrounds the core material.14 This simple modification allows a wide range of problems to be addressed. Most importantly, a barrier shell keeps carriers away from the nanocrystal surface, decreasing nonradiative recombination. Core
shell structures also allow one to obtain type-II band structures in which electrons and holes are confined to different regions of the dot.15 Core
shell structures may also allow control over doping by changing the spectrum of the dot. Doped nanocrystals do not behave the same as doped bulk materials, in part because carriers from ionized impurities go into confined states rather than a continuum of bulk states.16 Many dots display fluorescence r 2011 American Chemical Society

blinking,17 which occurs over time scales on the order of a second. The cause is not clear, and it has been attributed to Auger effects.18 Blinking has also been attributed to quenching of the fluorescence by fluctuating charges in surface traps,19 and for Stranski
Krastanow grown quantum dots, blinking has been attributed to fluorescence quenching by nearby metastable defects.20 Although the cause of blinking is not clear, it has been shown that blinking can be influenced or controlled by a shell.21 Finally, in biological applications a shell may provide a mechanical barrier to isolate toxic materials. The most toxic material used in dots is cadmium, which may be avoided altogether by using III
V materials rather than II
VI materials. A complication arrises in core
shell structures if the two materials have different bulk lattice constants since the mismatch between the two materials leads to strain which alters the simple potentials inferred from the bulk band edges. The strain will depend on the materials and the relative sizes of the core and shell, and will vary spatially, giving a complex effective potential for the carriers. While this would at first appear to require a complete strain-dependent electronic structure calculation for each material combination and geometry, accurate results may be Received: October 1, 2010 Revised: February 28, 2011 Published: May 16, 2011 10931

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Figure 1. Components of the strain tensor as a function of position along [001]-oriented and [111]-oriented lines, both through the center of an InAs/GaAs core
shell dot with a shell has a thickness 1/5 of the total dot radius. The shell is cut off in the figure to better show the strain near the interface.

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Figure 3. Band edges (X-bands, Γ-band, and valence bands) along lines through the center of a dot consisting of an InAs core and a GaAs shell. The core radius is 1/2 the total dot radius. The left panel shows the band edges along the [111]-direction, and the right panel shows the band edges along the [100]-direction. Due to symmetry the X-bands are not split along the [111]-direction in contrast to the case along the [100]-direction. The Γband and the valence bands behave very similarly in both cases.

Figure 4. Average band edge energies for nitride-based core
shell dots as a function of core radius.

Figure 2. Components of the strain tensor as a function of position along [001]-oriented and [111]-oriented lines, both through the center of an InAs/GaAs core
shell dot. The core is 1/5 the total dot radius.

obtained by calculating the strain-dependent band structure as a function of a small number of parameters. To compare different material systems, we have calculated band-edge diagrams for spherical core
shell dots involving all combinations of AlN, GaN, and InN, as well as all combinations of AlP, GaP, AlAs, GaAs, InP, InAs, AlSb, GaSb, and InSb, as a function of core radius. The crystal structure was taken to be zincblende. We have also computed the effective masses of the

electrons and the holes. This allows easy selection of structures that have desired properties, such as a chosen fluorescence energy, or containing or lacking certain elements. We identify a large set of unusual structures which have separation of charges when photoexcited, as well as structures having separation of charges without requiring photoexcitation. The latter structures are metallic, and we identify all structures that are metallic due to a negative gap. Such metallic semiconductor dots should support plasmons but without requiring elementary metals. We find that the band-edge diagrams for the core
shell dots are similar to the band-edge diagrams for core
shell wires. In the Supporting Information we provide tables of the effective masses and the energies that we have computed.

’ METHOD We consider spherical core
shell dots composed of binary zincblende III
V materials that are coherently strained without defects. We calculate the band structures using a method that has 10932

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Figure 5. Average band edge energies for nitride-based core
shell wires as a function of core radius.

Figure 7. Average band edge energies for core
shell dots as a function of core radius. The core is GaP.

tensor, e = exx þ eyy þ ezz is the hydrostatic strain, and ki is a unit vector in the direction of the relevant X-valley. The parameters for the calculations were taken from refs 24 and 25, and the temperature was taken to be 0 K. Since the calculated band energies are spatially dependent, we have computed the mean values for each band within the core and shell materials, respectively. Due to the scale invariance of the elasticity equations, the strain is independent of the overall scale of the structure, and therefore the band structure is completely determined for a given ratio of core and shell radii.

Figure 6. Average band edge energies for core
shell dots as a function of core radius. The core is AlP.

been used to calculate the band structure of coherently strained core
shell nanowires.22 We first compute the strain using a continuum elasticity approximation on a cubic grid of 120  120  120 sites. From the spatially dependent strain tensor, eij(x B), we compute the energy of the Γ band edge, the valence bands, and spin
orbit band using linear deformation potential theory including mixing among eight bands.23 To calculate the position of the X-band minimum, we used a single-band approximation with the energy of the X-band given by EX = Ξue þ Ξdkieijkj, where Ξu and Ξd are deformation potentials, eij is the strain

’ RESULTS AND DISCUSSION To illustrate the effects involved, we first consider the strain in core
shell dots in which the shell has a smaller lattice constant. For a thin shell, the general features of the strain may be understood by recognizing that the core undergoes approximately isotropic compression from the shell, and the shell itself resembles a planar slab on a substrate. Figure 1 shows the strain tensor as a function of position along [001] and [111]-oriented lines through the center of a dot with an InAs core and a GaAs shell with a thickness 1/5 the radius of the dot. Along the [001] line the shear components (exy,eyz,ezx) are zero and exx =eyy due to the spherical symmetry of the dot and the cubic symmetry of the materials. In the core, the three nonzero components exx, eyy, and ezz are approximately equal and constant. The core is under hydrostatic compression (e < 0) since InAs has a larger lattice constant than the surrounding GaAs. The uniformity of the strain 10933

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Figure 8. Average band edge energies for core
shell dots as a function of core radius. The core is AlAs.

Figure 9. Average band edge energies for core
shell dots as a function of core radius. The core is GaAs.

inside the core is attributable to the fact that the shell completely surrounds the core. There is some inhomogeneity near the interface where the negative in-plane strain of the core becomes less negative, and due to the Poisson effect, the strain perpendicular to the interface becomes more negative. The similarity with a slab is seen in the shell’s strain as exx and eyy are both positive, and ezz < 0 from the Poisson effect. Along the [111] direction the strain in the core is nearly identical to the [001] direction, with compressive hydrostatic strain. Also the hydrostatic strain in the shell is comparable to that along the [001] direction; however, the shear strain is nonzero along the [111] line which will affect the holes. With a thicker shell, the strain becomes more complex. Figure 2 shows the strain for an InAs/GaAs dot with a core that has radius 1/5 of the dot. Inside the core, the strain is almost entirely hydrostatic and nearly uniform, just as for the thin shell, but the hydrostatic strain is larger in magnitude for the thicker shell. Along the [001] direction, the strain components in the shell look qualitatively similar to the thin shell case, with exx = eyy > 0 and ezz < 0. This is understood by the same arguments used for the thin shell, even if now the shell is thick compared to the size of the core, and so is in no sense thin. While the relative signs of the strain components are simply understood, their magnitudes are such that the hydrostatic strain e = exx þ eyy þ ezz is actually negative near the interface, going positive further out. In Figure 3 we show the band edges as a function of position in a dot with an InAs/GaAs dot in which the InAs core radius is 1/2 the total for the dot. We have plotted the band edges along the [100]-direction and the [111]-direction to illustrate the

anisotropy. We find that the band edges in the core are remarkably flat, reflecting the uniform hydrostatic strain. In the shell the valence band is split quite strongly but the splitting is not very dependent on the direction. The X-band in the shell, however, is split when plotted along the [100]-direction but remains degenerate when plotted along the [111]-direction. In the core the X-band remains unsplit, independent of position. The complicated spatial variation of the band energies may be summarized by taking the mean values within the core and in the shell, giving a guide to the character of the dot. Our main results are collected in Figures 4
14, which give the mean energies of the band extrema in the core and shell as a function of Rcore/Rtotal for all material combinations. At first glance, the graphs look somewhat similar to the corresponding graphs for hexagonal core
shell nanowires in ref 22. This is to be expected since they share the same materials and a roughly circular core
shell geometry. The nanowire results would provide a very rough approximation to the nanocrystals, which would be better than using unstrained band structures. If the constituent materials have the same lattice constant, then the band structures of core
shell dots and core
shell wires must be exactly the same. However there are quantitative differences due the difference in geometry if there is a lattice mismatch between the core and the shell. These quantitative differences can sometimes lead to qualitative differences, and this is illustrated in Figures 4 and 5. Figure 4 shows the band edges for the core
shell dots made from the nitrides, and Figure 5 shows the same thing for core
shell wires made from the nitrides. It can be seen from 10934

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Figure 11. Average band edge energies for core
shell dots as a function of core radius. The core is InAs. Figure 10. Average band edge energies for core
shell dots as a function of core radius. The core is InP.

Figure 4 that dots with an AlN core and an InN shell confine electrons and holes to the shell for all core thicknesses. For wires, the situation is different. Figure 5 shows that for small core radius the holes are confined to the AlN core instead of to the InN shell. It can also be seen that the band gap in the structures can differ by almost a factor of 2 between wires and dots. This occurs for InN dots with a shell of AlN or GaN, when the core radius is small.To design a core
shell nanocrystal with a particular band structure, the kinds of quantitatively accurate results in Figures 4
14 will be necessary. The alignment of the bands is an important parameter, and the different possibilities are explained in Figure 15. For some material combinations the strain is sufficiently high that the conduction band minimum of one material has a lower energy than the valence band maximum of the other, in which case charge is transferred, giving a metallic structure. To give an overview, we include Tables 1 and 2 which contain the band alignments and range of gaps for each material combination. In some cases, such as AlP/GaP, no range is given since the core and shell lattice constants are so similar that the gap is nearly independent of Rcore/Rtotal. The antimonides often form type III and negative gap structures in which case the gap is given as zero. This situation only occurs with the antimonides. Not surprisingly, the largest gaps are found among the nitrides. It should be noted that some combinations, such as AlSb/GaP,

have an indirect gap. Such dots should have long-lived excitons which may be interesting or useful. Dots with a shell containing aluminum may present difficulties in practical applications since the shell is water sensitive. In the Supporting Information we also give the effective masses for the dots in the core and the shell, including the effects of strain. Confinement energies may be calculated using a single band model with the same effective mass in the core and shell and an infinitely thick shell, as shown in Figure 16. The data for Figure 16 are given in the Supporting Information. Determining the confinement energy requires solving the Schr€odinger equation for a particular spatially dependent potential energy and effective mass. However, by simplifying the potential and taking the effective mass to be a constant, we can tabulate confinement energies in a form that is applicable to any core
shell system. We take the spherically symmetric potential to be V ðrÞ ¼ V θðr
Rcore Þ

ð1Þ

where Rcore is the radius of the core and θ is the Heaviside step function. Taking the effective mass to be its value in the core, the radial Schr€odinger equation is ! p2 d2 p2 lðl þ 1Þ þ V θðr
Rcore Þ ψðrÞ ¼ EψðrÞ
 2þ 2mr 2 2m dr ð2Þ 10935

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Figure 12. Average band edge energies for core
shell dots as a function of core radius. The core is AlSb.

Figure 13. Average band edge energies for core
shell dots as a function of core radius. The core is GaSb.

where l is the angular momentum. We may rewrite this in terms of the dimensionless parameter x defined by r = xRcore and the energy parameter U0 = p2/2m*Rcore2 to obtain

’ APPLICATIONS Such a wide range of electronic structures provides possibilities for a variety of applications. Type I-c dots should be efficient fluorophores since their fluorescence occurs in the core away from the deleterious effects of the surface. Depending on the material system and geometry, it is possible to tune the direct gap energy from 0 (InSb/InP) to 3.6 eV (GaN/AlN). For the nitrides it is also possible to have a very small gap of approximately 0 eV, which occurs for a core of InN in a shell of AlN. The tunability range includes the infrared region in which human tissue is quite transparent,13 and reaches into the ultraviolet. Type II-c dots do not undergo charge separation at equilibrium; however, under resonant illumination electrons will go to the core and holes will go to the shell, producing a carrier concentration n. The dots are thus metallic, albeit having two types of charge carriers, and should support plasmon modes. Since the plasmon energy is proportional to (n)1/2, this will allow the plasmon energy to be tuned by the intensity of incident light. This is in contrast to plasmons in metal systems, for which it is very difficult to continuously tune the energy. Also, the carrier concentration of an illuminated dot will be lower than in metals, with a correspondingly lower plasmon energy. Dots with II-c alignment will also have slow radiative recombination and may thus have applications as whiteners and in phosphorescent light sources. Type II-s dots would act as artificial atoms since holes will collect in the core, acting as a positively charged nucleus, while electrons will collect in the

! d2 lðl þ 1Þ V E
2þ þ θðx
1Þ ψðxÞ ¼ ψðxÞ 2 x U0 U0 dx

ð3Þ

The dimensionless eigenvalues E/U0 may be computed numerically as a function of V/U0 and used to determine the confinement energy of a core
shell structure with particular values of m* and Rcore. In Figure 16 we show E/U0 for the five lowest states in a spherical quantum dot as a function of V/U0. As expected, for sufficiently small V there are no confined states. The confinement energies obtained using Figure 16 are typically accurate to within 10% compared with a full eight-band calculation. As an example, consider a dot with an InN core of radius Rcore = 5 nm and a 5 nm thick GaN shell. From Figure 16 we find for Rcore/Rtotal = 0.5 that the InN conduction band minimum (Γ-minimum) has an energy of about
1.3 eV and the GaN conduction band minimum (Γ-minimum) has an energy of about 0.5 eV, giving a conduction band offset of V = 1.8 eV. From the table of effective masses we find that such an InN/GaN dot has m* = 0.03me in the core. Using the fact that U0 = 38.1 meV for Rcore = 1 nm and m* = me, we obtain U0 = 50.8 meV for the InN/GaN dot. From the calculated eigenvalues (using Figure 16) we find E/U0 ≈ 7 for V/U0 ≈ 35, or E ≈ 350 meV for the ground state.

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shell forming an electron cloud, as shown schematically in Figure 17. The number of electrons in the cloud will be optically tunable, which could lead to some interesting experiments, in particular in single-dot phosphorescence spectroscopy using a tunable laser for excitation. There are also possibilities of using the charge separation under illumination for solar cells. Type III-c dots will undergo spontaneous charge separation with electrons collecting in the core and holes in the shell (and

Figure 14. Average band edge energies for core
shell dots as a function of core radius. The core is InSb.

Figure 15. Different types of band alignments in core
shell dots, with shading indicating the energy gap. Ec is the conduction band minimum, Ev is the valence band maximum, and Ef is the Fermi level. The basic type (I, II, III) follows the usual convention, followed by a letter indicating whether the core (c) or shell (s) has the lower conduction band. For type III structures charge is transferred from one region to the other, which will result in band banding which is not considered here. For a type III-c alignment the valence band maximum of the shell is above the conduction band minimum of the core. Electrons are transferred from the shell to the core, creating a metallic dot with a negative core and a positive shell.

Table 1. Band Alignments Expected in Different Non-nitride Core
Shell Dot Systemsa for given shell material core material

AlP

AlP 2.52 eV GaP

GaP

AlAs

GaAs

II-c

I-s

I-s, II-c

I-s, II-c

I-s

II-c, I-s

I-s

2.2 eV

2.1
2.5 eV

1.5
1.7 eV

1.4
1.6 eV

0.4
0.8 eV

0.8
1.6 eV

0.3
0.8 eV

0
0.2 eV

II-s

I-s

I-s

I-s

II-c, I-s

II-c, I-s

I-s, metallic-s

II-s

InP

InAs

AlSb

GaSb

InSb I-s, metallic-s

2.2 eV

2.35 eV

2
2.2 eV

1.5
1.8 eV

1.5
1.6 eV

0.4
0.8 eV

0.7
1.1 eV

0.3
0.7 eV

0
0.2 eV

AlAs

I-c 2.2
2.3 eV

II-c 2 eV

2.24 eV

I-s 1.5 eV

I-s 1.4
1.5 eV

I-s 0.4
0.7 eV

I-s 1.1
1.7 eV

I-s 0.5
0.8 eV

I-s, metallic-s 0
0.2 eV

GaAs

II-s, I-c

I-c

I-c

I-c

I-s

II-c

II-c

metallic-c

1.7 eV

1.6
2 eV

1.5 eV

1.519 eV

1.1
1.5 eV

0.3
0.7 eV

0.3
1 eV

0
0.6 eV

0 eV

II-s, I-c

II-s, I-c

I-c

I-s

I-s

II-c

II-c

III-c

1.5
1.8 eV

1.2
2.1 eV

1.5
1.8 eV

1
1.5 eV

1.4246 eV

0.4
0.6 eV

0.5
0.8 eV

0.2
0.4 eV

0 eV

I-c

I-c

I-c

I-c, II-c

I-c

II-c

III-c

III-c

0.5
1 eV

0.5
1 eV

0.4
0.8 eV

0.1
0.9 eV

0.4
0.6 eV

0.417 eV

0.1
0.2 eV

0 eV

0 eV

I-c, II-s 1.3 eV

I-c, II-s 1
1
5 eV

I-c, II-s 1.4
1.5 eV

II-s, I-s 0.4
1.3 eV

II-s 0.6
0.9 eV

II-s 0.1
0.2 eV

1.696 eV

I-s 0.7 eV

I-s 0.1
0.3 eV

I-c, II-s

I-c

I-c

II-s

II-s

III-s

I-c

0.7
0.8 eV

0.6
0.9 eV

0.8
1 eV

0.2
0.8 eV

0.3
0.5 eV

0 eV

0.7
0.8 eV

0.812 eV

metallic-c, I-c

metallic-c, I-c

I-c

I-c, II-s

I-c, III-s

III-s, metallic-s

I-c

I-c, I-s

0
0.5 eV

0 eV

0
0.5 eV

0
0.3 eV

0
0.3 eV

0 eV

0.3
0.5 eV

0
0.3 eV

InP InAs AlSb GaSb InSb

I-c, I-s 0
0.3 eV 0.235 eV

a

The notation is explained in Figure 1, except metallic-c (metallic-s) which means that the core (shell) has a negative gap. Multiple alignment types are ordered by the core radius. That is, I-c, I-s means that for small core radius the structure is type I-c and for larger core radius the structure is type I-s, as one example. The gap range of the dots is given in the table. Metallic combinations have been bold-faced. 10937

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Table 2. Band Alignments Expected in Different Nitride Core
Shell Dot Systemsa for given shell material core material

AlN

GaN

InN

AlN

4.9 eV

I-s, 3.3 eV

I-s, 0.3
0.8 eV

GaN

I-c, 3.3
3.6 eV

3.3 eV

I-s, 0.4
0.8 eV

InN

I-c, 0
1.4 eV

I-c, II-c; 0.1
1.2 eV

0.78 eV

a

The notation is the same as in Table 1. The gap range of the dots is given in the table.

’ SUMMARY We have calculated the gross electronic structure of most combinations of III
V semiconductor core
shell quantum dots. We have identified a large set of structures that are of physical and biological interest such as intrinsically metallic dots and dots having very small or very large gaps. ’ ASSOCIATED CONTENT

bS

Supporting Information. Tables showing the effective masses and the band-edge energies in III
V dots, text describing the method used in this work, and the data used in Figure 16. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected] (M.-E.P.); craig-pryor@ uiowa.edu (C.E.P.).

Figure 16. Confinement energies as a function of barrier height for a spherical core
shell dot with an infinitely thick shell, classified by angular momentum. Both the barrier height V and the confinement energy E are expressed in terms of the reference energy U0 = p2/ 2m*Rcore2, where m* is the effective mass in the core and shell and Rcore is the radius of the core. For m* = me and Rcore = 1 nm, U0 = 38.1 meV.

’ ACKNOWLEDGMENT We acknowledge Dr. S. Gray for interesting discussions. This work was performed within the nanometer structure consortium in Lund and supported by the Swedish Foundation for Strategic Research (SSF), the Swedish Research Council (VR), and E.ON. AG as part of the E.ON International Research Initiative. The responsibility for the content of this publication lies with the authors. ’ REFERENCES

Figure 17. Illustration of an artificial atom, having three holes in the core and a shell containing three electrons.

vice versa for the type III-s dots). Such dots should be metallic at equilibrium, and we expect them to be plasmonically active with a low plasmon energy due to the low carrier concentration compared with metals. The plasmon energy may be engineered by choosing the materials and radii to obtain a particular charge density, similar to gold shell dots in which the plasmon resonance can be tuned to the infrared region.26 Solid gold dots are known to have a plasmon energy in the infrared
visible range for some sizes.27

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