Exciton spectra in vertical stacks of triple and quadruple quantum dots in an electric field

We study an electron-hole pair in a stack of multiple quantum dots in the presence of an external electric field using the configuration-interaction approach. We find that the bright energy levels can be grouped into families which are associated with the hole localized in a specific dot of the stack. The exciton energy levels undergo avoided crossings as function of the external electric field with different patterns for each family. We show that the variation of the depths of the dots along the stack can be deduced from the exciton spectrum. For stronger coupling the families are mixed by a weak electric field due to hole tunneling. This results in a characteristic multiple avoided crossing of energy levels belonging to different families with an accompanying modulation of the recombination probabilities and an appearance of a single particularly bright state. Discussion of a simple modeling of these avoided crossings is provided.

Figure 1

(Color online) (a) The confinement potential (cross section at $y=0$) for a stack of dots with interdot barrier of $4\mathrm{nm}$—the lighter the shade of gray, the deeper the potential. Dots have identical size with diameter of $20\mathrm{nm}$ and height of $4\mathrm{nm}$. The dots are numbered from the lowest to the uppermost. The arrow at left (right) shows the electric field force acting on the electron (hole) for $F>0$. (b) The confinement potential for the electron plotted along the axis of the stack $(x=0,y=0)$.

Figure 2

(Color online) Exciton energy spectrum for two identical dots separated by barrier thickness of $5\mathrm{nm}$. The thickness of the lines is proportional to the recombination probability [see Eq. (7)]. The bright energy levels are denoted by the number 1 or 2 which indicates the dot in which the hole is localized (see Fig. 1). The two arrows indicate the energy levels that avoided cross when the electron is removed of the dot in which the hole is localized.

Figure 3

(Color online) Exciton energy spectrum for three identical dots separated by barrier thicknesses of 4 (a) and $5\mathrm{nm}$ (b). The thickness of the lines is proportional to the recombination probability [Eq. (8)]. The bright energy levels are denoted by the numbers 1, 2, or 3 which indicates the dot in which the hole is localized (see Fig. 1). Letters $(a,b,c)$ mark the bright energy levels of the same family (with the hole localized in the same dot) that appear at $F⪡0$, $F\simeq 0$, and $F⪢0$, respectively.

Figure 4

(Color online) Cross section $(y=0)$ of electron [(a), (c), and (e)–(h)] and hole [(b) and (d)] densities in the states of the $2x$ family, in which the hole is localized in the central (2) dot for three identical dots with interdot barrier thickness of $4\mathrm{nm}$ for various values of the electric field $F$. The corresponding energy levels are plotted in Fig. 3a. Thin solid boxes indicate the region in which the confinement potential is at least 10% of its maximal value with respect to the center of each dot.

Figure 5

(Color online) Same as Fig. 4 but for the $3c$ energy level of Fig. 3. Plots [(a), (c), and (d)] show the electron density and (b) the hole density.

Figure 6

(Color online) Same as Fig. 4 but for the $1x$ family of states in which the hole is localized in the lowest dot. All the plots show the electron density. The upper [(a)–(c)], middle [(d)–(g)], and lower [(h) and (i)] rows of plots correspond to $1a$, $1b$, and $1c$ energy levels of Fig. 3. The columns of plots from left to right correspond to $F=1$, 10, 20, and $50\mathrm{kV}∕\mathrm{cm}$.

Figure 7

(Color online) Same as Fig. 3 but for three nonidentical dots separated by barrier thicknesses of 4 (a), 5 (b), and $6\mathrm{nm}$ (c). The depth of the confinement potential of the dots 1, 2, and 3 is varied by $+10$, 0, and $-10\mathrm{meV}$ (constant gradient of potential depth), respectively, for both the electron and the hole. Schematic drawing of the valence and conduction band along the stack is presented in the inset of (c).

Figure 8

(Color online) Same as Fig. 7c but for varied depths of the dot with barrier thickness of $t=6\mathrm{nm}$. The depth of the confinement potential of the dots 1, 2, and 3 is varied by 0, $+10$, and $-10\mathrm{meV}$, respectively, for both the electron and the hole. The inset shows a sketch of conduction and valence band extrema along the growth direction.

Figure 9

(Color online) Same as Fig. 7c but for varied depths of the dot with barrier thickness of $t=6\mathrm{nm}$. The depth of the confinement potential of the dots 1, 2, and 3 is varied by $+20$, 0, and $-20\mathrm{meV}$, respectively, for the hole. Dots have equal depth for the electron. The inset shows a sketch of conduction and valence band extrema along the growth direction.

Figure 10

(Color online) Spectra for the strong-coupling case with $t=1\mathrm{nm}$ (a) correspond to identical dots and in (c) the depth of the confinement potentials of the dots 1, 2, and 3 is modified by $+10$ meV, and $-10\mathrm{meV}$, respectively. (b) and (d) present fragments of (a) and (c), respectively, in which avoided crossings related to the avoided crossings between different families are observed.

Figure 11

(Color online) Cross section $(y=0)$ of the electron and hole densities for the energy levels participating in the two avoided crossings presented in Fig. 10b. The upper group of plots [(a)–(l)] corresponds to the avoided crossing marked by the upper rectangle in Fig. 10b and the lower group [(m)–(y)] to the ground-state avoided crossing marked by the lower rectangle in Fig. 10b. In each group—higher-energy states correspond to the upper panels. The first (second) column of plots shows the electron (hole) density at $F=0$ (namely, a,c,e,m,o,q for electron, and b,d,f,n,p,r for the hole). The third (fourth) the electron (hole) density at $F=10\mathrm{kV}∕\mathrm{cm}$ (g,i,k,s,u,x for electron, and h,j,l,t,v,y, for the hole). Identities of the states when their energy levels emerge from the avoided crossing can be deduced from the plots for $F=10\mathrm{kV}∕\mathrm{cm}$. The densities are therefore labeled according to the “family” notation used in previous discussion.

Figure 12

(Color online) Same as Fig. 3a but for four identical dots separated by identical barriers of thickness $4\mathrm{nm}$.

Figure 13

(Color online) Electron densities for the stack of four identical dots (for the energy levels presented in Fig. 12). The left column of plots [(a)–(e)] corresponds to $F=0.5\mathrm{kV}∕\mathrm{cm}$ and the right column [(f)–(j)] to $F=40\mathrm{kV}∕\mathrm{cm}$. In each case the hole is totally localized within the dot specified by the family label (additionally marked a dot).

Figure 14

(Color online) The spectrum for four stacked dots for barrier thickness of $5.5\mathrm{nm}$. In (a) a constant gradient of the confinement potential depths of the dots is assumed: (see the schematic drawing above the plot) the depth of each subsequent dot is decreased by $10\mathrm{meV}$ for both the electron and the hole when one moves up the stack. In (b) it is assumed that the shifts of the dot depth for both carriers are from the bottom to the top of the stack: $+10$, $+20$, $-10$, and $0\mathrm{meV}$.

Figure 15

(Color online) Spectra for four dots with barrier width of $1\mathrm{nm}$; (a) and (b) correspond to identical dots and in (c) the depth of the confinement potentials of the dots 1–4 is varied by $+10$, $+0$, $-10$, and $-20\mathrm{meV}$, respectively. (b) is a zoom of (a).

Figure 16

(Color online) Cross section $(y=0)$ of the electron and hole densities corresponding to the energy levels participating in the two avoided crossings marked by rectangles in Fig. 15b. The lower group of plots [the ones marked by (e)–(h) for $F=0$, and by the family labels $1a$, $2b$, $3c$, and $4d$ for $F=20\mathrm{kV}∕\mathrm{cm}$] corresponds to the ground-state avoided crossing marked by the lower rectangle in Fig. 15b. The upper group of plots [(a)–(d) and $1b$, $2c$, $3b$, and $4c$]) corresponds to the avoided crossing marked by the upper rectangle in Fig. 15b. In each group—the higher-energy states correspond to the upper panels. The first (second) column shows the electron (hole) density at $F=0$ and the third (fourth) the electron (hole) density at $F=20\mathrm{kV}∕\mathrm{cm}$. Identities of the states when their energy levels emerge from the avoided crossing can be deduced from the plots for $F=20\mathrm{kV}∕\mathrm{cm}$. The corresponding densities are therefore labeled in the family notation.

Figure 17

(Color online) Energy eigenvalues of the hole Hamiltonian (10) modeling the multiple avoided crossing between energy levels of different families. Black lines correspond to the hopping matrix element of $c=2.5\mathrm{meV}$, and red lines to the neglected hole tunneling $c=0$. In (a) all the dots are equivalent $(\delta =0)$, in (b) the two central dots are effectively deeper due to the interaction with the electron $(\delta =3\mathrm{meV})$, and in (c) the outer dots are effectively deeper $(\delta =-3\mathrm{meV})$. Plot (b) well corresponds to the lowest avoided crossing of Figs. 15b, 15c to the second avoided crossing of Fig. 15b.