# Self-Assembled Quantum Dots

Richard J. Warburton, Professor of Experimental Physics (Ordinarius), Department of Physics, University of Basel, Switzerland. Corresponding author: richard.warburton@unibas.ch |

Semiconductor quantum dots are building blocks in quantum communication and information processing systems. A single quantum dot behaves in some ways like a single atom but with the huge advantages that the quantum dot is locked in position and can be functionalized by embedding the quantum dot into a sophisticated heterostructure.

Quantum dots for optical applications can be created by self-assembly using standard growth techniques, for instance molecular beam epitaxy. The work-horse system is InAs for the dot material and GaAs for the substrate material. The lattice constant of InAs is 7% larger than that of GaAs such that a monolayer of InAs on GaAs is highly strained. At a critical thickness of about 1.5 monolayers, instead of a smooth layer, InAs-rich islands form, the quantum dots [1]. This particular growth mode is referred to as the Stranski-Krastanow mode. There is a thermodynamic argument for the formation of the quantum dots: by clumping into islands the overall energy associated with the strain is reduced at the expense of an increase in surface energy. However, in practice, the growth is complex and a number of kinetic factors come into play. This complexity can be exploited: by tweaking conditions in the growth, the properties of the quantum dots can be changed. Typically, the quantum dots are about 20 nm in diameter and 5 nm high, sometimes adopting a lens-shape, sometimes a truncated pyramid. The strain-driven growth mode works also for other semiconductor systems, for instance InAs on InP. As the quantum dots can be formed with standard semiconductor growth techniques, they can be embedded into heterostructures, for instance a vertical tunneling device or a VCSEL-like cavity structure. The wafer can be processed post growth, creating Ohmic contacts, Schottky gates, micropillars, photonic crystals, depending on the particular application.

A semiconductor has a fundamental band gap separating the occupied valence band and the unoccupied conduction band. In a quantum dot, the bands are replaced by discrete levels. In the effective mass approximation, a conduction electron behaves as a free electron but with a much reduced mass (the “effective mass”) and the quantum dot represents a confinement potential in all three dimensions. The particle-in-a-box model of quantum mechanics then leads to quantized states. Typically, there are between 1 and 3 confined electron states, possibly a few more valence states (the “hole” states) [2]. At higher electron or hole energies, the semiconductor bands exist, initially associated with the so-called wetting layer, a thin InGaAs layer which connects all the dots, and at higher energy still, associated with the bulk GaAs. The important optical transition connects the first hole state (the valence state with highest energy) with the first electron state (the conduction state with lowest energy). This is an allowed electric dipole transition with optical dipole moment of d.e where d is about 1 nm [3]. Excitation of this transition creates a so-called exciton, an electron-hole pair in the quantum dot. Without a microcavity, the exciton decays by spontaneous emission with a radiative lifetime of about 1 ns [4]. In a resonant microcavity, the spontaneous emission can be accelerated [5], the Purcell effect, possibly by a factor of ~ 10 with a resonant, small-volume, high-quality microcavity.

A key signature of a two-level optical transition in a single quantum emitter is the antibunching of the emitted photons. This is the fundamental requirement for a single photon source. The emission from a single quantum dot demonstrates photon antibunching [6]. Furthermore, at least at low temperature, laser spectroscopy on single quantum dots has revealed very narrow optical lines: full-widths-at-halfmaxima of 500 MHz are achieved reasonably routinely [7], and in the best case, linewidths of about 200 MHz have been recorded, close to the “transform limit” where the linewidth is determined solely by the radiative recombination rate. The development of laser spectroscopy (both with transmission detection [3, 7] and resonance fluorescence [8]) has enabled other signatures of an atomic two-level system to be demonstrated: power broadening/power-induced transparency, the existence of dressed states, the Mollow triplet, etc. It is remarkable to have a textbook two-level system inside a semiconductor. Furthermore, a gate allows the quantum dot “periodic table” to be accessed just by applying a voltage to the device [9], and in fact many properties can be tuned in this way. However, in other experiments, the complexity of the semiconductor environment reveals itself. This interplay between atomic physics on the one hand and full-blown condensed matter physics has driven a lot of research in this area.

One of the major problems in turning the antibunched photons from a single quantum dot into a real single photon source is extracting the photons from the semiconductor with high efficiency. The semiconductor has a very high refractive index, 3.5 for GaAs for instance, meaning that most of the photons are refracted at the semiconductor-air interface to large angles where it is very difficult to collect them with a lens. One possible solution is to embed the quantum dots in a microcavity [5]. The Purcell effect causes the photons to be emitted preferentially into the cavity mode. In this way, micropillar devices have achieved efficiencies of about 40%. However, the cavity amplifies any number of interactions in the semiconductor and in many cases, the quality of the antibunching goes down as the quantum efficiency goes up. Other solutions are emerging, for instance, a tapered one-dimensional waveguide which has already demonstrated quantum efficiencies of 72% [10]. Quantum dots are therefore robust, high repetition rate, narrow linewidth sources of single photons, characteristics not shared presently by any other emitter.

A self-assembled quantum dot is also an attractive host for a spin qubit [11]. Applications include an ultra-sensitive magnetometer, quantum repeaters (to extend the distance over which quantum cryptography can operate) and possibly also quantum information processing. There are two main points. First, a spin confined to a self-assembled quantum dot can be initialized, manipulated and read-out optically. The use of pulsed lasers allow these operations to be carried out quickly, with spin rotations on sub-ns time scales for instance [12]. Secondly, the strong quantization induces a huge mismatch between the “size” of the electron wave function in the quantum dot and the phonon wavelengths corresponding to the Zeeman splitting, particularly at small magnetic fields. Spin relaxation in a quantum dot is therefore highly suppressed: relaxation times longer than 1 s are possible. However, the coherence times are much shorter: by suppressing the interaction with the phonons, the hyperfine interaction (electron spin-nuclear spin interaction) has been enhanced such that the dephasing is dominated by noise in the nuclear spins. Taming the nuclear spins, or even exploiting them, are possibilities as is taking them out of the game as best one can by switching from an electron spin to a hole spin [13, 14].

*Figure A. Atomic force microscope image of InAs self-assembled quantum dots on a GaAs
surface. Image recorded by Axel Lorke. *

*Figure B. Cross-sectional scanning tunneling microscope image of an
InGaAs quantum dot. The image is 80 X 40 nm ^{2}. *

*Figure C. Photoluminescence from a single quantum dot in
a vertical transistor structure as a function of voltage. There are clear charging events, from the neutral
exciton X ^{0} to the charged exciton X^{1-} to the doubly-charged exciton X^{2-} [9]. *

*Figure D. Laser spectroscopy
on a single quantum dot at 4.2 K showing a transmission dip at the resonance [7].*

**References**

[1] D. Leonard, K. Pond, and P. M. Petroff, Phys. Rev. B **50**, 11687 (1994).

[2] R. J. Warburton, C. S. Dürr, K. Karrai, J. P. Kotthaus, G. Medeiros-Ribeiro, and P. M. Petroff, Phys. Rev. Lett. **79**, 5282 (1997).

[3] K. Karrai and R. J. Warburton, Superlattices and Microstructures **33**, 311 (2003)

[4] P. A. Dalgarno, J. M. Smith, J. McFarlane, B. D. Gerardot, K. Karrai, A. Badolato, P. M. Petroff, and R. J. Warburton, Phys. Rev. B **77**, 245311 (2008).

[5] J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, Phys. Rev. Lett. **81**, 1110 (1998).

[6] P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff1, L. Zhang, E. Hu, and A. Imamoglu, Science **290**, 2282 (2000).

[7] A. Högele, S. Seidl, M. Kroner, K. Karrai, R. J. Warburton, B. D. Gerardot, and P. M. Petroff, Phys. Rev. Lett. **93**, 217401 (2004).

[8] A. N. Vamivakas, C. -Y. Lu, C. Matthiesen, Y. Zhao, S. Fält, A. Badolato, and M. Atatüre, Nature **467**, 297 (2010).

[9] R. J. Warburton, C. Schäflein, D. Haft, F. Bickel, A. Lorke, K. Karrai, J. M. Garcia, W. Schoenfeld, and P. M. Petroff, Nature **405**, 926 (2000).

[10] J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J. -M. Gérard, Nature Photonics **4**, 174 (2010).

[11] D. Loss and D. P. DiVincenzo, Phys. Rev. A **57**, 120 (1998).

[12] D. Press, T. D. Ladd, B. Zhang, and Y. Yamamoto, Nature **456**, 218 (2008).

[13] B. D. Gerardot, D. Brunner, P. A. Dalgarno, P. Öhberg, S. Seidl, M. Kroner, K. Karrai, N. G. Stoltz, P. M. Petroff, and R. J. Warburton, Nature **451**, 441 (2008).

[14] D. Brunner, B. D. Gerardot, P. A. Dalgarno, G. Wüst, K. Karrai, N. G. Stoltz, P. M. Petroff, and R. J. Warburton, Science **325**, 70 (2009).