Condensed-matter physics

The dance of electrons and holes

How many pairs of electrons and 'missing electrons' can sustain collective motion in a semiconductor? The limits of this electron–hole dance are found by probing the dance floor using ultrashort laser pulses.

When light is absorbed by a semiconductor, its energy causes an electron to jump from the material's filled valence energy band to the empty conduction band, leaving behind a 'hole' of opposite electric charge to that of the electron. In a solar cell, these fundamental charge carriers — the electron and hole — move completely independently and are swept towards opposite electrodes to produce electricity. But in semiconductors at low temperature (about 10 kelvin), the electron and hole experience a weak Coulombic attraction to each other, and become bound into a stable entity known as an exciton1. Rather than moving independently, the motions of the electron and hole are now correlated — just as an electron orbits a proton in the Bohr model of a hydrogen atom.

On page 1089 of this issue, Turner and Nelson2 describe experiments that provide insights into how electrons and holes interact at low temperature in 'quantum wells' of the semiconductor gallium arsenide (GaAs). Understanding the way carriers interact is not only important for testing fundamental theories of many-body physics, but also has major practical implications for physics and chemistry.

Returning to fundamentals, what happens if a second electron–hole pair is added to the semiconductor? Just as two hydrogen atoms bond to form an H2 molecule, the two excitons bind weakly to form a species called a biexciton. In a biexciton, all four carriers are executing correlated motions, presenting a challenging problem for theory. What particularly interests researchers about these motions are non-classical phenomena3. First, electrons and holes are fermions, that is, they possess half-integer spin, and certain combinations of these spin configurations mix quantum mechanically to form the system's many-particle wavefunction. The way in which these spins are paired among the particles can help electrons to avoid each other by means of the Pauli exclusion principle; but quantum mechanics guarantees that the Coulomb repulsion between the electrons cannot become infinite. Second, the wavefunction describing the collective motions of several electrons and holes reflects the fact that these particles do not fly along classical trajectories, but instead follow several alternative routes simultaneously. Like waves, the amplitudes of those alternatives sum to determine the most probable motions.

Although excitons and biexcitons in GaAs have been investigated in some detail, there remains the question of how many excitons can interact coherently such that the motions of all the electrons and holes are correlated. Turner and Nelson2 show that, in this material, sets of three electron–hole pairs dance in step, but sets of four pairs do not. Addressing such a seemingly innocent question required significant developments in a technique known as coherent nonlinear optical spectroscopy. This form of spectroscopy has long held promise for providing a systematic way to explore a hierarchy of electron–hole systems and their correlations, because the 'order' of the spectroscopy — the number of times an incident laser pulse strikes the sample — can be used to control the number of electron–hole pairs3,4. More recently, it has been realized that two-dimensional photon-echo techniques can determine the energy required to form an exciton compared with that required to form a biexciton, thereby directly revealing the energy that binds excitons to form biexcitons5,6. Turner and Nelson combine these strategies, but then extend them to an heroic seventh-order spectroscopic level — that is, their femtosecond laser (1 femtosecond = 10−15 seconds) interacts seven times with the sample before a signal is radiated.

Because of the correlations between electrons and holes, photoexcitation results in the formation of collective electronic states that, in turn, contribute significantly to phenomena such as nonlinear optical properties, optical gain in laser media, the dynamics of highly excited states and the production of 'entangled' photons. Although the authors' experiments2 indicate the presence of correlations (Fig. 1), do they reveal how the details or implications of such correlations relate to these kinds of phenomena? It seems not.

Figure 1: The apparatus of Turner and Nelson's experiments2.
figure1

A single laser beam that produces pulses of femtosecond duration is split into a pattern of multiple laser beams by a beam shaper. The timings between the femtosecond pulses in these beams are precisely tuned by using a pulse shaper. The beams are then focused on the sample, which is made of gallium arsenide semiconducting material, through a lens system. As a result of the way the laser pulses drive electronic transitions in the sample, a signal beam is produced. This beam is detected by a specially constructed spectrometer. From their analysis of the resulting spectrum, Turner and Nelson find that sets of three electron–hole pairs in the semiconductor display collective behaviour, whereas sets of four pairs do not.

The energy shifts recorded in the experiments (the binding energies of biexcitons and triexcitons) arise from a combination of two major effects. The first, which probably dominates, is explained by theories such as the Hartree–Fock method. These theories evaluate the average electron–hole attractions, electron–electron (hole–hole) repulsions and quantum-mechanical exchange corrections to these Coulomb interactions by assuming that the charges are spread out in space according to the probability that the particle could be found there.

The second effect, called electron correlation in the quantum-chemical literature7, describes the fascinating way in which electrons and holes tend to coordinate their motions to minimize repulsions. For example, two electrons might move in such a fashion that they avoid crossing paths. The complexity of this problem scales steeply with the number of particles involved. Although current investigations into many-body effects provide an important first step to understanding how groups of carriers interact collectively, they cannot quantify the average Coulomb repulsions and attractions relative to how these are modified by correlations in multiparticle motions.

There are many reasons to investigate how electrons (and holes) interact8. For instance, chemical structure–property relationships that govern the nonlinear response of organic molecules to light depend on the energy of multiexciton states relative to singly excited states9. The factors dictating electron–hole binding in organic solar cells are also under intense investigation now. The challenge in this area is that electrons and holes in organic materials interact much more strongly than those in semiconductors such as GaAs or silicon because of the low dielectric constant of organic materials. Dissociating the carriers efficiently to produce electricity is therefore a much harder task in organic solar cells than in conventional silicon ones.

Predicting chemical reactivity is another salient example. Mean-field models in chemistry — that is, theories based on molecular-orbital energies, symmetries and shapes — have been enormously successful in predicting electronic structure, explaining reactivity and qualitatively describing spectroscopy. To a significant extent, for example, our understanding of chemical reactivity is guided by Pauling's electronegativity scale; electrons move from electron-rich to electron-poor regions of reactants. However, the correlated motions of electrons in molecules are significant. Can we use this correlated response of electrons to promote efficient, concerted chemical change? Elucidating these kinds of quantum-mechanical details in condensed-matter physics would be transformative and have far-reaching consequences.

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Scholes, G. The dance of electrons and holes. Nature 466, 1047–1048 (2010). https://doi.org/10.1038/4661047a

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