Here, we focus on another type of QD array in which the coherent coupling between QDs forms a band structure in the energy spectrum. Kouwenhoven et al. studied the magnetotransport properties of an artificial one-dimensional (1D) crystal of 15 QDs defined in the 2DEG of GaAs/AlGaAs heterostructures by means of a split-gate technique [227]. They observed large oscillations with two deep dips enclosing 15 small oscillations. They argued that the larger oscillations correspond to large energy gaps of the 1D crystal of QDs and the smaller ones to the discrete levels in the miniband. Haug et al. [228] built linear chains of QDs consisting of one to four dots. Transport measurements of these devices show sharp resonant conductance peaks around threshold, which are attributed to the Coulomb charging and quantum confinement effects.

Phase transition [229-235], correlated electron transport [184,190,236-238], and addition spectra [185,229,239] in linear chains of coupled QDs have been theoretically studied. Stafford and Das Sarma [181] studied the addition spectra in a 1D array of QDs and obtained quantum phase transitions, such as the Mott-Hubbard metal-insulator transition. Ugajin [230-232] predicted a Mott-Hubbard metal-insulator transition driven by an external electric field. This effect provides a method of modulating collective excitations locally and can be applied to new field-effect devices.

Engineering of electronic and magnetic properties of a lattice of QDs has also been investigated [240-247]. Sugimura [240] theoretically proposed the possibility of magnetic ordered states in a semiconductor quantum dot array. Describing the GaAs/AlGaAs QD system by the one-band Hubbard model, he showed that magnetic orderings, such as ferromagnetism and antiferromagnetism, can be expected in

Figure 15. (a) Schematic of a QCA half cell. (b) Scanning electron microscopy image of the device utilizing Al/AlO^/Al tunnel junctions. (c) Schematic diagram of the experimental circuit. A single electron moves between the top (D1) and the bottom (D3) quantum dots through the middle dot (D2), which acts as a barrier and locks the electron in either D1 or D3 when it is biased. (d)-(f) Measured (solid lines) and calculated (dotted lines) responses to the input and clocking signals together with applied input signals (dashed lines) of the top (d), middle (e), and bottom (f) dot. Reprinted with permission from [225], A. O. Orlov et al., Appl. Phys. Lett. 77, 295 (2000). ©2002, American Institute of Physics.

Figure 15. (a) Schematic of a QCA half cell. (b) Scanning electron microscopy image of the device utilizing Al/AlO^/Al tunnel junctions. (c) Schematic diagram of the experimental circuit. A single electron moves between the top (D1) and the bottom (D3) quantum dots through the middle dot (D2), which acts as a barrier and locks the electron in either D1 or D3 when it is biased. (d)-(f) Measured (solid lines) and calculated (dotted lines) responses to the input and clocking signals together with applied input signals (dashed lines) of the top (d), middle (e), and bottom (f) dot. Reprinted with permission from [225], A. O. Orlov et al., Appl. Phys. Lett. 77, 295 (2000). ©2002, American Institute of Physics.

QD arrays. Khurgin et al. showed that exchange interactions between holes confined in arrays of strained In1-x GaxAs QDs (see Fig. 16) can lead to ferrimagnetic arrangement of magnetic moments. The ferrimagnetism is induced by coupling two neighboring QDs having different values of orbital angular momenta. The Curie temperature for QDs with thickness of t1 = 7 nm and t2 = 3 nm is of the order of 25 K (see Fig. 16d). Khurgin et al. also proposed a method of engineering ferroelectric arrays of coupled strained QDs [242].

Lattice structure and carrier concentration are essential factors determining the electronic and magnetic properties in solids. Carrier concentration in solids can be controlled by chemical or gate-induced doping [248,249]. On the other hand, building a desired lattice of atoms is very hard, because the lattice structure is strictly determined by the atomic nature of constituent elements. The design of materials with desired lattice is possible by putting QD atoms on the lattice points and coupling the QDs with each other to form artificial crystals called quantum dot superlattices (QDSLs). Semiconductor fabrication technology enables us to form QDSLs in various patterns as we wish, including lattices not existing in nature. The electron concentration can be tuned by the gate voltages. Therefore, QDSLs are suitable for achieving various interesting effects, such as flat-band ferromagnetism [244-247], which have been considered unlikely to occur in semiconductors.

Figure 16. (a) One period of a QD array of In1-x GaxAs and In1-y Gay As buried in In0 52Al048As. (b) Schematic of the valence-band alignment of the QD array and (c) directions of spin and orbital angular momenta. (d) Curie temperature Tc and the thickness of unstrained QD2-t2 as functions of the thickness of strained QD1-i1 for different compositions x of strained In1-x GaxAs. Lateral size d = 120 Â. Barrier thickness tb = 20 Â. Reprinted with permission from [241], J. B. Khurgin et al., Appl. Phys. Lett. 73, 3944 (1998). ©1998, American Institute of Physics.

Figure 16. (a) One period of a QD array of In1-x GaxAs and In1-y Gay As buried in In0 52Al048As. (b) Schematic of the valence-band alignment of the QD array and (c) directions of spin and orbital angular momenta. (d) Curie temperature Tc and the thickness of unstrained QD2-t2 as functions of the thickness of strained QD1-i1 for different compositions x of strained In1-x GaxAs. Lateral size d = 120 Â. Barrier thickness tb = 20 Â. Reprinted with permission from [241], J. B. Khurgin et al., Appl. Phys. Lett. 73, 3944 (1998). ©1998, American Institute of Physics.

It has been proved that the Hubbard model on specific types of lattice having a flat band exhibits ferromagnetism in the presence of on-site Coulomb interaction [203,250-255]. The flat band is a dispersionless subband in the single-particle band structure. Figure 17 shows two examples of lattices having flat bands, where (a) is a Lieb lattice and (b) is a Kagome lattice. The Hubbard Hamiltonian for these lattices is

(i,k)a where t (t') is the transfer integral between the (next) nearest neighbor sites (i,j) ((i,k)), U is the on-site Coulomb energy, c+a(cia) is the creation (annihilation) operator of an electron with spin a =\ or \ on site i, and na = c+acia. The single-particle band diagrams calculated for tf = U = 0 are shown in Figure 18, where flat subbands can be seen. In the presence of on-site Coulomb repulsion, it has been mathematically proven that the Hubbard model exhibits flat-band ferromagnetism on the Lieb [250] and Kagome lattice [251] at zero temperatures. The mechanism of ferromagnetism in these lattices has already been explained in Figure 13. In the bipartite Lieb lattice, kinetic exchange interactions make the neighboring spins align in opposite directions (see Fig. 13a) and the different numbers of up- and down-spins result in ferrimagnetism. The ferromagnetism in the Kagome lattice is caused by a ring exchange interaction of the third-order process cycling two electrons with the opposite spins within the triangular lattice of three sites (see Fig. 13b). When the next nearest neighbor transfer t' is included, the flat band is broken as shown in Figure 18. It is very important to know how t' affects the flat-band ferromagnetism, because the next nearest neighbor transfer t' is not negligible in QDSLs [247]. To include the effect of t', we consider a realistic potential in QDSL. The transfer and on-site Coulomb energies for QDSL are evaluated by assuming that electrons are

Figure 17. The Lieb lattice (a) and Kagome lattice (b). Solid lines represent the nearest neighbor transfer t. Dotted and dashed lines indicate one and 2 x 2 unit cell(s). A unit cell contains three sites i = 1, 2, 3.

Figure 17. The Lieb lattice (a) and Kagome lattice (b). Solid lines represent the nearest neighbor transfer t. Dotted and dashed lines indicate one and 2 x 2 unit cell(s). A unit cell contains three sites i = 1, 2, 3.

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