Title: Colloidal Quantum Dots Intraband Photodetectors Authors: Zhiyou Deng, Kwang Seob Jeong, and Philippe Guyot-Sionnest* Supporting Information: I. Considerations on the optimal detectivity of interband or intraband CQDs. The specific detectivity is defined as (1) where R is the responsivity, A is the area, f is the bandwidth and I n is the current noise (A). The responsivity is the number of carriers per second collected per Watt of input power. In a device with a unity gain, (2) where is the quantum efficiency for charge separation, e is the electron charge, is the frequency of light detected and h is Planck s constant. Increasing the gain such as in phototransistors increases the responsivity but the detectivity is not improved since the noise increases by the same gain. The noise is given by many factors, including Johnson noise, 1/f noise, shot noise and thermal background noise. In the shot noise limit, the noise arises from the constant production and recombination of thermal carriers. The rate of thermal carriers produced in a device of area A and thickness 1/ where is the absorption coefficient is 1 (3) where is the carrier lifetime. A thicker device introduces unnecessary noise without increasing the photocurrent. The shot noise associated with G th is then and D* is given by 1 (4) The background limited performance (BLIP) is achieved when the incident background flux is larger than the thermal carrier generation rate, in which case. The 77K cooling requirements for bulk MCT detectors is due to the increase of the carrier density and the dominance of Auger process at higher temperatures. 2 Carrier Lifetime in QDs films: Assuming that there are no traps, carriers that are in the QD films can recombine by geminate recombination of an electron and a hole in a QD in a time 1 or by Auger recombination if there are three or more carriers in the QD. The dots with three carriers have an Auger recombination lifetime of A0. The number of QDs per unit volume is N. For intrinsic interband CQDs and doped (n=2) intraband CQDs, the average number of electrons and holes per dot is such that the carrier geminate recombination lifetime is, and the carrier Auger lifetime is If, the geminate recombination dominates Auger recombination. The Auger lifetime has not yet been measured in the doped HgSe CQDs but it could possibly be much longer than for interband CQDs given the much sparser density of states. Detectivity in the radiative lifetime limit for Interband and Intraband CQDs. In the radiative limit, the lifetime and absorption depth are related through Einstein s A and B coefficients, where a fast radiative lifetime implies a short absorption depth, therefore Eq.4 may be simplified further since the product becomes material independent. The absorption depth is, where V is the volume occupied per nanocrystal and is the peak cross section. The
integrated cross section is so that where is the absorption bandwidth and is the wavelength. Then using and, (4) the maximum detectivity becomes (5) In the intrinsic limit, where N 1 and N 2 are the states degeneracy at the lower and higher level. Equation (5) shows that differences between interband and intraband detectivity with quantum dots are going to be small since the density of states of the ground and excited states will be rather similar. With non-radiative processes, the detectivity is lower. Using,. II. k.p Model The 2-band k.p model and an infinite potential well are used to get the energy vs size shown in Fig. S7. 5 The oscillator strength is calculated as, where m 0 is the free electron mass, is the intraband angular frequency, and Z 1S1P is the matrix element taken as ~ <1S z 1P z > 4 where 1S and 1P z are the envelope functions of the particle in the spherical box. Z 1S1P ~ 0.306R where R is the nanoparticle radius. The oscillator strength is shown in Fig.S7. The radiative lifetime is estimated as (CGS units), where here n is now the medium index of refraction (n~ 1.5 for C 2 Cl 4 ) and L is the local field factor where and for C 2 Cl 4. For a 6 nm diameter HgSe dot, the intraband energy is 0.30 ev, (2450 cm -1 ) the oscillator strength is 6.7, and the radiative lifetime is 640 ns in C 2 Cl 4.
Figure S1. TEM image and size analysis of the 1 min sample. Figure S2. TEM image and size analysis of the 4 min sample. Figure S3. TEM image and size analysis of the 16 min sample.
Absorption Difference (O.D.) Responsivity (ma/w) Dark current ( A) 4 30 20 Figure S4. Responsivity and dark current for a HgSe CQD film under higher bias at 80K. 2 10 0 0 25 50 75 0 100 Bias (V) Figure S5. Absorption and photocurrent of a HgS CQD film. The absorption (red line) is taken on a ZnSe ATR window at 300K and exhibits the strong intraband absorption. The Photocurrent (PC) (blue line) is taken at 80K. The HgS samples also exhibit a blue shift of the 80K photocurrent compared to the 300 K absorption. The HgS sample is made by reacting. HgCl 2 and thioacetamide in Oleylamine at 35 C for 10 min. 0.3 0.2 0.1 0.0-0.1-0.2 A -2.2 V 1.0 V - 2.2 V 2000 4000 6000 8000 10000 12000 Wavenumber(cm -1 ) 1.0 V 0.6 V 0.2 V - 0.6 V - 1.8 V - 2.2 V Figure S6. Difference absorption spectra of a HgSe CQD film on an evaporated gold slide under electrochemical potential, vs a Ag wire pseudoreference. At increasingly reducing potentials, both the mid-ir intraband absorption and the near-ir interband bleach increase. The sample is pressed against a CaF 2 window to minimize solvent absorption. At the more negative potential, the 1Pe state is charged as evidenced by the interband bleach around 8000 cm -1.
Figure S7. Two-band k.p results for the 1S e 1P e transition energy and the oscillator strength. 5 The Kane parameter is E p =18 ev and the bulk gap is chosen to be either -0.1 ev ( red lines, ~ room temperature) or -0.3 ev (blue lines, ~low temperature. E 1Se1Pe (solid lines) and E 1Se (dashed lines) are shown for the two values of the negative gap showing little effect on E 1Se1Pe. The model predicts an intraband transition energy of 0.31 ev (~ 2500 cm -1 ) for a spherical particle of 6 nm diameter, in good agreement with the experimental result. The temperature dependence of E 1Se1Pe is however not explained by this simple model since the change of the gap has apparently little effect. The oscillator strength of the 1S e 1P e transition (black solid line) is shown as well and it is comparable to that of the interband transition (black dashed line). 4 (1) Philipps, J. Evaluation of the Fundamental Properties of Quantum Dot Infrared Detectors, J. Appl. Phys. 2002. 91, 4590-4594. (2) Chang, Y.; Grein, C. H.; Zhao, J.; Sivananthan, S.; Flatte, M. E.; Liao, P. K.; Aqariden, F. Carrier Recombination Lifetime Characterization of MBE-Grown HgCdTe, Appl. Phys. Lett. 2008, 93, 192111. (3) Robel, I.; Gresback, R.; Kortshagen, U.; Klimov, V. I. Universal Size-Dependent Trend in Auger Recombination in Direct-Gap and Indirect-Gap Semiconductor Nanocrystals, Phys Rev Lett. 2009, 102, 177404 (4) Khurgin, J. Comparative Analysis of the Intersubband versus Band-to-Band Transitions in Quantum Wells, Appl. Phys. Lett. 1993, 62, 1390 (5) Lhuillier, E.; Keuleyan, S.; Guyot-Sionnest, P. Optical Properties of HgTe Colloidal Quantum Dots. Nanotechnology 2012, 23, 175705.