Quantum dot and quantum well solar energy converters

Abstract

We review designs of a solar cell constructed from pn junctions, quantum dots and quantum wells. In the first instance we show that quantum wells of varying size embedded in the depletion region yields spatial variation of the energy gap that can be controlled. An advantage of the proposed structure is efficient utilization of the broad solar spectrum, lessening of lattice matching problems and generation of electron-hole pairs in narrow depletion regions which yields fast spatial separation of charges and, thus, reduces recombination losses. In another model we show how quantum coherence can be used, in principle, to eliminate radiative recombination and increase photocell power.

Appendices

Flow of electrons through depletion region

Electrons (holes) generated in quantum wells (dots) by photon absorption can propagate through the depletion region by quantum tunneling mechanism. Namely, electrons (holes) flow from one quantum well (dot) to another by tunneling through the potential barrier in the region between adjacent wells or dots (see Fig. 9). Next we estimate spacing between wells (dots) for which tunneling is efficient.

Fig. 9

Electron tunneling between quantum wells

To estimate electron tunneling rate we approximate spacing between wells (dots) as one dimensional potential barrier

$$\begin{aligned} V(x)=\left\{ \begin{array}{c} U,\qquad 0<x<a \\ 0,\qquad \text {otherwise} \end{array} \right. . \end{aligned}$$

(A1)

Tunneling probability for electron is given by

$$\begin{aligned} D=\frac{1}{1+\frac{U^{2}}{4\varepsilon \Delta }\sinh ^{2}\left( \frac{\sqrt{ 2m\Delta }a}{\hbar }\right) }, \end{aligned}$$

(A2)

where m is the electron mass, $\Delta =U-\varepsilon$ is the height of the potential barrier and $\varepsilon$ is the electron energy in the dot. For $a\gg \hbar /\sqrt{2m\Delta }$ Eq. (A2) yields

$$\begin{aligned} D\approx \frac{16\varepsilon \Delta }{U^{2}}\exp \left( -\frac{2\sqrt{ 2m\Delta }}{\hbar }a\right) . \end{aligned}$$

(A3)

One can think about the electron bouncing back and forth within the potential well of the dot with an average velocity $v=\sqrt{2\varepsilon /m}$ . The average time between its collisions with the “wall” of the potential is approximately 2l/v, so the frequency of collisions is v/2l, where l is the size of the quantum dot (well). The probability that the electron tunnels during any particular collision is D, so the probability of tunneling at any given time is vD/2l. Thus the tunneling time of electron from one dot to another is about

$$\begin{aligned} \tau =\frac{2l}{vD}\sim \tau _{\text {coll}}\cdot \frac{U^{2}}{16\varepsilon \Delta }\exp \left( \frac{2\sqrt{2m\Delta }}{\hbar }a\right) , \end{aligned}$$

(A4)

where

$$\begin{aligned} \tau _{\text {coll}}=2l\sqrt{\frac{m}{2\varepsilon }} \end{aligned}$$

(A5)

is the collision time of electron with the walls of the dot. For $l=8$ nm and $\varepsilon =0.1$ eV we obtain $\tau _{\text {coll}}=0.85\times 10^{-13}$ s.

The tunneling time $\tau$ should be much smaller than time of electron-hole recombination in the quantum dot

$$\begin{aligned} \tau \ll \tau _{\text {rec}} \end{aligned}$$

(A6)

which yields limitation

$$\begin{aligned} a\lesssim \frac{\hbar }{2\sqrt{2m\Delta }}\ln \left( \frac{\tau _{\text {rec}} }{\tau _{\text {coll}}}\frac{16\varepsilon \Delta }{U^{2}}\right) . \end{aligned}$$

(A7)

For $\varepsilon =0.1$ eV, $\Delta =0.9$ eV, $U=1$ eV, $\tau _{\text {coll} }=10^{-13}$ s and $\tau _{\text {rec}}=10^{-8}$ s Eq. (A7) yields

$$\begin{aligned} a\lesssim 1\text { nm.} \end{aligned}$$

(A8)

So, in order to have efficient tunneling the quantum wells (dots) must be very close to each other.

High precision quantum dot processing

Semiconductor quantum dot nanocrystals are prepared by a variety of techniques and are available commercially. For example, solutions of CdS quantum dots covering the spectrum from 360 nm to 460 nm and CdSe dots going from 460 nm to 650 nm can be purchased. For some quantum solar cells it may suffice to use these semiconductor quantum dots “as-is”. However, in the case we envision, it will be useful to have a finer control on the dot sizes.

We here present a simple and potentially inexpensive means of precision sorting of an ensemble of quantum dots. In Fig. 10 we show an inhomogeneous powder of quantum dots having a wide distribution of sizes. The dots are dropped in vacuum or other medium (e.g. inert gas) and subjected to a dot band gap sequence of (e.g. semiconductor diode) laser beams tuned to a particular quantum resonant frequency.

Now when a dot absorbs or emits a photon of frequency $\nu$ from a light beam, a transfer of recoil momentum $\Delta p=\hbar k=\hbar \nu /c$ takes place between the dot and the field. If absorption is followed by spontaneous emission, there is a net momentum transfer to the dot as the spontaneous emission goes in $4\pi$ steradians and gives no average contribution. Hence, as shown in Appendix C, the force on a dot due to this absorption and emission of laser photons is given by

$$\begin{aligned} F=\frac{1}{2}\Gamma \hbar k~, \end{aligned}$$

(B1)

where $\Gamma$ is the radiative decay rate and $\hbar k$ is the momentum of a photon of wave vector k.

Then, as is shown in the following example, a substantial deflect of $\approx$ 1 cm can be achieved with inexpensive diode lasers.

Fig. 10

Laser deflection of falling quantum dots according to optical resonant energy $\hbar \nu =\epsilon _{g}$

To estimate the deflection we note that the vertical position of a quantum dot falling in vacuum in the earth’s gravitational field is given by $\frac{1 }{2}gt^{2}$ and the deflection is then $x=\frac{1}{2}\frac{F}{m}t^{2}=\frac{F }{m}\frac{y}{g}$. Hence, using Eq. (B1) for F, we obtain

$$\begin{aligned} x=\frac{\Gamma \hbar k}{2mg}y. \end{aligned}$$

(B2)

The buffer gas molecules would exert a viscous force on the quantum dot proportional to the quantum dot velocity. This changes x(t) and y(t), but not the ratio x(t)/y(t). As a consequence, presence of the viscous force does not change Eq. (B2).

We may then calculate the deflection from Eq. (B2) by taking the reasonable case of a 10 nm dot of CdSe having mass $m\simeq 10^{-19}$ kg, $g\simeq 10\text { m}/\text {s}^{2}$, $\Gamma \sim 10^{8}\text { s}^{-1}$ and a 500 nm photon having momentum $\hbar k\simeq \tfrac{1}{2}10^{-27} \text { kg}\cdot \text {m}/\text {s}$ for which the ratio of radiation to gravitational force is of order $10^{-2}$. Hence we find a substantial 1 cm deflection when falling 1 m due to its interaction with an array of 100 milliwatt lasers each focused to $1\text { mm}^{2}$ for which the Rabi frequency $\Omega _{R}>\Gamma$. We note that this is well within the state of the art using inexpensive semiconductor diode lasers.

One should mention that non-destructive dispersion of quantum dots into a buffer gas has been experimentally demonstrated by Kumakura et al. [29], while size-separation of quantum dots by laser ablation has been demonstrated in liquid helium [30].

Radiative pressure on a quantum dot

Upon absorbing a laser photon a dot experiences a momentum recoil of $\delta p=\hbar k$ upon each event and the force on the dot F is given by

$$\begin{aligned} F=r\hbar k, \end{aligned}$$

(C1)

in which r is the rate of radiation decay given by

$$\begin{aligned} r=\Gamma \rho _{ee}, \end{aligned}$$

(C2)

where $\Gamma$ is the spontaneous emission rate from the excited state $|e\rangle$ to the ground state $|g\rangle$.

The interaction with a radiation field of frequency $\nu$ is described by the following set of equations for the density matrix of an effective two level atom describing the quantum dot exciton

$$\begin{aligned}&{\dot{\rho }}_{eg}=-\left( i\Delta +\frac{\Gamma }{2}\right) \rho _{eg}+i\Omega _{R}\rho _{ee}-\frac{i}{2}\Omega _{R}, \end{aligned}$$

(C3)

$$\begin{aligned}&{\dot{\rho }}_{ee}=-\Gamma \rho _{ee}+\frac{i\Omega _{R}}{2}(\rho _{eg}-\rho _{ge}), \end{aligned}$$

(C4)

$$\begin{aligned}&{\dot{\rho }}_{ge}=\left( i\Delta -\frac{\Gamma }{2}\right) \rho _{ge}-i\Omega _{R}\rho _{ee}+\frac{i}{2}\Omega _{R}, \end{aligned}$$

(C5)

where the detuning $\Delta =\omega -\nu$ and $\omega$ is the transition frequency. Here $\Omega _{R}$ is the Rabi frequency associated with the intensity of the light beam and the matrix element coupling states e and g. The steady-state solution of Eqs. (C3)-(C5) in the saturated limit $\Omega _{R}>>\Gamma$ is

$$\begin{aligned} \rho _{ee}=\frac{\Omega _{R}^{2}}{4\Delta ^{2}+\Gamma ^{2}+2\Omega _{R}^{2}} \rightarrow \frac{1}{2} \end{aligned}$$

(C6)

and from Eqs. (C1), (C2) and (C6) the absorptive force is found to be

$$\begin{aligned} F=\frac{1}{2}\hbar k\Gamma , \end{aligned}$$

(C7)

which is in the same direction as the deflecting laser.

Light absorption by quantum wells

Absorption and emission of light by quantum wells have been studied in details in connection with quantum well lasers. In particular, formulas obtained for the laser gain g (in cm$^{-1}$) can be used to estimate light absorption since absorption length $l_{\text {abs}}$ is related to gain as $l_{\text {abs}}=-1/g$. When gain is negative the incident light is being absorbed by the medium.

Here we estimate $l_{\text {abs}}$ using Eq. (35) on page 36 of Ref. [31] which gives a general answer for gain valid for bulk semiconductors and quantum wells. In terms of the absorption length this equation reads

$$\begin{aligned} \frac{1}{l_{\text {abs}}}=\frac{1}{\hbar \omega }\frac{\pi \hbar e^{2}}{ \varepsilon _{0}cm_{e}^{2}}\frac{{\bar{n}}_{g}}{{\bar{n}}^{2}}|M_{T}|^{2}\rho _{ \text {red}}(f_{v}-f_{c}), \end{aligned}$$

(D1)

where e is the electron charge, $m_{e}$ is the mass of bare electron, $\varepsilon _{0}$ is the permittivity of free space, c is the speed of light in vacuum, ${\bar{n}}$ is the index of refraction in the crystal, $\bar{n }_{g}$ is the group index of refraction

$$\begin{aligned} {\bar{n}}_{g}={\bar{n}}+\omega \frac{d{\bar{n}}}{d\omega }, \end{aligned}$$

(D2)

$M_{T}$ is the transition matrix element, $\rho _{\text {red}}$ is the reduced density of states at the energy $E=\hbar \omega -E_{g}$:

$$\begin{aligned} \frac{1}{\rho _{\text {red}}}=\frac{1}{\rho _{c}}+\frac{1}{\rho _{v}}, \end{aligned}$$

(D3)

$f_{c}$ and $f_{v}$ are Fermi-Dirac distribution factors for electrons in the conduction and valence bands

$$\begin{aligned}&f_{c}=\frac{1}{1+\exp \left( \frac{E_{e}-E_{f}}{k_{B}T}\right) }, \end{aligned}$$

(D4)

$$\begin{aligned}&f_{v}=\frac{1}{1+\exp \left( \frac{E_{h}-E_{f}}{k_{B}T}\right) }. \end{aligned}$$

(D5)

For parabolic conduction and valence bands the electron (hole) energy reads

$$\begin{aligned}&E_{e}=E_{c}+\frac{\hbar ^{2}k_{e}^{2}}{2m_{c}}, \end{aligned}$$

(D6)

$$\begin{aligned}&E_{h}=E_{v}-\frac{\hbar ^{2}k_{h}^{2}}{2m_{v}}, \end{aligned}$$

(D7)

where $E_{c}$ and $E_{v}$ are the band edge energies, $m_{c}$ and $m_{v}$ are the effective masses in the two bands and $k_{e}$, $k_{h}$ are the magnitudes of the wavevectors of a given electron or hole.

For parabolic conduction and valence bands the reduced density of states $\rho _{\text {red}}$ for a bulk 3D material is [31]

$$\begin{aligned} \rho _{\text {red}}=\left( \frac{2m_{\text {red}}}{\hbar ^{2}}\right) ^{3/2} \frac{\sqrt{E}}{4\pi ^{2}}, \end{aligned}$$

(D8)

where $m_{\text {red}}$ is the reduced mass

$$\begin{aligned} \frac{1}{m_{\text {red}}}=\frac{1}{m_{c}}+\frac{1}{m_{v}}, \end{aligned}$$

(D9)

while for a quantum well (2D)

$$\begin{aligned} \rho _{\text {red}}=\frac{m_{\text {red}}}{2\pi \hbar ^{2}}\frac{1}{L_{\text {QW }}}, \end{aligned}$$

(D10)

where $L_{\text {QW}}$ is the quantum well thickness.

To estimate the absorption length one can take $f_{v}\approx 1$, $f_{c}\approx 0$ and ${\bar{n}}_{g}\approx {\bar{n}}$. Then Eq. (D1) reduces to

$$\begin{aligned} \frac{1}{l_{\text {abs}}}=\frac{\alpha }{{\bar{n}}}\frac{|M_{T}|^{2}}{\hbar \omega m_{e}}\frac{4\pi ^{2}\hbar ^{2}\rho _{\text {red}}}{m_{e}}, \end{aligned}$$

(D11)

where $\alpha$ is the fine-structure constant

$$\begin{aligned} \alpha =\frac{e^{2}}{4\pi \varepsilon _{0}\hbar c}\approx \frac{1}{137}. \end{aligned}$$

Substituting $\rho _{\text {red}}$ for quantum well from Eq. (D10) we obtain

$$\begin{aligned} \frac{1}{l_{\text {abs}}}=\frac{2\pi \alpha }{{\bar{n}}}\frac{|M_{T}|^{2}}{ \hbar \omega m_{e}}\frac{m_{\text {red}}}{m_{e}}\frac{1}{L_{\text {QW}}}. \end{aligned}$$

(D12)

So that the number of wells necessary to absorb incident light is independent of well thickness and given by

$$\begin{aligned} \frac{l_{\text {abs}}}{L_{\text {QW}}}=\frac{{\bar{n}}}{2\pi \alpha }\frac{m_{e} }{m_{\text {red}}}\frac{\hbar \omega m_{e}}{|M_{T}|^{2}}. \end{aligned}$$

(D13)

Let us consider quantum wells made of GaAs. We approximate $|M_{T}|^{2}$ by its bulk value (see Table 2 on page 49 of Ref. [31])

$$\begin{aligned} \frac{2|M_{T}|^{2}}{m_{e}}\approx 29\text { eV,} \end{aligned}$$

(D14)

and take $m_{c}=0.067m_{e}$, $m_{v}=0.082m_{e}$ (that is $m_{\text {red} }=0.037m_{e}$). Then for $\hbar \omega =1.6$ eV photons (${\bar{n}}=3.7$) we obtain that the necessary number of GaAs wells is

$$\begin{aligned} \frac{l_{\text {abs}}}{L_{\text {QW}}}=240 \end{aligned}$$

(D15)

which for 5 nm thick wells gives the absorption length of $1.2\mu$m. Thus, the absorption length is of the order of the size of the depletion region.

For InP

$$\begin{aligned} \frac{2|M_{T}|^{2}}{m_{e}}\approx 20\text { eV,} \end{aligned}$$

(D16)

$m_{c}=0.077m_{e}$, $m_{v}=0.64m_{e}$ ($m_{\text {red}}=0.069m_{e}$). Then for $\hbar \omega =1.6$ eV photons (${\bar{n}}=3.7$) the necessary number of InP wells is

$$\begin{aligned} \frac{l_{\text {abs}}}{L_{\text {QW}}}=187 \end{aligned}$$

(D17)

which for 5 nm thick wells gives the absorption length of $0.9\mu$m.

Absorption of photons by a bulk semiconductor is different due to different value of the density of states. However, difference is not substantial. Indeed, Eqs. (D8) and (D10) give that the ratio of the density of states for a bulk material and a quantum well is given by

$$\begin{aligned} \frac{\rho _{\text {bulk}}}{\rho _{\text {QW}}}=\sqrt{\frac{\hbar \omega -E_{g} }{E_{\text {QW}}}}, \end{aligned}$$

(D18)

where

$$\begin{aligned} E_{\text {QW}}=\frac{\pi ^{2}\hbar ^{2}}{2m_{\text {red}}L_{\text {QW}}^{2}}. \end{aligned}$$

(D19)

For 5 nm thick well with $m_{\text {red}}=0.069m_{e}$ Eq. (D19) yields $E_{\text {QW}}=0.2$ eV. Therefore, e.g., for $\hbar \omega -E_{g}=0.5$ eV Eq. (D18) gives

$$\begin{aligned} \frac{\rho _{\text {bulk}}}{\rho _{\text {QW}}}=1.6, \end{aligned}$$

(D20)

that is absorption cross section of a bulk material is of the same order of magnitude as for quantum wells. The absorption length we estimated for GaAs (1.2 $\mu$m) and InP (0.9 $\mu$m) quantum wells of 5 nm thickness approximately agrees with those measured for the bulk materials near the absorption edge. One can say roughly that in order to absorb light by quantum wells the length of QW stack must match the absorption length of the bulk material.