According to our earlier estimations, the densities of photogenerated charges (

*n* and

*p* for electrons and holes, respectively) are much smaller than 1 per NC, while the density of mid-gap states (

*N*_{0}) is at least on the order of 1 because they form a percolated conductive network. Because

*p, n**N*_{0}, the relaxation of photogenerated carriers is dominated not by band-to-band recombination but trapping at the mid-gap levels. From charge neutrality, the density of non-equilibrium electrons in the MGB under photoexcitation is

*n*_{m}=

*p*−

*n*. Alternatively, the effect of photoexcitation on occupancy of MGB states can be characterized in terms of photogenerated holes with density

*p*_{m}=

*n*−

*p*. Under steady state, the trapping rates are equal to the photogeneration rate:

where

*α* and

*β* are electron- and hole-capture coefficients, respectively.

When MGB is completely empty (

*n*_{0}=0),

equations (1) and

(2) yield

*n*~

*G*/(α

*N*_{0}) and

*βp*^{2}~

*G*(1+

*βα*^{−1}*p*/

*N*_{0}); these expressions are obtained assuming that

*p, n**N*_{0}. As we show below, in our structures

*β**α*, therefore,

*p*~(

*G*/

*β*)

^{1/2}, and consequently,

*n* can be expressed as

*n*~

*p*(

*p*/

*N*_{0})(

*β*/

*α*). The latter relationship suggests that

*n**p*, and therefore,

*n*_{m}~

*p*. In this case, photoconduction is dominated by holes and the photoconductivity can be expressed as

*σ*_{ph,h}=

*eμ*_{h}*n*_{NC}*p*=

*eμ*_{h}*n*_{NC}(

*G*/

*β*)

^{1/2}, where

*μ*_{h} is hole mobility and

*n*_{NC} is the density of NCs in the film. As MGB gets filled, the contribution from holes to photoconduction decreases whereas the contribution from electrons increases until the latter reaches the value of

*σ*_{ph,e}=

*eμ*_{e}*n*_{NC}*n*=

*eμ*_{e}*n*_{NC}(

*G*/

*α*)

^{1/2} (

*μ*_{e} is electron mobility) when the MGB is completely full (

*n*_{0}=

*N*_{0}). The above two limits describe the situations of, respectively, large negative and flat-band gate biases, when the Fermi level is either below or above the MGB ().

Because of symmetry between the conduction and valence bands in PbS

28,

29,

*μ*_{e} and

*μ*_{h} in our films are likely similar (

*μ*_{h}~

*μ*_{e}=

*μ*). Therefore, if the electron- and hole-capture probabilities were the same, the filling of the MGB would not lead to changes in total photoconductivity,

*σ*_{ph}=

*σ*_{ph,e}+

*σ*_{ph,h}, as the decrease in

*σ*_{ph,h} would be compensated by the increase in

*σ*_{ph,e}. However, our experimental data ( and ) indicate a decrease in the photoconductivity by a factor (

*ξ*) of ca 20 to 50 as

*V*_{g} changes from large negative values to the flat-band voltage. This implies that the electrons provide a smaller contribution to photoconduction than holes and, further, suggests that

*α* is much greater than

*β*. Specifically, in the limits of empty and full MGB, the total photoconductivity can be expressed as

*σ*_{ph}(empty)~

*σ*_{ph,h}=

*eμn*_{NC}(

*G*/

*β*)

^{1/2} and

*σ*_{ph}(full)=

*e*μ*n*_{NC}(

*G*/α)

^{1/2}+

*σ*_{ph,h}(full), respectively, which yields the ratio

*ξ*=

*eμn*_{NC}(

*G*/

*β*)

^{1/2}/[

*e*μ

*n*_{NC}(

*G*/α)

^{1/2}+σ

_{ph,h}(full)]. From this expression, we obtain (

*α*/

*β*)

^{1/2}=

*ξ*[1+

*σ*_{ph,h}(full) (

*eμn*_{NC})

^{−1}(

*α*/

*G*)

^{−1/2}], which suggests that (

*α*/

*β*)

^{1/2}>

*ξ*. On the basis of the measured ratio of the photocurrents (), we can conclude that

*α* is at least 400 times greater than

*β*. This result indicates that band-edge electrons in our films are much shorter lived than holes (

*α* *β*) and therefore provide a smaller contribution to the photocurrent. This is probably because of close proximity of the MGB to the conduction band (~0.4 eV versus ~0.9 eV separation from the valence band), which makes trapping through multiphonon emission easier for electrons than for holes

31.

To quantify the relative contributions of band-edge electrons and holes to photoconduction, we solve

equations (1) and

(2) for the situation when the density of MGB equilibrium charges is much greater than the density of photogenerated carriers (

*n*_{0}*n*_{m} and

*N*_{0}–

*n*_{0}*n*_{m}), and hence, the dynamics of both electrons and holes are controlled by the equilibrium occupancy of the MGB. In this case, σ

_{ph} is

where

*u*=β/α

1 and

*x*=

*n*_{0}/

*N*_{0}.

Equation (3) indicates a linear dependence of σ

_{ph} on

*G* and it approximates the behaviour of σ

_{ph} between the limits of almost complete filling and almost complete depletion of the MGB. The first and the second terms in the right-hand side of

equation (3) correspond, respectively, to electron and hole contributions. In , we compare them for β/α=0.002 (black dashed and red dashed-and-dotted lines for holes and electrons, respectively). Interestingly, up to near-unity values of

*x*, the photoconductivity is dominated by valence-band holes and even for

*x*=0.95, the hole contribution to σ

_{ph} is more than 20 times greater than that of electrons. This is a direct result of a high capture probability for electrons that quickly relax from the band-edge states into the low-mobility MGB. The short electron lifetime also explains why optical transitions, where MGB electrons are promoted to the conduction band (M'

_{1S} and M'

_{X} transitions in ; red dashed arrows), are not prominent in the photocurrent. The dominant role of hole photoconduction in our devices helps also to rationalize the overall

*V*_{g}-dependent trends observed in the photocurrent spectra (), namely the increase in

*I*_{sd} when

*V*_{g} changes from positive to progressively more negative values. This behaviour reflects the change in the lifetime of photogenerated holes, which becomes progressively longer as MGB gets more depleted of electrons under increasing negative gate bias.

Assuming hole-dominated photoconductance, from

equations (1) and

(2), we can obtain the following expression for

*σ*_{ph}, which in addition to the regime described by

equation (3) also allows us to treat the transition to a completely depleted MGB, that is, the range of

*x* from 1 –

to 0 (

is a small quantity defined by the condition

*N*_{0}–

*n*_{0}*n*_{m} or

*n*_{m}/

*N*_{0}):

We further use this expression (the corresponding dependence is shown by the blue solid line in ) to model the measured

*V*_{g}-dependence of photocurrent for different excitation intensities (lines in ). In our modelling, we relate

*n*_{0} to

*N*_{0} and

*V*_{g} by

where

*k*_{B} is the Boltzmann constant,

*T* is the temperature, and

*E*_{F}=

*γ*(

*V*_{g}−

*V*_{0}) is the Fermi energy (

*γ* is the dimensionless constant and

*V*_{0} is the flat-band voltage).

Equation (5) correctly predicts that under the flat-band condition (

*V*_{g}=

*V*_{0}),

*n*_{0}=

*N*_{0}, which corresponds to a fully occupied MGB, and

*n*_{0} exponentially decreases for progressively more negative gate voltages. Using

equations (4) and

(5), we can accurately describe both the gate-voltage and the illumination-intensity dependence of the measured photocurrent (; compare calculations shown by lines with measurements shown by symbols). This confirms the validity of our model and provides an extra piece of evidence that photoconduction in our structures is dominated by valence-band holes.

An interesting prediction of

equation (4) is that in addition to the magnitude of photoconductivity, the type of light-intensity-dependence of photoconduction can also be controlled by

*n*_{0}, and, hence, gate voltage. For example, in the regime of a depleted MGB, when

*n*_{0}2(

*G*/

*β*)

^{1/2}, which realized in our devices for large negative

*V*_{g}, the band-edge electrons quickly relax into the mid-gap states and hole dynamics are dominated by bimolecular recombination with these non-equilibrium MGB electrons. This results in a square-root dependence of

*σ*_{ph} on

*G*:

*σ*_{ph,h}=

*eμ*_{h}*n*_{NC}(

*G*/

*β*)

^{1/2}. However, as MGB becomes occupied with even a moderate number of electrons (

*n*_{0}2(

*G*/

*β*)

^{1/2}), the hole dynamics are dominated by recombination with 'pre-existing' MGB electrons, which leads to a linear dependence of

*σ*_{ph} on

*G*:

*σ*_{ph,h}=

*eμ*_{h}*n*_{NC}*G*/(

*βn*_{0}).

Our data indeed indicate an approximately square-root scaling of the photocurrent with

*G* for large negative

*V*_{g} (; open red circles). We also observe that the

*σ*_{ph} versus

*G* dependence becomes progressively steeper as

*V*_{g} gets less negative, and the number of MGB electrons increases (compare data shown by solid black squares and open green triangles in ; see also

Supplementary Fig. S2 for a different device, which shows a similar change in the scaling of photocurrent with light intensity). This behaviour is consistent with the predicted transition to the linear scaling described by

equation (4).

The demonstrated control of recombination dynamics by gate voltage is useful in photodetection, where it can be employed for controlling both device sensitivity and its dynamic range. A significant increase in device sensitivity can be obtained by combining contributions from photoconductive and photovoltaic effects. This is illustrated in (inset) where we compare a purely photoconductive response measured at *V*_{g}=*V*_{gmin} (the photocurrent minimum) with one obtained for *V*_{g}=0; the latter signal exhibits a much faster growth with *G* due to a photovoltaic contribution. The enhancement in sensitivity arising from the photovoltaic effect is also pronounced in the spectral dependence of a photoresponse (main panel of ; compare traces shown by solid and dashed lines).

The observed difference in hole and electron relaxation times results in a peculiar, mixed character of photoconduction, when holes are transported through quantized states whereas electrons through the MGB. One implication of this transport mechanism is that the photovoltage, which can be derived from such a structure, is determined by the energy difference between the valence-band edge and the MGB. As a result, while displaying correlation with the band-gap energy it should be always lower than the ideal

*V*_{oc} calculated for a given

*E*_{g}. This trend has indeed been observed in NC solar cells

13,

14,

15,

17,

27,

32,

33 (

Supplementary Fig. S3).

To summarize, we have investigated OFETs fabricated from PbS NCs. Spectrally resolved studies of

*V*_{g}-dependent photocurrent reveal the existence of mid-gap states in NCs films that form a weakly conducting MGB. This band serves as a charge conduit in dark, however, the mechanism of charge transport changes under illumination when it becomes dominated by a more conductive network of quantized band-edge states. In this case, the MGB still has an important role as its occupancy controls photoconduction by controlling recombination dynamics of band-edge charges. The mechanisms for 'dark' and 'light' transport are schematically illustrated in . The results of our studies together with the developed model should help one to predictably engineer both charge transport properties of NC films in dark as well as under illumination. Interesting opportunities could also be opened by engineered placement of MGB states within the NC energy gap. This could allow, for example, switching between electron- and hole-dominated photoconductivity and, perhaps, help in practical realization of advanced photovoltaic concepts such as intermediate-band solar cells

34.