Supporting Information THz conductivity within colloidal CsPbBr3 ...
Supporting Information THz conductivity within colloidal CsPbBr3 perovskite nanocrystals: remarkably high carrier mobilities and large diffusion lengths Gurivi Reddy Yettapu,§ Debnath Talukdar,‡ Sohini Sarkar,§ Abhishek Swarnkar,§ Angshuman Nag*,§, Prasenjit Ghosh, *,‡§ and Pankaj Mandal*,§ §
Department of Chemistry, Indian Institute of Science Education and Research (IISER), Pune, India, 411008
‡
Department of Physics, Indian Institute of Science Education and Research (IISER), Pune, India, 411008
*To whom correspondence should be addressed: [email protected] (PM), [email protected] (PG), [email protected] (AN).
1. Synthesis and characterization of colloidal CsPbBr3 nanocrystals: 1.A Preparation of Cs-oleate: Mixture of Cesium carbonate (0.4 g), Oleic acid (1.25 mL) and Octadecene (20 mL) were loaded into a 50 mL three neck flask and dried under vacuum for 1hr at 120oC. Then the temperature was increased to 150oC and kept until the solution became clear. 1.B. Colloidal CsPbBr3 NCs were synthesized following the procedure reported by Kovalenko et al.1 In brief, mixture of PbBr2 (0.188 mmol) with 5 mL Octadecene was dried under vacuum for 1 hour. Then anhydrous Oleylamine (0.5mL) and Oleic acid (0.5mL) were added to the mixture at 120oC. After the solution became clear, temperature was increased to 190oC and pre-heated Cs-oleate solution (0.1M, 0.4 mL) was swiftly injected to the reaction mixture. The reaction was stopped after 5 sec and cooled down to room temperature in icewater bath. The as synthesized CsPbBr3 NCs were precipitated by adding n-butanol and centrifuging at 10000 rpm for 10 min. It was then re-dissolved in toluene to form a long-term colloidal stable solution. 1
1.C Characterization: UV-visible absorption spectrum was recorded using a Perkin Elmer, Lambda- 45 UV/Vis spectrometer. Steady state photoluminescence (PL) of NCs was measured using FLS 980 (Edinburgh Instruments). Powder x-ray diffraction (XRD) data were recorded using a Bruker D8 Advance x-ray diffractometer using Cu Kα radiation (1.54 Å). Transmission electron microscopy (TEM) studies were carried out using a JEOL JEM 2100 F field emission transmission electron microscope at 200 kV. The sample preparation for TEM was done by putting a drop of the colloidal solution of NCs in hexane on the carbon coated copper grids.
CsPbBr3 NC's
Absorbance
0.4 0.3 0.2 0.1 0.0 350
400
450
500
550
PL Intensity (a.u.)
0.5
600
Wavelength (nm)
Figure S1: UV-visible absorption and photoluminescence (PL) spectra of CsPbBr3 NCs.
2
2. THz time domain spectroscopy (THz-TDS) and Optical Pump-THz Probe (OPTP) Spectroscopy
Figure S2: Experimental setup for Time Resolved THz spectroscopy. THz radiation is generated by air plasma and detected by air biased coherent detection (ABCD) method. The schematic of our experimental setup to generate2 THz from air plasma and detect it with ABCD method3, 4 is shown in figure S2. A regenerative amplifier (Spitfire Pro XP, Spectra Physics) seeded by oscillator (Tsunami, Spectra Physics) generates pulses of 50 fs (FWHM), 800nm central wavelength at 1 kHz repetition rate. The laser beam is split (BS 1) into two parts: one part is utilised to generate and detect THz radiation and another part is sent through the optical parametric amplifier (OPA) to generate the optical pump pulses of varying wavelengths. 2.A. THz Time Domain Measurement The first part is again split (BS 2) into two beams. The transmitted beam ( the BBO crystal that creates second harmonic (
; and on focusing (
passes through and (
in air,
plasma is generated. The plasma gives THz radiation along with other radiations. The THz radiation, filtered out by Si window, is collimated and focused to a spot size of ~1 mm
3
diameter on to the sample with the first pair of parabolic mirrors. The THz radiation transmitted through the sample is re-collimated and focused between two electrodes by another set of parabolic mirrors. The reflected beam from the BS2, called probe or gate beam which is reflected further by the retro reflector kept on a delay line (50 mm) is also focused between the electrodes. An AC voltage of 1.5 kV from high voltage modulator (HVM) is applied across the electrodes. The modulation frequency of HVM is kept at half (500 Hz) the laser repetition rate. The bias field acts as local oscillator and together with THz and fundamental beam generate second harmonic of the probe beam. The second harmonic signal is filtered from residual fundamental beam with band pass filter and detected by a photomultiplier tube (PMT). The PMT current is amplified using the current preamplifier and fed into the lock-in amplifier. THz electric field as a function of time delay between the gate pulse and THz pulse, is recorded by scanning the optical delay stage. 2.B. THz Time Resolved Measurement For time resolved THz study, the sample is excited by pump beam generated from the OPA. The delay between optical pump and THz probe beam is varied by scanning the 300 mm length stage marked as Delay 3 in figure S2. For controlling the pump power, variable neutral density filter is placed in the pump path. The pump beam is chopped using a chopper rotating at a frequency of 333 Hz. The pump beam passes through a hole in the off axis parabolic mirror and becomes collinear with the THz beam. The pump beam diameter on to the sample is ~ 3.5 mm, 3.5 times greater than the THz beam diameter to guarantee uniform sample excitation. The optical pump and THz probe diameters were measured by knife-edge technique. A black polyethylene sheet was used to block the pump light transmitted through the sample from reaching the PMT. The pump probe experiment is carried out using double lock in technique in which the pump induced change in THz transmission (-ΔE(tp)) and the corresponding THz transmission through the non-photoexcited NCs (E0(tp)) are recorded simultaneously.5 The signal from current preamplifier is split and sent to two separate lock in amplifiers. The THz probe beam is modulated at 500 Hz and this frequency acts as reference for one lock in amplifier. The reference frequency for the second lock in amplifier is 333 Hz, the frequency at which the optical pump beam is modulated. To minimise the crosstalk between the signals measured at 500 Hz and 333 Hz, the frequencies are not harmonic with respect to each other.
4
The pump induced signal can be collected in two ways.6 a) Frequency averaged experiments are performed by varying the delay between the optical pump and THz probe by scanning the 300 mm length stage after fixing the 50 mm stage at THz peak maximum (tmax). This gives the temporal evolution of the sample after photoexcitation by collecting the alteration in THz field amplitude, pulses,
, as a function of delay between THz probe and optical pump
. b) Frequency resolved experiments which are performed by scanning the 50 mm
stage at a fixed pump probe delay, yield the pump induced changes in THz waveform, , at a fixed pump probe delay (
.
To avoid THz absorption by water vapour, the entire experimental setup is enclosed in a transparent box and purged continuously with N2 or dry air. 2.C. Sample Preparation for THz Experiments Colloidal CsPbBr3 NC’s are dissolved in HMN solvent, which is quite transparent in broad band THz region. The solution is then injected into the demountable liquid cell (Harrick Scientific Products, DLC-M25), in which two TPX windows are placed separated by a Teflon spacer of 950 um thickness. Path length of the sample is equal to the spacer thickness.
3. THz TDS Data Analysis: The THz electric field passing through the sample and reference is given by Esample(t) and Eref(t) respectively. Fourier transform of these electric fields gives power (P) and phase (Φ) which are related to the refractive index and absorption coefficients and is given by
The complex refractive index can be written as to complex valued permittivity as
, where
, and it is related
The real and imaginary parts of the
permittivity are given by
5
The intrinsic frequency dependent complex dielectric function of the NCs was calculated using a simple effective medium approach (EMA) for dielectric inclusion embedded in a host medium: 7 (S5) where ɛeff(ν), ɛi(ν) and ɛh(ν) are the complex dielectric functions of the composite (NC solution, experimentally determined from THz-TDS), the inclusion (NCs) and the host medium (HMN) respectively; and f is the volume fraction of the NCs in the solution. The simple linear effective medium approach (EMA) has been adopted in the present work for the following reasons: (1) the size of the inclusions (nanocrystals, ~11 nm) is significantly less than the wavelength of the probe light (0.5 to 7 THz, 43 to 60 micron), (2) the volume fraction being very small (~10-3) makes Bruggeman method inapplicable, this being used for very large volume fraction where inclusions come together and form domains; moreover linear EMA seems to be working reasonably well for such dilute solution,7 (3) the NCs of this study have a cubic/orthorhombic geometry and the geometry factor for cubic/orthorhombic structure required for Maxwell-Garnet method is not known and determining the same is beyond the scope of the present work, and (4) the linear EMA yields real dielectric values which are in between the static and high-frequency limit of dielectric constants predicted in our GGA+vdW level of theory.
Figure S3: (A) THz absorption and (B) real refractive indices of HMN solvent and the CsPbBr3 perovskite NC solution.
6
Figure S4: (A) Real and (B) imaginary components of dielectric function of HMN and CsPbBr3 perovskite NC solution.
4. Recombination Dynamics via optical-pump THz-probe studies Charge carriers (electrons & holes) are generated by optical-pump pulses whose energies exceed the band gap (~2.5 eV) of the NCs. Here, we have used optical-pump pulses of wavelengths 504nm (energy equivalent to band gap), 480 nm and 400 nm (energy more than band gap). Optical pulses at 400nm were generated by passing the fundamental beam (800 nm) through BBO crystal and other two pulses were generated through optical parametric amplifier (OPA). The optical chopper was placed in pump beam path to modulate optical pump pulses. Carrier dynamics was monitored by measuring the change in THz field amplitude
as a function of pump-probe delay ( ). Note that
is
proportional to change in conductivity, which in turn depends on charge carriers. We have fitted the normalized photoconductivity decays to a multi-exponential function convoluted with a Gaussian function of the form: 8 (S6) where
is a Gaussian function centered at t0 with FWHM of ~300 fs which
represents the instrument response time of our OPTP experiment, and ai is the coefficient of the ith exponential decay channel with time constant τi.
7
5. Determination
of
transient photoconductivity from THz
electric field
transmission at different time delays The transient photoconductivity Δσ of NCs dissolved in HMN solvent placed between the is expressed as 6
two TPX windows, under the condition that
Where
is the refractive index of the NCs solution and
is given by
Here, l is the path length of the sample, n1 is the refractive index of the TPX window (n1= 1.524), n2 is the refractive index of the NCs solution (n2=1.61), MR is multiple reflection term (here MR= 0) and
are pump-induced and pump-off Fourier transforms of
transmitted THz electric fields. The dielectric constant of the pump induced NCs solution obtained from the conductivity (calculated from eq. S7) is
The effective dielectric constant of the photoexcited NCs can be calculated from the effective dielectric constant of the non photoexcited NCs and pump induced change in dielectric constant of NCs solution using the following equation:
After calculating
using eq. S10 and using this value in eq. S5 (effective medium
theory), the intrinsic dielectric constant of the photoexcited NCs can be calculated ( difference between
and
. The
(intrinsic dielectric constant of the non-photoexcited NCs)
is given by,
which is the pump induced dielectric constant of NCs. Using conductivity
of NCs can be calculated from eq. S9.
8
, the pump induced
In the above calculation for THz conductivity of NCs it is assumed that the entire sample (thickness, d) gets photoexcited. But true THz conductivity should be calculated for the thickness (skin depth ) till which the sample gets excited. So it must be corrected by a factor
if d> . The photoexcited sample thickness d for 504 nm and 480 nm pump was
taken to be same as the sample cell thickness (950 µm) as the penetration depth (1/OD) of the pump light at these two wavelengths are longer than the sample cell thickness. However, for 400 nm pump the photoexcited sample thickness d was considered to be equal to the penetration depth (δ; 546 µm), which is significantly shorter than the sample cell thickness (950 µm). Thus the real photo induced THz conductivity of NCs is given by,9
For 480 and 504 nm pump beam, sample thickness are more than the corresponding skin depths, so multiplication of the conductivity with the scaling factor
is not required. In
case of 400 nm, the skin depth is less than the sample thickness, so conductivity is multiplied with 1.74 (
.
6. Carrier density and carrier mobility by fitting Drude-Smith model to THz conductivity obtained from THz scans We have collected the pump induced change in THz electric field, ΔE (t, tp), and pump-off THz electric field, E0 (t, tp), at different pump-probe delays (tp). Fourier transform of the electric fields allowed us to calculate the change in THz conductivity of photo excited NCs at different pump-probe delays (tp). The THz conductivities can be well fitted by Drude-Smith model10 and is given by
where
is plasma frequency. It is related to carrier density through being the average effective mass of the carriers,
related to mobility as
and
with is scattering rate
represents the fraction of electrons having the
original velocity after first collision. The obtained mobilities are attributed to both electrons and holes, since THz radiation cannot distinguish charge carrier type.
9
7. Calculation of effective mass, exciton binding energy and Bohr exciton radius: Based on the assumption that in this material the excitons are Wannier type excitons, according to the effective mass theory the Bohr exciton radius is given by: ,
(S14)
and the exciton binding energy is given by: (S15) In equation S14 and S15,
is the dielectric constant,
is the radius of H atom and
reduced mass of the exciton. The reduced mass is given by
,
is the
being the
effective masses of electron and hole respectively. Since the photoconductivity measured is obtained from averaging along x, y and z direction, we consider the conductivity effective masses
. The effective masses of electrons and holes along
the [100], [010] and [001] directions are determined by fitting the band structure (within kBT of band edges) at the valence band maximum (VBM) and conduction band minimum (CBM) at Γ point of the Brillouin zone. Determining the correct values of Bohr exciton radius (equation S14) and exciton binding energy (equation S15) are challenging because the dielectric constant variation with frequency and temperature. The correct choice of
shows large
depends on the exciton
binding energy (which itself is unknown) and the energy of the optical phonons.11 Hence we have given estimates for the upper and lower limits of the exciton binding energy and Bohr exciton radius by computing them using values of the dielectric constant in the high frequency limit (
) and in the static ( ) limit. ,
where
is given by: (S16)
is the phonon frequency of mode p, V is the unit cell volume and
is the mass-
weighted mode effective Born vector. These quantities that go into the calculation of the dielectric constant of CsPbBr3 are obtained from density functional perturbation theory calculations.
10
8. DFT calculations (effect of van der Waals interaction): 8.A. Structure: For orthorhombic bulk CsPbBr3, within the GGA approximation, we obtain the lattice parameters a, b and c to be 8.52, 12.10, and 8.55 Å. In comparison to the experimental values, the lattice parameters are overestimated by 3-4 %, resulting in about 11% overestimation in the unit cell volume. On including the van der Waals (vdW) interaction, we find the lattice parameters b and c are significantly reduced to 11.89 and 7.56 Å while a remains unchanged. This results in about 3.3% compression in the unit cell volume compared to the experimentally measured one. In addition to the lattice parameters, the in-plane and apical Pb-Br bond lengths and Pb-Br-Pb angles are shortened on inclusion of the vdW interaction. A comparison of the structural parameters obtained from calculations with GGA and GGA+vdW with the experimental ones are given in Fig. S5 (A) and (B). 8.B. Phonons and dielectric constant: Consistent with the overall compression in the structural parameters on inclusion of vdW interactions, we find that the phonon modes are stiffened. Fig. S6 shows the computed IR spectra with and without vdW. We observe vdW interactions result in an overall shift of the phonon modes towards high frequency side because of the stiffening of the bonds and compression of the unit cell. This effect is further reflected when we calculate the static dielectric constant. Though we find similar values for
with (4.0) and without (4.14) vdW,
the static dielectric constants reduces to almost half of the value we obtain with GGA. While with GGA the static dielectric constant is 30.79, after incorporating vdW interactions we obtain
. We find that this value of dielectric constant is very similar to what is
obtained for bulk CsPbCl3, measured at room temperature at
Hz.12 This suggests
that it is indeed important to incorporate vdW interactions in order to compute structure and electronic properties of lead halides. A detailed analysis of the phonon frequencies, born effective charges and effective plasma frequencies of the phonon modes indicate that the large ionic contribution to the static dielectric constant in absence of vdW results primarily from the overall softening (low phonon frequencies) of the phonon modes.
11
Figure S5: Structure of orthorhombic bulk CsPbBr3 with (A) GGA and with (B) GGA+vdW. The Pb-Br computed (experimental) bond lengths are given in Å. The angles made by the axis of two octahedra along the b direction and in the ac plane are also shown. The electronic structure (with and without SOC) along high symmetry directions of the BZ for (C) GGA and (D) GGA+vdW. The brown, black and green spheres represent Br, Pb and Cs atoms respectively.
12
Figure S6: Computed IR spectra with (red lines) and without (black lines) vdW showing an overall shift in the phonon frequencies due to strengthening of bonds on inclusion of vdW interactions.
13
Figure S7: (A) and (B) shows the displacement patterns (denoted by blue arrows) corresponding to the two peaks in the computed spectra at 2.33 and 2.9 THz in Fig. 1(D) in the main manuscript. Contribution to the charge density from the valence band maxima (C) and conduction band minima (D) at Γ point of the BZ. The brown, black and green spheres represent Br, Pb and Cs atoms respectively. 8.C. Electronic structure: van der Waals interactions also affect the electronic properties of the system implicitly through the change in structure. Fig. S5 (C) and (D) show the band dispersion with and without vdW. In order to highlight the importance of spin orbit coupling (SOC), we have also included the band structure without spin orbit coupling for both the cases. A comparison of the red curves in Fig. S5 (C) and (D) shows that vdW reduces the dispersion of the bands, thereby making them flatter. This results in larger effective masses for both electrons and holes at the Γ-point of the Brillouin zone. The effective masses for both the cases are listed in table S1. Additionally there is also an increase in the band gap (1.01 eV with vdW+GGA compared to 0.76 eV with GGA). In both the cases we find a SOC splitting of about 0.81 eV. The isosurfaces of the wavefunction corresponding to the VBM and CBM with vdW interaction are shown in Figure S7(C) and (D) respectively. 14
Table S1: Effective masses of electrons and holes along the three high symmetric directions and the conductivity effective mass. The numbers in parenthesis are those obtained without the inclusion of van der Waals interaction. Direction [100]
0.24 (0.15)
0.26 (0.16)
[010]
0.17 (0.14)
0.20 (0.15)
[001]
0.27 (0.15)
0.26 (0.15)
Conductivity
0.22 (0.15)
0.24 (0.15)
effective mass
Table S2: Effect of van der Waals interaction and choice of dielectric constant in calculation of Bohr exciton radius and exciton binding energies. The numbers in parenthesis are computed with static dielectric constant. Property Bohr exciton radius (Å) Exciton binding energy (meV)
Level of theory GGA vdW + GGA 28.21 (217.17) 19.39 (73.08) 64.0 (1.1) 90.0 (6.0)
8.D. Exciton binding energies and Bohr exciton radius: Using
, the exciton binding energies with and without vdW interactions are 64 and 90 meV
respectively. The higher exciton binding energy upon inclusion of vdW can be attributed to the increase in the electron and hole effective masses, which in turn increases the exciton mass to 0.113 from 0.075 (with only GGA). Using the static value of the dielectric constant we find the exciton binding energies to be 1.1 meV and 6.3 meV with and without vdW respectively. Our calculations predict that within GGA the Bohr exciton radius will lie between 28.21 (with
) and 217.17 Å (with
), depending on the value of the dielectric constant. On
inclusion of vdW, we find that the lower and upper limit of the Bohr exciton radius reduces to 19.39 and 73.08 Å respectively.
15
Figure S8: Initial effective mobility (φµ) from peak photoconductivity and carrier density (assuming φ = 1) vs. fluence at different pump wavelengths.
Figure S9: Diffusion length as a function of initial carrier density. 16
(A)
(B)
Figure S10: Carrier density vs pump-probe delay at different fluences for (A) 400 nm and (B) 480 nm pump. The solid lines are the fits to the kinetic equation: ; where k1, k2, and k3 are the rate constants for mono-molecular trap-assisted
recombination, bi-molecular electron-hole recombination, and tri-molecular Auger recombination, respectively. The rate constants are given in Table S4.
17
Figure S11: Frequency resolved (A) real and (B) imaginary conductivity spectra at 480 nm pump wavelength with fluence of 166 µJ/cm2. Solid lines are fit to Drude-Smith model. The spectra at all delays except at longest delays are vertically offset for clarity.
Figure S12: Frequency resolved real (A & C) and imaginary (B & D) conductivity spectra at fluence of 94 (upper panels) and 57 µJ/cm2 (lower panels) for 400 nm pump wavelength. Solid lines are fit to Drude-Smith model. The spectra at all delays except at longest delays are vertically offset for clarity.
18
Figure S13: Mobility vs delay for 480 nm 166 µJ/cm2 fluence, 400 nm 57 µJ/cm2 fluence and 400 nm 94 µJ/cm2 fluence. The error bars are the uncertainties in calculated values of mobility obtained from the corresponding uncertainties in scattering rate ( ) in DS fits.
Figure S14: Carrier density vs pump-probe delay for 400 nm pump at 57 and 94 µJ/cm2 fluence. The solid lines are the fits to the kinetic equation (see Figure S10). The error bars are the uncertainties in calculated values of carrier density obtained from the corresponding uncertainties of the plasma frequency (ωp) in DS fits. 19
Table S3: Details of parameters obtained from fitting the frequency averaged transient photoconductivity data to multi-exponential decay function (equation S6) at excitation pulse energy. λpump
Fluence
Nph/1015
(nm)
(µJ/cm2)
(cm3)
41
1.50
57 400
72
94
24.9
39.5
72.7
480
93.5
114.3
135.1
166.3
42.6 504
63.4
2.07
2.62
3.42
0.55
0.87
1.61
2.07
2.53
2.98
3.67
0.97
1.44
a1 15000
15000
15000
15000
15000
15000
15000
15000
15000
15000
15000
15000
15000
a2
0.62
209.7
0.27
(0.001)
(0.9)
(0.001)
0.53
234.8
(0.00)
a3 ..
..
0.32
32.9
0.13
(0.9)
(0.001)
(0.7)
(0.001)
0.49
214.0
0.36
22.9
0.14
(0.00)
(0.5)
(0.001)
(0.3)
(0.001)
0.46
214.4
0.37
32.2
0.18
(0.00)
(0.5)
(0.001)
(0.3)
(0.001)
0.60
178.2
0.25
..
..
(0.001)
(1)
(0.001)
0.56
162.8
0.37
..
..
(0.001)
(0.4)
(0.001)
0.49
159.2
0.46
..
..
(0.00)
(0.2)
(0.001)
0.38
183.4
0.44
25.3
0.20
(0.00)
(0.3)
(0.001)
(0.2)
(0.001)
0.28
153.6
0.50
19.8
0.21
(0.00)
(0.02)
(0.001)
(0.2)
(0.001)
0.26
157.7
0.44
23.2
0.26
(0.00)
(0.2)
(0.001)
(0.1)
(0.001)
0.25
167.7
0.44
24.1
0.29
(0.00)
(0.2)
(0.001)
(0.1)
(0.001)
0.46
329.5
0.14
..
..
(0.002)
(8.5)
(0.001)
0.48
225.4
0.42
..
..
(0.001)
(1.1)
(0.001)
20
78
104
135
166
176
1.77
2.36
3.06
3.76
3.99
15000
0.43
214.6
0.46
(0.00)
(0.6)
(0.001)
0.43
247.3
0.5
(0.001)
(0.5)
(0.00)
0.44
260.3
0.53
(0.00)
(0.6)
(0.00)
0.43
224.5
0.55
(0.00)
(0.3)
(0.00)
0.38
221.5
(0.00)
(0.3)
15000
15000
15000
15000
..
..
..
..
..
..
..
..
0.53
37.4
0.07
(0.00)
(1.1)
(0.001)
Table S4: Kinetic parameters obtained from fitting the frequency averaged and frequency resolved data to the kinetic equation. The frequency averaged data evaluates k1, φk2 and φ2k3, whereas frequency resolved data evaluates k1, k2 and k3 directly. λpump (nm) Frequency 400
Resolved Frequency Averaged
480
Frequency Resolved Frequency Averaged
k1 (s-1)
k2 or φk2 (cm3 s-1)
k3 or φ2k3 (cm6 s-1)
1.7±0.14 x 109
1.35±0.8 x 10-7
9.55±2.2 x 10-22
2.37±0.8 x 109
2.99±0.24 x 10-7
3.14±0.18 x 10-23
0.9±0.7 x 109
9.3±0.14 x 10-8
1.016±0.06 x 10-21
2.93±0.39 x 109
9.14±2.14 x 10-7
4.53±0.45 x 10-23
21
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Department of Chemistry, Indian Institute of Science Education and Research (IISER), Pune, India, 411008
‡
Department of Physics, Indian Institute of Science Education and Research (IISER), Pune, India, 411008
*To whom correspondence should be addressed: [email protected] (PM), [email protected] (PG), [email protected] (AN).
1. Synthesis and characterization of colloidal CsPbBr3 nanocrystals: 1.A Preparation of Cs-oleate: Mixture of Cesium carbonate (0.4 g), Oleic acid (1.25 mL) and Octadecene (20 mL) were loaded into a 50 mL three neck flask and dried under vacuum for 1hr at 120oC. Then the temperature was increased to 150oC and kept until the solution became clear. 1.B. Colloidal CsPbBr3 NCs were synthesized following the procedure reported by Kovalenko et al.1 In brief, mixture of PbBr2 (0.188 mmol) with 5 mL Octadecene was dried under vacuum for 1 hour. Then anhydrous Oleylamine (0.5mL) and Oleic acid (0.5mL) were added to the mixture at 120oC. After the solution became clear, temperature was increased to 190oC and pre-heated Cs-oleate solution (0.1M, 0.4 mL) was swiftly injected to the reaction mixture. The reaction was stopped after 5 sec and cooled down to room temperature in icewater bath. The as synthesized CsPbBr3 NCs were precipitated by adding n-butanol and centrifuging at 10000 rpm for 10 min. It was then re-dissolved in toluene to form a long-term colloidal stable solution. 1
1.C Characterization: UV-visible absorption spectrum was recorded using a Perkin Elmer, Lambda- 45 UV/Vis spectrometer. Steady state photoluminescence (PL) of NCs was measured using FLS 980 (Edinburgh Instruments). Powder x-ray diffraction (XRD) data were recorded using a Bruker D8 Advance x-ray diffractometer using Cu Kα radiation (1.54 Å). Transmission electron microscopy (TEM) studies were carried out using a JEOL JEM 2100 F field emission transmission electron microscope at 200 kV. The sample preparation for TEM was done by putting a drop of the colloidal solution of NCs in hexane on the carbon coated copper grids.
CsPbBr3 NC's
Absorbance
0.4 0.3 0.2 0.1 0.0 350
400
450
500
550
PL Intensity (a.u.)
0.5
600
Wavelength (nm)
Figure S1: UV-visible absorption and photoluminescence (PL) spectra of CsPbBr3 NCs.
2
2. THz time domain spectroscopy (THz-TDS) and Optical Pump-THz Probe (OPTP) Spectroscopy
Figure S2: Experimental setup for Time Resolved THz spectroscopy. THz radiation is generated by air plasma and detected by air biased coherent detection (ABCD) method. The schematic of our experimental setup to generate2 THz from air plasma and detect it with ABCD method3, 4 is shown in figure S2. A regenerative amplifier (Spitfire Pro XP, Spectra Physics) seeded by oscillator (Tsunami, Spectra Physics) generates pulses of 50 fs (FWHM), 800nm central wavelength at 1 kHz repetition rate. The laser beam is split (BS 1) into two parts: one part is utilised to generate and detect THz radiation and another part is sent through the optical parametric amplifier (OPA) to generate the optical pump pulses of varying wavelengths. 2.A. THz Time Domain Measurement The first part is again split (BS 2) into two beams. The transmitted beam ( the BBO crystal that creates second harmonic (
; and on focusing (
passes through and (
in air,
plasma is generated. The plasma gives THz radiation along with other radiations. The THz radiation, filtered out by Si window, is collimated and focused to a spot size of ~1 mm
3
diameter on to the sample with the first pair of parabolic mirrors. The THz radiation transmitted through the sample is re-collimated and focused between two electrodes by another set of parabolic mirrors. The reflected beam from the BS2, called probe or gate beam which is reflected further by the retro reflector kept on a delay line (50 mm) is also focused between the electrodes. An AC voltage of 1.5 kV from high voltage modulator (HVM) is applied across the electrodes. The modulation frequency of HVM is kept at half (500 Hz) the laser repetition rate. The bias field acts as local oscillator and together with THz and fundamental beam generate second harmonic of the probe beam. The second harmonic signal is filtered from residual fundamental beam with band pass filter and detected by a photomultiplier tube (PMT). The PMT current is amplified using the current preamplifier and fed into the lock-in amplifier. THz electric field as a function of time delay between the gate pulse and THz pulse, is recorded by scanning the optical delay stage. 2.B. THz Time Resolved Measurement For time resolved THz study, the sample is excited by pump beam generated from the OPA. The delay between optical pump and THz probe beam is varied by scanning the 300 mm length stage marked as Delay 3 in figure S2. For controlling the pump power, variable neutral density filter is placed in the pump path. The pump beam is chopped using a chopper rotating at a frequency of 333 Hz. The pump beam passes through a hole in the off axis parabolic mirror and becomes collinear with the THz beam. The pump beam diameter on to the sample is ~ 3.5 mm, 3.5 times greater than the THz beam diameter to guarantee uniform sample excitation. The optical pump and THz probe diameters were measured by knife-edge technique. A black polyethylene sheet was used to block the pump light transmitted through the sample from reaching the PMT. The pump probe experiment is carried out using double lock in technique in which the pump induced change in THz transmission (-ΔE(tp)) and the corresponding THz transmission through the non-photoexcited NCs (E0(tp)) are recorded simultaneously.5 The signal from current preamplifier is split and sent to two separate lock in amplifiers. The THz probe beam is modulated at 500 Hz and this frequency acts as reference for one lock in amplifier. The reference frequency for the second lock in amplifier is 333 Hz, the frequency at which the optical pump beam is modulated. To minimise the crosstalk between the signals measured at 500 Hz and 333 Hz, the frequencies are not harmonic with respect to each other.
4
The pump induced signal can be collected in two ways.6 a) Frequency averaged experiments are performed by varying the delay between the optical pump and THz probe by scanning the 300 mm length stage after fixing the 50 mm stage at THz peak maximum (tmax). This gives the temporal evolution of the sample after photoexcitation by collecting the alteration in THz field amplitude, pulses,
, as a function of delay between THz probe and optical pump
. b) Frequency resolved experiments which are performed by scanning the 50 mm
stage at a fixed pump probe delay, yield the pump induced changes in THz waveform, , at a fixed pump probe delay (
.
To avoid THz absorption by water vapour, the entire experimental setup is enclosed in a transparent box and purged continuously with N2 or dry air. 2.C. Sample Preparation for THz Experiments Colloidal CsPbBr3 NC’s are dissolved in HMN solvent, which is quite transparent in broad band THz region. The solution is then injected into the demountable liquid cell (Harrick Scientific Products, DLC-M25), in which two TPX windows are placed separated by a Teflon spacer of 950 um thickness. Path length of the sample is equal to the spacer thickness.
3. THz TDS Data Analysis: The THz electric field passing through the sample and reference is given by Esample(t) and Eref(t) respectively. Fourier transform of these electric fields gives power (P) and phase (Φ) which are related to the refractive index and absorption coefficients and is given by
The complex refractive index can be written as to complex valued permittivity as
, where
, and it is related
The real and imaginary parts of the
permittivity are given by
5
The intrinsic frequency dependent complex dielectric function of the NCs was calculated using a simple effective medium approach (EMA) for dielectric inclusion embedded in a host medium: 7 (S5) where ɛeff(ν), ɛi(ν) and ɛh(ν) are the complex dielectric functions of the composite (NC solution, experimentally determined from THz-TDS), the inclusion (NCs) and the host medium (HMN) respectively; and f is the volume fraction of the NCs in the solution. The simple linear effective medium approach (EMA) has been adopted in the present work for the following reasons: (1) the size of the inclusions (nanocrystals, ~11 nm) is significantly less than the wavelength of the probe light (0.5 to 7 THz, 43 to 60 micron), (2) the volume fraction being very small (~10-3) makes Bruggeman method inapplicable, this being used for very large volume fraction where inclusions come together and form domains; moreover linear EMA seems to be working reasonably well for such dilute solution,7 (3) the NCs of this study have a cubic/orthorhombic geometry and the geometry factor for cubic/orthorhombic structure required for Maxwell-Garnet method is not known and determining the same is beyond the scope of the present work, and (4) the linear EMA yields real dielectric values which are in between the static and high-frequency limit of dielectric constants predicted in our GGA+vdW level of theory.
Figure S3: (A) THz absorption and (B) real refractive indices of HMN solvent and the CsPbBr3 perovskite NC solution.
6
Figure S4: (A) Real and (B) imaginary components of dielectric function of HMN and CsPbBr3 perovskite NC solution.
4. Recombination Dynamics via optical-pump THz-probe studies Charge carriers (electrons & holes) are generated by optical-pump pulses whose energies exceed the band gap (~2.5 eV) of the NCs. Here, we have used optical-pump pulses of wavelengths 504nm (energy equivalent to band gap), 480 nm and 400 nm (energy more than band gap). Optical pulses at 400nm were generated by passing the fundamental beam (800 nm) through BBO crystal and other two pulses were generated through optical parametric amplifier (OPA). The optical chopper was placed in pump beam path to modulate optical pump pulses. Carrier dynamics was monitored by measuring the change in THz field amplitude
as a function of pump-probe delay ( ). Note that
is
proportional to change in conductivity, which in turn depends on charge carriers. We have fitted the normalized photoconductivity decays to a multi-exponential function convoluted with a Gaussian function of the form: 8 (S6) where
is a Gaussian function centered at t0 with FWHM of ~300 fs which
represents the instrument response time of our OPTP experiment, and ai is the coefficient of the ith exponential decay channel with time constant τi.
7
5. Determination
of
transient photoconductivity from THz
electric field
transmission at different time delays The transient photoconductivity Δσ of NCs dissolved in HMN solvent placed between the is expressed as 6
two TPX windows, under the condition that
Where
is the refractive index of the NCs solution and
is given by
Here, l is the path length of the sample, n1 is the refractive index of the TPX window (n1= 1.524), n2 is the refractive index of the NCs solution (n2=1.61), MR is multiple reflection term (here MR= 0) and
are pump-induced and pump-off Fourier transforms of
transmitted THz electric fields. The dielectric constant of the pump induced NCs solution obtained from the conductivity (calculated from eq. S7) is
The effective dielectric constant of the photoexcited NCs can be calculated from the effective dielectric constant of the non photoexcited NCs and pump induced change in dielectric constant of NCs solution using the following equation:
After calculating
using eq. S10 and using this value in eq. S5 (effective medium
theory), the intrinsic dielectric constant of the photoexcited NCs can be calculated ( difference between
and
. The
(intrinsic dielectric constant of the non-photoexcited NCs)
is given by,
which is the pump induced dielectric constant of NCs. Using conductivity
of NCs can be calculated from eq. S9.
8
, the pump induced
In the above calculation for THz conductivity of NCs it is assumed that the entire sample (thickness, d) gets photoexcited. But true THz conductivity should be calculated for the thickness (skin depth ) till which the sample gets excited. So it must be corrected by a factor
if d> . The photoexcited sample thickness d for 504 nm and 480 nm pump was
taken to be same as the sample cell thickness (950 µm) as the penetration depth (1/OD) of the pump light at these two wavelengths are longer than the sample cell thickness. However, for 400 nm pump the photoexcited sample thickness d was considered to be equal to the penetration depth (δ; 546 µm), which is significantly shorter than the sample cell thickness (950 µm). Thus the real photo induced THz conductivity of NCs is given by,9
For 480 and 504 nm pump beam, sample thickness are more than the corresponding skin depths, so multiplication of the conductivity with the scaling factor
is not required. In
case of 400 nm, the skin depth is less than the sample thickness, so conductivity is multiplied with 1.74 (
.
6. Carrier density and carrier mobility by fitting Drude-Smith model to THz conductivity obtained from THz scans We have collected the pump induced change in THz electric field, ΔE (t, tp), and pump-off THz electric field, E0 (t, tp), at different pump-probe delays (tp). Fourier transform of the electric fields allowed us to calculate the change in THz conductivity of photo excited NCs at different pump-probe delays (tp). The THz conductivities can be well fitted by Drude-Smith model10 and is given by
where
is plasma frequency. It is related to carrier density through being the average effective mass of the carriers,
related to mobility as
and
with is scattering rate
represents the fraction of electrons having the
original velocity after first collision. The obtained mobilities are attributed to both electrons and holes, since THz radiation cannot distinguish charge carrier type.
9
7. Calculation of effective mass, exciton binding energy and Bohr exciton radius: Based on the assumption that in this material the excitons are Wannier type excitons, according to the effective mass theory the Bohr exciton radius is given by: ,
(S14)
and the exciton binding energy is given by: (S15) In equation S14 and S15,
is the dielectric constant,
is the radius of H atom and
reduced mass of the exciton. The reduced mass is given by
,
is the
being the
effective masses of electron and hole respectively. Since the photoconductivity measured is obtained from averaging along x, y and z direction, we consider the conductivity effective masses
. The effective masses of electrons and holes along
the [100], [010] and [001] directions are determined by fitting the band structure (within kBT of band edges) at the valence band maximum (VBM) and conduction band minimum (CBM) at Γ point of the Brillouin zone. Determining the correct values of Bohr exciton radius (equation S14) and exciton binding energy (equation S15) are challenging because the dielectric constant variation with frequency and temperature. The correct choice of
shows large
depends on the exciton
binding energy (which itself is unknown) and the energy of the optical phonons.11 Hence we have given estimates for the upper and lower limits of the exciton binding energy and Bohr exciton radius by computing them using values of the dielectric constant in the high frequency limit (
) and in the static ( ) limit. ,
where
is given by: (S16)
is the phonon frequency of mode p, V is the unit cell volume and
is the mass-
weighted mode effective Born vector. These quantities that go into the calculation of the dielectric constant of CsPbBr3 are obtained from density functional perturbation theory calculations.
10
8. DFT calculations (effect of van der Waals interaction): 8.A. Structure: For orthorhombic bulk CsPbBr3, within the GGA approximation, we obtain the lattice parameters a, b and c to be 8.52, 12.10, and 8.55 Å. In comparison to the experimental values, the lattice parameters are overestimated by 3-4 %, resulting in about 11% overestimation in the unit cell volume. On including the van der Waals (vdW) interaction, we find the lattice parameters b and c are significantly reduced to 11.89 and 7.56 Å while a remains unchanged. This results in about 3.3% compression in the unit cell volume compared to the experimentally measured one. In addition to the lattice parameters, the in-plane and apical Pb-Br bond lengths and Pb-Br-Pb angles are shortened on inclusion of the vdW interaction. A comparison of the structural parameters obtained from calculations with GGA and GGA+vdW with the experimental ones are given in Fig. S5 (A) and (B). 8.B. Phonons and dielectric constant: Consistent with the overall compression in the structural parameters on inclusion of vdW interactions, we find that the phonon modes are stiffened. Fig. S6 shows the computed IR spectra with and without vdW. We observe vdW interactions result in an overall shift of the phonon modes towards high frequency side because of the stiffening of the bonds and compression of the unit cell. This effect is further reflected when we calculate the static dielectric constant. Though we find similar values for
with (4.0) and without (4.14) vdW,
the static dielectric constants reduces to almost half of the value we obtain with GGA. While with GGA the static dielectric constant is 30.79, after incorporating vdW interactions we obtain
. We find that this value of dielectric constant is very similar to what is
obtained for bulk CsPbCl3, measured at room temperature at
Hz.12 This suggests
that it is indeed important to incorporate vdW interactions in order to compute structure and electronic properties of lead halides. A detailed analysis of the phonon frequencies, born effective charges and effective plasma frequencies of the phonon modes indicate that the large ionic contribution to the static dielectric constant in absence of vdW results primarily from the overall softening (low phonon frequencies) of the phonon modes.
11
Figure S5: Structure of orthorhombic bulk CsPbBr3 with (A) GGA and with (B) GGA+vdW. The Pb-Br computed (experimental) bond lengths are given in Å. The angles made by the axis of two octahedra along the b direction and in the ac plane are also shown. The electronic structure (with and without SOC) along high symmetry directions of the BZ for (C) GGA and (D) GGA+vdW. The brown, black and green spheres represent Br, Pb and Cs atoms respectively.
12
Figure S6: Computed IR spectra with (red lines) and without (black lines) vdW showing an overall shift in the phonon frequencies due to strengthening of bonds on inclusion of vdW interactions.
13
Figure S7: (A) and (B) shows the displacement patterns (denoted by blue arrows) corresponding to the two peaks in the computed spectra at 2.33 and 2.9 THz in Fig. 1(D) in the main manuscript. Contribution to the charge density from the valence band maxima (C) and conduction band minima (D) at Γ point of the BZ. The brown, black and green spheres represent Br, Pb and Cs atoms respectively. 8.C. Electronic structure: van der Waals interactions also affect the electronic properties of the system implicitly through the change in structure. Fig. S5 (C) and (D) show the band dispersion with and without vdW. In order to highlight the importance of spin orbit coupling (SOC), we have also included the band structure without spin orbit coupling for both the cases. A comparison of the red curves in Fig. S5 (C) and (D) shows that vdW reduces the dispersion of the bands, thereby making them flatter. This results in larger effective masses for both electrons and holes at the Γ-point of the Brillouin zone. The effective masses for both the cases are listed in table S1. Additionally there is also an increase in the band gap (1.01 eV with vdW+GGA compared to 0.76 eV with GGA). In both the cases we find a SOC splitting of about 0.81 eV. The isosurfaces of the wavefunction corresponding to the VBM and CBM with vdW interaction are shown in Figure S7(C) and (D) respectively. 14
Table S1: Effective masses of electrons and holes along the three high symmetric directions and the conductivity effective mass. The numbers in parenthesis are those obtained without the inclusion of van der Waals interaction. Direction [100]
0.24 (0.15)
0.26 (0.16)
[010]
0.17 (0.14)
0.20 (0.15)
[001]
0.27 (0.15)
0.26 (0.15)
Conductivity
0.22 (0.15)
0.24 (0.15)
effective mass
Table S2: Effect of van der Waals interaction and choice of dielectric constant in calculation of Bohr exciton radius and exciton binding energies. The numbers in parenthesis are computed with static dielectric constant. Property Bohr exciton radius (Å) Exciton binding energy (meV)
Level of theory GGA vdW + GGA 28.21 (217.17) 19.39 (73.08) 64.0 (1.1) 90.0 (6.0)
8.D. Exciton binding energies and Bohr exciton radius: Using
, the exciton binding energies with and without vdW interactions are 64 and 90 meV
respectively. The higher exciton binding energy upon inclusion of vdW can be attributed to the increase in the electron and hole effective masses, which in turn increases the exciton mass to 0.113 from 0.075 (with only GGA). Using the static value of the dielectric constant we find the exciton binding energies to be 1.1 meV and 6.3 meV with and without vdW respectively. Our calculations predict that within GGA the Bohr exciton radius will lie between 28.21 (with
) and 217.17 Å (with
), depending on the value of the dielectric constant. On
inclusion of vdW, we find that the lower and upper limit of the Bohr exciton radius reduces to 19.39 and 73.08 Å respectively.
15
Figure S8: Initial effective mobility (φµ) from peak photoconductivity and carrier density (assuming φ = 1) vs. fluence at different pump wavelengths.
Figure S9: Diffusion length as a function of initial carrier density. 16
(A)
(B)
Figure S10: Carrier density vs pump-probe delay at different fluences for (A) 400 nm and (B) 480 nm pump. The solid lines are the fits to the kinetic equation: ; where k1, k2, and k3 are the rate constants for mono-molecular trap-assisted
recombination, bi-molecular electron-hole recombination, and tri-molecular Auger recombination, respectively. The rate constants are given in Table S4.
17
Figure S11: Frequency resolved (A) real and (B) imaginary conductivity spectra at 480 nm pump wavelength with fluence of 166 µJ/cm2. Solid lines are fit to Drude-Smith model. The spectra at all delays except at longest delays are vertically offset for clarity.
Figure S12: Frequency resolved real (A & C) and imaginary (B & D) conductivity spectra at fluence of 94 (upper panels) and 57 µJ/cm2 (lower panels) for 400 nm pump wavelength. Solid lines are fit to Drude-Smith model. The spectra at all delays except at longest delays are vertically offset for clarity.
18
Figure S13: Mobility vs delay for 480 nm 166 µJ/cm2 fluence, 400 nm 57 µJ/cm2 fluence and 400 nm 94 µJ/cm2 fluence. The error bars are the uncertainties in calculated values of mobility obtained from the corresponding uncertainties in scattering rate ( ) in DS fits.
Figure S14: Carrier density vs pump-probe delay for 400 nm pump at 57 and 94 µJ/cm2 fluence. The solid lines are the fits to the kinetic equation (see Figure S10). The error bars are the uncertainties in calculated values of carrier density obtained from the corresponding uncertainties of the plasma frequency (ωp) in DS fits. 19
Table S3: Details of parameters obtained from fitting the frequency averaged transient photoconductivity data to multi-exponential decay function (equation S6) at excitation pulse energy. λpump
Fluence
Nph/1015
(nm)
(µJ/cm2)
(cm3)
41
1.50
57 400
72
94
24.9
39.5
72.7
480
93.5
114.3
135.1
166.3
42.6 504
63.4
2.07
2.62
3.42
0.55
0.87
1.61
2.07
2.53
2.98
3.67
0.97
1.44
a1 15000
15000
15000
15000
15000
15000
15000
15000
15000
15000
15000
15000
15000
a2
0.62
209.7
0.27
(0.001)
(0.9)
(0.001)
0.53
234.8
(0.00)
a3 ..
..
0.32
32.9
0.13
(0.9)
(0.001)
(0.7)
(0.001)
0.49
214.0
0.36
22.9
0.14
(0.00)
(0.5)
(0.001)
(0.3)
(0.001)
0.46
214.4
0.37
32.2
0.18
(0.00)
(0.5)
(0.001)
(0.3)
(0.001)
0.60
178.2
0.25
..
..
(0.001)
(1)
(0.001)
0.56
162.8
0.37
..
..
(0.001)
(0.4)
(0.001)
0.49
159.2
0.46
..
..
(0.00)
(0.2)
(0.001)
0.38
183.4
0.44
25.3
0.20
(0.00)
(0.3)
(0.001)
(0.2)
(0.001)
0.28
153.6
0.50
19.8
0.21
(0.00)
(0.02)
(0.001)
(0.2)
(0.001)
0.26
157.7
0.44
23.2
0.26
(0.00)
(0.2)
(0.001)
(0.1)
(0.001)
0.25
167.7
0.44
24.1
0.29
(0.00)
(0.2)
(0.001)
(0.1)
(0.001)
0.46
329.5
0.14
..
..
(0.002)
(8.5)
(0.001)
0.48
225.4
0.42
..
..
(0.001)
(1.1)
(0.001)
20
78
104
135
166
176
1.77
2.36
3.06
3.76
3.99
15000
0.43
214.6
0.46
(0.00)
(0.6)
(0.001)
0.43
247.3
0.5
(0.001)
(0.5)
(0.00)
0.44
260.3
0.53
(0.00)
(0.6)
(0.00)
0.43
224.5
0.55
(0.00)
(0.3)
(0.00)
0.38
221.5
(0.00)
(0.3)
15000
15000
15000
15000
..
..
..
..
..
..
..
..
0.53
37.4
0.07
(0.00)
(1.1)
(0.001)
Table S4: Kinetic parameters obtained from fitting the frequency averaged and frequency resolved data to the kinetic equation. The frequency averaged data evaluates k1, φk2 and φ2k3, whereas frequency resolved data evaluates k1, k2 and k3 directly. λpump (nm) Frequency 400
Resolved Frequency Averaged
480
Frequency Resolved Frequency Averaged
k1 (s-1)
k2 or φk2 (cm3 s-1)
k3 or φ2k3 (cm6 s-1)
1.7±0.14 x 109
1.35±0.8 x 10-7
9.55±2.2 x 10-22
2.37±0.8 x 109
2.99±0.24 x 10-7
3.14±0.18 x 10-23
0.9±0.7 x 109
9.3±0.14 x 10-8
1.016±0.06 x 10-21
2.93±0.39 x 109
9.14±2.14 x 10-7
4.53±0.45 x 10-23
21
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