Terahertz Ionization of Highly Charged Quantum Posts in a Perforated

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Terahertz Ionization of Highly Charged Quantum Posts in a Perforated Electron Gas Christopher M. Morris,*,†,‡ Dominik Stehr,‡,z Hyochul Kim,‡,# Tuan-Anh Truong,|| Craig Pryor,^ Pierre M. Petroff,§,|| and Mark S. Sherwin†,‡ Department of Physics, §Department of Electrical and Computer Engineering, and Department of Materials Science, University of California, Santa Barbara, Santa Barbara, California 93106, United States ‡ Institute for Terahertz Science and Technology, Santa Barbara, California, 93106, United States ^ Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242-1479, United States # Department of Electrical and Computer Engineering, IREAP, University of Maryland, College Park, Maryland 20742, United States z Institute for Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, P.O. Box 510119, 01314 Dresden, Germany )

†

bS Supporting Information ABSTRACT: “Quantum posts” are roughly cylindrical semiconductor nanostructures that are embedded in an energetically shallower “matrix” quantum well of comparable thickness. We report measurements of voltage-controlled charging and terahertz absorption of 30 nm thick InGaAs quantum wells and posts. Under flat-band (zero-electric field) conditions, the quantum posts each contain approximately six electrons, and an additional ∼2.4  1011 cm
2 electrons populate the quantum well matrix. In this regime, absorption spectra show peaks at 3.5 and 4.8 THz (14 and 19 meV) whose relative amplitude depends strongly on temperature. These peaks are assigned to intersubband transitions of electrons in the quantum well matrix. A third, broader feature has a temperature-independent amplitude and is assigned to an absorption involving quantum posts. Eight-band k 3 p calculations incorporating the effects of strain and Coulomb repulsion predict that the electrons in the posts strongly repel the electrons in the quantum well matrix, “perforating” the electron gas. The strongest calculated transition, which has a frequency close to the center of the quantum post related absorption at 5 THz (20 meV), is an ionizing transition from a filled state to a quasi-bound state that can easily scatter to empty states in the quantum well matrix. KEYWORDS: Terahertz, quantum dots, quantum posts, electron gas, ionization

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uantum dots have excited considerable interest since their inception due to their ability to function as artificial atoms.1 Unlike real atoms, their optoelectronic properties can be engineered through materials selection and geometry, allowing for production of dots with properties suited to their application. As in atoms, discrete energy levels are formed for confined carriers. However, due to their significantly increased size, their energy levels are usually much more closely spaced. Coulomb effects play a more important role in these structures as their dimensions grow larger and the Coulomb energy scales become a significant fraction of the intersublevel energy spacing. Considerable study has gone into the way in which electrons fill and interact in quantum dots, and their corresponding infrared and optical spectroscopic properties.2
6 Recently, we introduced the quantum post, a nanostructure based on quantum dots in the InGaAs materials system. r 2011 American Chemical Society

Quantum posts are roughly cylindrical in shape and have diameters similar to quantum dots (∼30 nm). While the heights of typical quantum dots are ∼5 nm, quantum posts have been grown with heights of up to 60 nm. This height is controlled with nanometer precision during growth. If they are sufficiently tall, their lowest energy levels can be controlled directly by selection of their height,7,8 rather than being defined by the radial confinement as in quantum dots.9 This produces an energy level structure with spacings that are tunable from the mid-IR to far-IR depending on the height of the post. Their increased height also increases the buildup of indium lattice mismatch strain, which significantly alters the energetic depth of quantum posts compared to dots. Quantum posts are offset from the GaAs conduction Received: December 17, 2010 Revised: March 19, 2011 Published: April 26, 2011 1115

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Figure 1. Sample structure. (a) Sample structure used for capacitance and absorption experiments. Terahertz light for the absorption experiments is polarized along the growth direction, and coupled directly into the cleaved side facets of the sample. This gives multiple passes through the post region, increasing absorption. Normal incidence spectra (not shown) with a 15 nm transparent Schottky contact, passing through the active region only once, did not show any detectable absorption. Voltage bias between the Al Schottky front contact and the nþ back contact is used to controllably load electrons into the quantum post region embedded between the two contacts. (b) Band diagram in the vertical (growth) direction of a 30 nm quantum post and the quantum well matrix under flat band conditions. At the bottom and top of the posts there are quantum dots with slightly higher indium content, making the post a few nanometers taller than the quantum well. These dots produce the features seen at the top and bottom of the post conduction band. Doping related band bending has not been included here.

band edge by only 120 meV, shallower than dots of similar indium content, which are offset by ∼300 meV.10
12 A final important difference between the two is that the quantum posts are embedded in an energetically shallower InGaAs quantum well matrix of the same physical height as the posts. This quantum well is similar to the wetting layer in quantum dots and is an unavoidable byproduct of the post growth process. Like the wetting layer, it acts as a charge trap, but it is far thicker and its spectral and charge transport properties are accordingly different.13 Related systems have been studied by placing quantum dots in quantum wells,14 although the height of the quantum well is significantly larger than that of the quantum dots in this case. Here, we present an investigation of terahertz absorption due to electrons confined in quantum post and quantum well nanostructures. In interpreting the absorption of the structures, single particle calculations are found to be insufficient to explain the observed behavior. Full many electron calculations must be performed to predict the shift of the post energy levels due to the Coulomb blockade and to understand the electronic filling of the posts. The energetically shallow nature of the posts and surrounding well means that both can be charged at the same time. Coulomb interactions within the posts produce a significant change to the post energy level structure. In addition, significant Coulomb interactions between electrons in the posts and matrix quantum well play an important role in the electronic structure of the system. Consequently, the normal two-dimensional electron gas present in a quantum well is perforated by highly charged quantum posts that can locally alter the density of electrons in the well.15,16 The samples studied are shown schematically in Figure 1a. A single active layer containing quantum posts is grown between two layers of GaAs. As Figure 1b shows, the quantum posts and surrounding quantum well matrix are of approximately the same height. Thus, the energy spacings between conduction subbands of the quantum well matrix are similar to the energy spacings between conduction-band levels arising from vertical confinement in the quantum posts. In the 30 nm tall quantum posts investigated here, the vertical confinement defines the smallest level spacing, and in the chosen experimental geometry (Figure 1a)

only height controlled states are involved in the terahertz response. To distinguish between effects occurring in the quantum well and those occurring due to the presence of the quantum posts, we compare experimental results on a sample containing a layer of 30 nm high quantum posts with results from a reference sample that contains a 30 nm In0.1Ga0.9As quantum well with no posts. The reference quantum well resembles the matrix of the quantum post sample. To control the charging and electric field, the active layer (quantum posts or quantum well) is inserted into the metal
insulator
semiconductor (MISFET) structure shown in Figure 1a.17 On a semi-insulating GaAs (100) substrate, the following layer sequence is deposited: a heavily doped GaAs nþ back contact layer to pin the Fermi level in the conduction band; a GaAs tunneling barrier; the quantum post or quantum well active layer; a GaAs layer with a heavily n-doped monolayer (delta-doped) to bring the conduction band close to the Fermi level, greatly reducing the built-in electric field across the active layer; an AlGaAs barrier to prevent tunneling to the top contact; and a thin GaAs cap. A 300 nm thick Al layer is deposited after growth to serve as a Schottky contact, and indium is used to electrically contact the nþ layer. Application of a bias between the Al Schottky contact and the nþ back contact enables the modification of the electric field across the device. The posts are grown at a high density of ∼1  1010/cm2, and experiments presented here always probe an ensemble of approximately 2.5  109 posts. Both the capacitance measurements and voltage dependent absorption reported in this Letter are taken at 10 K, close to the zero-temperature limit for these samples where the thermal energy (kBT = 0.86 meV) is much less than the energy of the terahertz transitions being studied (3 THz = 12.9 meV). Capacitance
voltage (C
V) spectroscopy is used to investigate the charging behavior of the samples as a function of the voltage between the Al Schottky gate and the nþ back contact.18 Integration of the C
V curve gives the charge in the active layers. The top panel of Figure 2a shows the C
V spectrum of the quantum well sample. Below
1.5 V the conduction band of the active layer remains above the Fermi level. As the voltage is increased above
1.5 V, the lowest subband of the quantum well reaches the Fermi level, leading to 1116

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Figure 2. Terahertz absorption of quantum well reference and quantum post sample. (a) Quantum well reference sample capacitance and absorption spectra as a function of applied voltage. As the ground subband of the quantum well fills, the first absorption peak appears and shifts to lower energies as the bias is increased and the electric field across the reference well decreases. When the delta-doped region fills at
0.75 V, the absorption frequency shift stops due to screening of the electric field by the charge in the delta doped region. At
0.5 V a second absorption peak at 19 meV appears, representing absorption from the now partially filled second subband to the empty third subband. Below
1.7 V there is no charging of the well and no absorption spectra were recorded. (b) Quantum post capacitance and absorption spectra. As the posts fill from
2.0 to
1.5 V, the absorption peak is Stark shifted above 6 THz, frequencies where the sample does not transmit. At
1.3 V the absorption has shifted into the experimentally accessible range. As in (a), two absorption peaks appear at the higher voltages. These are assigned to intersubband absorptions from the quantum well matrix. The additional broad underlying absorption is assigned to transitions involving quantum posts.

the electronic filling of the quantum well. This increases the capacitance by reducing the distance between the front gate and the leading edge of the charge distribution. As the bias voltage is further increased above
0.75 V, the delta-doped region of the sample reaches the Fermi level and begins to fill with electrons, causing another steplike increase in the capacitance. This charging behavior of the sample was simulated using a 1D Schr€odinger
Poisson solver,19 which confirms the interpretation of these capacitance features. The C
V spectrum of the quantum post sample (Figure 2b) is similar to that of the quantum well. The two peaks found in the quantum well reference sample are present, again caused by electronic filling of the quantum well matrix and delta-doped region. In addition, at
2.0 V there is a capacitance increase associated with the onset of charging in the quantum posts. Due to their higher indium content, the posts are energetically deeper than the quantum well, and their bound states reach the Fermi level and begin to fill at lower voltages. However, integration of the capacitance in this region between
2.0 and
1.5 V reveals that the posts have only filled with ∼2.5  109/cm2 electrons, significantly less than their areal density. Decreasing the frequency of the oscillating voltage in the C
V measurement shows no increase in the electronic filling down to 50 Hz. This behavior is not fully understood. However, as the first subband in the well reaches the Fermi level, charge is able to efficiently transfer into the well and then immediately into the posts.20 Comparison of the C
V from the quantum well and quantum post samples shows that the post electron density is ∼6  1010/cm2, or six electrons per post by the time the modulation doped region begins to fill at
0.75 V (Supporting Information). To investigate the terahertz electronic resonances of the quantum posts, we compare the voltage-dependent absorption spectra of the quantum well reference and the quantum post

samples, shown in the lower panels of Figure 2. When the quantum well reference sample begins to load at
1.5 V, a single absorption line appears near 5 THz. As the voltage is further increased, the dc electric field at the well decreases (Figure 2a, inset), and the absorption red shifts strongly due to the dc Stark effect. Since the posts are energetically deeper than the well in the reference sample, they begin to load at
2 V rather than at
1.5 V. The Stark shift at this voltage is so large that it pushes the frequency of the post transition above 6 THz, where phonon absorption makes the sample opaque to terahertz light. With the current sample design, it is not possible to significantly reduce the electric field, and hence the Stark shift, at which the posts load: a longer tunneling barrier would be required to reduce the electric field, and the filling of posts is already inefficient across the large 45 nm tunneling barrier. Absorption spectra were investigated in detail at 0.5 V, when the charge in each active region is nearly independent of voltage and the quantum well and quantum posts are close to flat-band (zero electric field) conditions. The quantum well spectrum (Figure 3a) shows two clear peaks at 14 and 19 meV, as well as a linear background absorption starting at 10 meV. The origin of the linear background absorption is unclear, although it may be associated with TO phonon absorption in InAs or possibly weak bound to continuum absorption from a subband of the well. We assign the peak at 14 meV as the ground to first excited intersubband transition in the quantum well. Quantum mechanical selection rules forbid the ground to second excited state transition, and theoretical calculations predict the first to second excited state transition to occur near 19 meV, and the second peak is assigned to this transition. Selfconsistent simulations that include the dynamic electron
electron interaction in the random phase approximation21 match these observed transition energies at the well electron density derived from the C
V data (∼2.4  1011/cm2). 1117

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Figure 3. Temperature-dependent terahertz absorption. (a) 10 K absorption spectra from 30 nm quantum post (black solid) and quantum well (blue dashed) reference samples at 0.5 V. (b) 30 nm quantum post absorption at 0.5 V with fits. The solid blue line shows the remaining absorption after the Lorentzian peaks from the quantum well (red dashed) and linear background are subtracted from the original spectrum. The green dot-dashed line shows the total fit. (c) Subbands of the quantum well, along with the fractional occupation due to the temperature-dependent Fermi distribution and the constant density of states for quantum well subbands. (d) Temperature-dependent absorption of the quantum well. The top panel shows the expected relative strength of the transitions from calculations (solid lines) along with the experimental transition strengths based on fits of the data. (e) Temperature-dependent absorption of the quantum post sample. The quantum well transitions show the same relative temperature-dependent features, while the strength of the absorption from the quantum posts shows virtually no sensitivity to temperature up to 100 K as expected from simple temperature-dependent electronic filling models of quantum posts.

The absorption of the quantum post sample shows the two peaks characteristic of the quantum well reference sample, which are assigned to the quantum well matrix surrounding the posts. In addition, there is a large underlying absorption that is absent in the quantum well sample, which is assigned to transitions involving the posts. Figure 3b shows a deconvolution of the spectrum in which the two Lorentzian peaks due to the quantum well transitions and the linear background are subtracted. The remaining absorption is fit with a Gaussian line shape, the expected absorption line shape for a transition dominated by inhomogeneity as in quantum dots. The simplest assignment of the Gaussian line would be to single-electron transitions between sublevels of the quantum posts, which are at terahertz frequencies. However, C
V experiments show that the posts have accepted their maximum number of electrons and the well has started to fill. This leaves all intersublevel transitions within the posts Pauli blocked. To further investigate and clarify the origin of the quantum post related absorption, temperature-dependent absorption measurements were performed on both samples. The post states are deeper below the Fermi level than the well states and as such should have a different temperature-dependent filling fraction.

Panels d and e of Figure 3 show the temperature-dependent absorption of the quantum well and quantum post sample at a voltage of 0.5 V, where conduction band states in the posts are full and both the first and second subbands of the quantum well have begun to fill. The same Lorentzian/Gaussian fits were done at each temperature, and the transition strengths derived from the fits are plotted in the top panel of the figures. Self-consistent calculations were used to predict the quantum well absorption strengths based on the transition oscillator strengths and the electronic filling of the well, with the results shown as solid lines. As expected in the quantum well, the first subband quickly depopulates with increasing temperature, and a corresponding decrease in absorption is seen from the ground to first excited subband transition. The population of the second subband increases with temperature, and correspondingly the absorption from the first to second excited subband gains strength. The underlying Gaussian transition is nearly unaffected by the temperature increase, consistent with an energy level that is deeper below the Fermi level than the quantum well subbands, implying the involved energy level must be in the posts. To identify the involved states, eight band k 3 p calculations of the 30 nm quantum post bound states and their energies were 1118

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Figure 4. Eight band k 3 p calculations of post energy levels with multiple electrons. The in-plane (x
y) conduction band profile corresponds to flat band conditions, moving parallel to the plane of the quantum well matrix and cutting through a post. The conduction band and associated energy levels for a post with no electrons are shown in black. When the posts are filled with their maximum of six electrons, the conduction band profile and the resulting electron energy levels (red) are significantly modified. The lowest three levels are below the quantum well band edge and are occupied, with higher lying states above the Fermi energy (blue dotted) remaining unoccupied. In the well, the band edge near the post is raised enough to locally depopulate the subbands of the quantum well near the posts. The observed absorption is due to transitions from the highest filled state to an unoccupied state 20.5 meV above. From this unoccupied state, electrons quickly scatter into the locally depopulated quantum well states. Both strain and Coulomb electric field effects are included in this calculation.

performed as a function of the number of electrons in the posts.22
24 As found experimentally from the capacitance, the calculations predict that the charging energy from the Coulomb blockade prevents more than six electrons from occupying each post. Although the quantum posts are physically much larger than quantum dots, the total charge they can hold is found to be similar because of their smaller band offset. With six electrons in a post, the strongest transition is from the filled third conduction band level to an unoccupied higher post level as shown in Figure 4. The associated transition energy is 20.5 meV, matching very well with the observed value of 21 meV found for the background Gaussian. Electrons from the third level absorb a terahertz photon and transition to the excited state and then scatter into the lowest subband of the quantum well. This is reminiscent of the transition used in quantum dot infrared photodetectors (QDIPs) and quantum dot-in-a-well photodetectors (DWELLs), but with significantly reduced energies.25 The inhomogeneity of the posts leads to the spread in transition energies seen in the Gaussian in Figure 3b. The post modeled in the calculations is the average post, but there is a variation in the size and shape of the posts as seen in quantum dots. This size variation also translates into different charging of posts which in turn additionally broadens the absorption line: posts with different numbers of electrons have energy levels that are modified differently by the charging energy from adding additional electrons. This charging energy effect works to spread the post energies out in addition to the broadening expected from variation in energy levels for posts of different sizes and shapes. A remaining question is how scattering from the highest filled state of the quantum post to the first quantum well subband is

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possible with the given electron distribution. On the basis of the capacitance and comparisons of the observed quantum well energies and transition strengths with theoretical calculations, the well has an electronic sheet density of approximately 2  1011/cm2 in the first subband, and 4  1010/cm2 in the second subband. With the first subband of the well filled, scattering from the posts to these states should be blocked. However, calculations show that the quantum well matrix does not obey the ideal two-dimensional parabolic dispersion relation everywhere. Far from the posts this is a good approximation, but near the posts electrons in the well feel a strong Coulomb repulsion from the six electrons in the posts. This pushes the local level for the well above the Fermi level, leaving open states for the quantum posts electrons to scatter into. The continuous electron gas normally found in a quantum well is perforated by these regions of strong Coulomb repulsion near the posts. Previously, two-dimensional electron gases have been perforated by “antidots”.26,27 In antidot systems, the perforations in the electron gas are created by etching regular arrays of cylindrical holes near an electron gas and thus depleting it by the surface field. In quantum post systems described here, the perforations in the electron gas in the quantum well matrix have a very different origin—they emerge as a consequence of the Coulomb repulsion by the high concentration of electrons in each quantum post. The length scales in the post systems (spacing and radius of perforations) are also significantly smaller than those typically investigated in artificial antidot systems, which are limited by processing restrictions. Normal incidence spectroscopy of the quantum post perforated electron gas system did not reveal any associated resonances, as observed in antidot systems. However, the small size of the posts may push the associated resonances above the experimental frequency range presented here, and their random distribution may also inhomogeneously broaden it enough to push the signal below the noise floor of the experimental setup. Further experimental investigation and calculations beyond the effective medium theory normally used in the antidot systems could prove to be an interesting area of investigation in the quantum post perforated electron gas system. The development of the quantum post materials system was initially motivated by the goal of creating nanostructures which resonate at terahertz frequencies, with potential applications including terahertz quantum cascade lasers28 based on a threedimensionally confined active region, improved terahertz detectors, and even quantum information processing.29 With further development, these applications of quantum posts may come to fruition. However, the central discovery of this Letter—that charging quantum posts and their quantum well matrix can result in a perforated electron gas—was completely unanticipated. While semiconductor quantum dots are often described as artificial atoms, the electron gas perforated by highly charged quantum posts has no analogue of which we are aware in atomic physics. The physics of this new electronic system, with intimate coupling between atom-like and quasi-two-dimensional states, multiple length scales, and important electron
electron interactions, is likely to yield new surprises in years to come.

’ ASSOCIATED CONTENT

bS

Supporting Information. Details of the structure of the sample grown, the experimental methods used for the FTIR spectroscopy, the capacitance
voltage charge integration, and

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Nano Letters the theoretical calculations that were performed. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Institute for Terahertz Science and Technology, 3410 Broida Hall, Mail Code 4170, Santa Barbara, CA 93106-4170.

’ ACKNOWLEDGMENT The authors thank the NSF Nanoscale Interdisciplinary Research Team Grant CCF-0507295, NSF DMR 1006603, DARPA, and the Alexander von Humboldt Foundation for supporting this research and Jim Allen for invaluable assistance with interpreting the C
V spectra. ’ REFERENCES (1) Ashoori, R. C. Electrons in artificial atoms. Nature 1996, 379 413–419. (2) Fricke, M.; Lorke, A.; Kotthaus, J. P.; Medeiros-Ribeiro, G.; Petroff, P. M. Shell Structure and electron-electron interaction in selfassembled InAs quantum dots. Europhys. Lett. 1996, 36, 197. (3) Warbutron, R. J.; Miller, B. T.; Durr, C. S.; Bodefeld, C.; Karrai, K.; Kotthaus, K. P.; Medeiros-Ribeiro, G.; Petroff, P. M.; Huant, S. Coulomb interactions in small charge-tunable quantum dots: A simple model. Phys. Rev. B 1998, 58, 16221–16231. (4) Petroff, P. M. Single Quantum Dots. In Topics in Applied Physics; Michler, P., Ed.; Springer: Berlin, 2003; Vol. 90; p 1. (5) Bras, F.; Boucaud, P.; Sauvage, S.; Fishman, G.; Gerard, J.-M. Temperature dependence of intersublevel absorption in InAs/GaAs selfassembled quantum dots. Appl. Phys. Lett. 2002, 80, 4620–4622. (6) Carpenter, B. A.; Zibnik, E. A.; Sadowski, M. L.; Wilson, L. R.; Whittaker, D. M.; Cockburn, J. W.; Skolnick, M. S.; Potemski, M.; Steer, M. J.; Hopkinson, M. Intraband magnetospectroscopy of singly and doubly charged n-type self-assembled quantum dots. Phys. Rev. B 2006, 74, 161302. (7) Krenner, H. J.; Pryor, C. E.; He, J.; Petroff, P. M. A Semiconductor Exciton Memory Cell Based on a Single Quantum Nanostructure. Nano Lett. 2008, 8, 1750–1755. (8) Krenner, H. J.; Pryor, C.; He, J.; Zhang, J. P.; Wu, Y.; Morris, C. M.; Sherwin, M. S.; Petroff, P. M. Growth and optical properties of self-assembled InGaAs quantum posts. Physica E 2008, 40, 1785–1789. (9) Zibik, E. A.; Grange, T.; Carpenter, B. A.; Porter, N. E.; Ferreira, R.; Bastard, G.; Stehr, D.; Winnerl, S.; Helm, M.; Liu, H. Y.; Skolnick, M. S.; Wilson, L. R. Long lifetimes of quantum-dot intersublevel transitions in the terahertz range. Nat. Mater. 2009, 8, 803–807. (10) He, J.; Notzel, R.; Offermans, P.; Koenraad, P. M.; Gong, Q.; Hamhuis, G. J.; Eijkemans, T. J.; Wolter, J. H. Formation of columnar (In,Ga)As quantum dots onGaAs(100). Appl. Phys. Lett. 2004, 85, 2771–2774. (11) He, J.; Krenner, H. J.; Pryor, C.; Zhang, J. P.; Wu, Y.; Allen, D. G.; Morris, C. M.; Sherwin, M. S.; Petroff, P. M. Growth, Structural, and Optical Properties of Self-Assembled (In,Ga)As Quantum Posts on GaAs. Nano Lett. 2007, 7, 802–806. (12) Krenner, H. J.; Petroff, P. M. Quantum posts with tailored structural, electronic and optical properties for optoelectronic and quantum electronic device applications. Solid State Commun. 2009, 149, 1386–1394. (13) Volk, S.; Schulein, F. R. J.; Knall, F.; Reuter, D.; Wieck, A. D.; Truong, T. A.; Kim, H. C.; Petroff, P. M.; Wixforth, A.; Krenner, H. J. Enhanced Sequential Carrier Capture into Individual Quantum Dots and Quantum Posts Controlled by Surface Acoustic Waves. Nano Lett. 2010, 10, 3399–3407.

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(14) Aivaliotis, P.; Menzel, S.; Zibik, E. A.; Cockburn, J. W.; Wilson, L. R.; Hopkinson, M. Energy level structure and electron relaxation times in InAs/InGaAs quantum dot-in-a-well structures. Appl. Phys. Lett. 2007, 91, 253502. (15) Sakaki, H.; Yusa, G.; Someya, T.; Ohno, Y.; Noda, T.; Akiyama, H.; Kadoya, Y.; Noge, H. Transport properties of two - dimensional electron gas in AlGaAs/GaAs selectively doped heterojunctions with embedded InAs quantum dots. Appl. Phys. Lett. 1995, 67, 3444–3446. (16) Ribeiro, E.; Muller, E.; Heinzel, T.; Auderset, H.; Ensslin, K.; Medeiros-Ribeiro, G.; Petroff, P. M. InAs self-assembled quantum dots as controllable scattering centers near a two-dimensional electron gas. Phys. Rev. B 1998, 58, 1506–1511. (17) Medeiros-Ribeiro, G.; Leonard, D.; Petroff, P. M. Electron and hole energy levels in InAs self-assembled quantum dots. Appl. Phys. Lett. 1995, 66, 1767–1769. (18) Medeiros-Ribeiro, G.; Pikus, F. G.; Petroff, P. M.; Efros, A. L. Single-electron charging and Coulomb interaction in InAs selfassembled quantum dot arrays. Phys. Rev. B 1997, 55, 1568–1573. (19) Snider, G. L. Computer Program: 1D Poisson/Schrodinger: A Band Diagram Calculator. (http://www.nd.edu/∼gsnider, University of Notre Dame, Notre Dame, IN). (20) Stehr, D.; Morris, C. M.; Talbayev, D.; Wagner, M.; Kim, H. C.; Taylor, A. J.; Schneider, H.; Petroff, P. M.; Sherwin, M. S. Ultrafast carrier capture in InGaAs quantum posts. Appl. Phys. Lett. 2009, 95, 251105–251107. (21) Ando, T.; Fowler, A. B.; Stern, F. Electronic properties of twodimensional systems. Rev. Mod. Phys. 1982, 54, 436–672. (22) Pryor, C. Eight-band calculations of strain InAs/GaAs quantum dots compared with one-, four-, and six-band approximations. Phys. Rev. B 1998, 57, 7190–7195. (23) Vurgaftman, I.; Meyer, J. R.; Ram-Mohan, L. R. Band parameters for III-V compound semiconductors and their alloys. Appl. Phys. Lett. 2001, 6089, 5815–5875. (24) Pryor, C. Geometry and material parameter dependence of InAs/GaAs quantum dot electronic structure. Phys. Rev. B 1999, 60, 2869–2874. (25) Kim, E.-T.; Madhukar, A.; Ye, Z.; Campbell, J. C. High Detectivity InAs quantum dot infrared photodetectors. Appl. Phys. Lett. 2004, 84, 3277. Sellers, I. R.; Mowbray, D. J.; Badcock, T. J.; Wells, J.-P. R.; Phillips, P. J.; Carder, D. A.; Liu, H. Y.; Groom, K. M.; Hopkinson, M. Infrared modulated interlevel spectroscopy of 1.3 μm self-assembled quantum dot lasers using a free electron laser. Appl. Phys. Lett. 2006, 88, 081108. (26) Hochgrafe, M.; Krahne, R.; Heyn, Ch.; Heitmann, D. Anticyclotron motion in antidot arrays. Phys. Rev. B 1999, 60, 10680–10682. (27) Kern, K.; Heitmann, D.; Grambow, P.; Zhang, Y. H.; Ploog, K. Collective excitations in antidots. Phys. Rev. Lett. 1991, 66, 1618–1621. (28) Williams, B. S. Terahertz quantum-cascade lasers. Nat. Photonics 2007, 1, 517–525. (29) Sherwin, M. S.; Imamoglu, A; Montroy, T. Quantum computation with quantum dots and terahertz cavity quantum electrodynamics. Phys. Rev. A 1999, 60, 3508–3514.

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