Excitons in Orthorhombic and Tetragonal Hybrid OrganicâInorganic

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Excitons in Orthorhombic and Tetragonal Hybrid Organic-Inorganic Perovskites Zhi-Gang Yu J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b00246 • Publication Date (Web): 24 Jan 2017 Downloaded from http://pubs.acs.org on January 28, 2017

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The Journal of Physical Chemistry

Excitons in Orthorhombic and Tetragonal Hybrid Organic-Inorganic Perovskites Zhi-Gang Yu∗ ISP/Applied Sciences Laboratory, Washington State University, Spokane, Washington 99210, United States E-mail: [email protected]

1 ACS Paragon Plus Environment


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Abstract The small exciton binding energy Eb in hybrid organic-inorganic perovskites (HOIPs) enables their extraordinary photovoltaic performance. The measured Eb in MAPbI3 (MA=CH3 NH3 ), the most studied HOIP, is 50% smaller in the orthorhombic phase than estimated from the high-frequency dielectric constant and drops an additional 25% in the tetragonal phase. Here we show that these puzzling exciton behaviors can be quantitatively explained by incomplete screening arising from polar optical phonons (POPs) and randomly oriented MA ions. Emission/absorption of POPs introduces a virtual radius of electron (hole), within which the screening due to POPs is ineffective. Randomly oriented dipoles of MA ions in the tetragonal phase mediate an extra longrange coupling between the electron and hole. The exciton Hamiltonian is accurately solved by using variational hydrogenic wave functions as the basis set. Our results consistently account for the observed exciton properties and reveal the impact of polar coupling on excitons in HOIPs.

Hybrid organic-inorganic perovskites (HOIPs) represent a revolutionary breakthrough in low-cost solar cells, 1–4 in which sunlight is converted into free electrons and holes. The Coulomb attraction between electrons and holes suggests that excitons, or bound electronhole (e-h) pairs, be present as well. In equilibrium, the densities of free electrons (holes) ne and of excitons nex are related via the law of mass action 5,6 for the exciton binding e+h → ex, µk T 3/2 n2e B = e−Eb /kB T , nex h ¯

(1)

where Eb is the exciton binding energy, µ = me mh /(me + mh ) the reduced mass of exciton with me (mh ) being the effective mass of electron (hole), kB the Boltzmann constant, and T the temperature. Equation (1) states that Eb controls the percentage of free carriers in photo excitations and strongly influences the photovoltaic efficiency. The confined exciton wave function manifests itself with distinct signatures in a variety of optical spectra, from which Eb can be estimated. The estimated values, however, fluctuate 2 ACS Paragon Plus Environment

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significantly, due partly to the unreliable models used for fitting experimental data. 7–16 For example, Eb in MAPbI3 (MA=CH3 NH3 ), the most extensively studied HOIP, ranges from 2 to 60 meV. Among various experimental methods, cyclotron resonance (CR) in unique in that it can definitively determine Eb as well as µ. 15–17 For MAPbI3 , Eb = 16 meV in the orthorhombic phase (OP) at T = 2 K, 15 which is considerably smaller than Eb0 = 34.3 meV, estimated from the hydrogenic spectrum of a Wannier exciton, 18

¯ 2) Eb0 = µe4 /(22∞ h

(2)

by using the measured material’s high-frequency dielectric constant ∞ = 6.5 7 (and µ = 0.104m0 from CR 15 with m0 being the free electron mass). In the tetragonal phase (TP) at T = 161 K, Eb further drops to 12 meV. 16 Effectively, ∞ in Eq. (2) increases to 9.4 and 11 at T = 2 and 161 K; i.e., the Coulomb interaction is screened. The screening, however, is far from complete, as the low-frequency dielectric constant reaches 0 = 36 in the OP and 0 = 130 in the TP at T = 161 K. 19 Despite the critical role of excitons in HOIPs, there has been little theoretical effort to date to examine this incomplete screening, other than attributing it generally to phonons and/or electric dipoles of MA ions. 11 Recently, the exciton states in MAPbI3 20 were calculated by using the screened Haken potential, 21 which, however, is not directly applicable to HOIPs due to their multiple phonon modes. Another intriguing observation is that Eb in MAPbBr3 is not much affected by the phonons or MA dipoles. 22,23 Here we present a microscopic model to elucidate how the phonons and MA dipoles screen the e-h Coulomb interaction in HOIPs. In addition to a quantitative account for the measured Eb in MAPbI3 , the model provides a systematic description of temperature dependence in Eb and a consistent explanation of the disparate exciton behaviors in MAPbI3 and MAPbBr3 . First we consider the OP, where the MA ions are ordered with zero spontaneous polarization, 24,25 as reflected in the temperature independent 0 , 19 and can therefore be ignored



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when studying excitons. In addition, to highlight the major physics, we neglect the weak anisotropy in both carrier effective masses and dielectric response for the OP and TP. The effect of anisotropy on the exciton binding is discussed in the Supporting Information and the relative error due to neglecting the anisotropy is found to be less than 1%. In an exciton, the electron and hole, apart from their mutual Coulomb interaction, Vc = −e2 /(∞ |re − rh |) with re (rh ) being the location of electron (hole), simultaneously interact with polar optical phonons (POPs), 26 which create a time-dependent long-range electric field. The polar coupling in an HOIP with multiple modes was recently evaluated, 27 and for an e-h pair (see Supporting Information),

Vep =

X jq

iUj [bjq (eiq·rh − eiq·re ) − b†jq (e−iq·rh − e−iq·re )], q

(3)

where b†jq (bjq ) creates (destroys) a longitudinal POP of the jth mode with momentum q −1/2

and frequency ωlj and Uj = (2πe2 h ¯ ωlj /Ω)1/2 sj

, with Ω being the material volume and

ωt2j Y ωl2j − ωt2i 1 1 = , 1− 2 sj ∞ ωlj i6=j ωl2j − ωl2i

(4)

where ωtj is the transverse optical frequency of the jth mode. −1 sj measures the total polarization due to mode j and is connected to the dimensionless polar-coupling strength αe(h)j for electron (hole) via αe(h)j =

1/2 1 e2 me(h) ( h 2¯ ¯ hωlj sj

and αej /αhj = (me /mh )1/2 . αe(h)j characterizes

the nature of electron (hole) polarons, being small (heavy) when αe(h)j 1 or large (light) when αe(h)j ≤ 1. 26,27 Among the multitude of phonon modes in HOIPs the Pb-X stretching and Pb-X bending can contribute significantly to the polar coupling. 27 The much smaller αe(h) associated with the Pb-X bending 27 allows us to consider only the Pb-X stretching mode and drop the mode index. The polar coupling Vep can induce an effective screening potential Vs between the electron and hole, which can be derived by extending the approach by Haken 21 to HOIPs at finite temperatures. A free electron, described by a wave function of Ω−1/2 eik·re |0i with k being 4 ACS Paragon Plus Environment

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its momentum and |0i the vacuum, can emit or absorb a phonon with momentum h ¯ q after Vep is turned on, and recoil to state of k − q or k + q, yielding a perturbed wave function, |φek i =

h

1−

X q

+

X q

−1 ¯ 2k2 iU h ¯ 2 (k − q)2 h − +h ¯ ωl e−iq·re b†q q 2me 2me

−1 i eik·re iU h ¯ 2 (k + q)2 h ¯ 2k2 − −h ¯ ωl eiq·re bq 1/2 |0i, q 2me 2me Ω

(5)

which represents an electron dressed by a phonon cloud. A phonon-dressed hole, |φhk i, can be similarly expressed. The exciton wave function is a linear combination of e-h pairs with different momenta, |ψex i =

P

k1 k2

C(k1 , k2 )|φek1 i|φhk2 i. Since these e-h pairs come mainly

from states near the conduction- and valence-band edges, k1 , k2 → 0, the screening potential Vs ≡

1 hψex |Vep |ψex i 2 hψex |ψex i

is found to be (see Supporting Information)

e2 h e−r12 /λe + e−r12 /λh cos(r12 /λe ) + cos(r12 /λh ) e−r12 /λe + e−r12 /λh i Vs (r12 ) = (1− )+nph ( − ), s r12 2 2 2 (6) where r12 = |re − rh |, λe(h) = [¯ h/2me(h) ωl ]1/2 , and nph ≡ [exp(¯ hωl /kB T ) − 1]−1 is the average phonon number at temperature T . For a system with a single optical mode, Eq. (6) reduces to the Haken potential at T = 0. 21 We emphasize that s in Vs depends on ∞ and optical phonon frequencies, as shown in Eq. (4), but not on the low-frequency dielectric response, 0 . The frequency-dependent dielectric response in the HOIPs is discussed in the Supporting Information. For r12 λe(h) , Vs (r12 ) has a 1/r12 dependence, and the total e-h −1 2 potential, Vc + Vs , becomes a screened Coulomb potential, −(−1 ∞ − s )e /r12 . When r12 is

comparable to λe(h) , however, Vs (r12 ) diminishes. Intuitively, λe(h) outlines a “virtual” radius of electron (hole), within which atoms are unable to provide the screening. This virtual radius can be understood from the quantum uncertainty rule. 28 In the presence of Vep , the electron’s momentum is fluctuating while absorbing or emitting a phonon, and for k → 0, (∆p)2 /2me = h ¯ ωl . Accordingly, the electron’s position is blurred, ∆r ' h ¯ /∆p = λe , as if the electron has a finite radius of λe . Furthermore, the finite phonon number nph in Eq. (6)



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adds an oscillatory component to Vs , suggesting that the exciton wave functions would be temperature-dependent. The effective Hamiltonian for the relative e-h motion in an exciton is then

H = −∇2 −

a0 a 2 ∞ h a0 a0 i − 0r + 2 − e− λe r + e λh (1 + nph ) + nph cos r + cos r , r s r λe λh

(7)

¯ 2 ∞ /(e2 µ), where the energy and length units are Eb0 and the unscreened exciton radius a0 ≡ h which is 33.1 ˚ A for MAPbI3 , whose parameters are summarized in Table I. Table 1: Parameters for MAPbI3 . me /mh 0.84 17,29

αh 1.1 27

h ¯ ωl (meV) ∞ 16.5 27 6.5 7

N (nm−3 ) 3.95 19

p (D) 0.88 19,30

γ (u30 ) 17.07

The long-range potential in Eq. (7) renders direct integration of the Schrödinger equation numerically unreliable. Instead, we employ the following ortho-normal functions as our basis set, v u

2 u (n − l − 1)! −βr/n 2βr l 2l+1 2βr (β) φnlm (r) = β 3/2 2 t e Ln−l−1 Ylm (θ, φ), n (n + l)! n n

(8)

where Ylm (θ, φ) is the spherical harmonic function and Lkn (x) is the generalized Laguerre poly(β)

nomial. Equation (8) is the eigen function of the Schrödinger equation (−∇2 − 2β/r)φnlm = (β)

−(β 2 /n2 )φnlm ; i.e., the hydrogenic equation with the nuclear charge of β|e|, where the variational β is introduced to enhance the solutions’ accuracy. These basis functions offer two salient advantages: 1) they have correct asymptotic behaviors at both r → 0 and r → ∞; and 2) the matrix elements of Hamiltonian (7) between basis functions {φ(β) } can be analytically evaluated (see Supporting Information). For calculating Eb , it is sufficient to find the energies and wave functions of the s orbitals (l = m = 0). Minimizing the ground-state energy E1 with respect to β, ∂E1 /∂β = 0, we find β as well as energy En and wave function ψn for the ns exciton. The accuracy of the numerical results is systematically studied in the Supporting Information. Figures 1a and 1b plot the effective e-h potential and the wave functions of 1s and 2s 6 ACS Paragon Plus Environment

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excitons of MAPbI3 at T = 2 K. In this case, nph ' 0 and Vs (r12 ) decays monotonically over distance. Vs (r12 ) makes the total potential less negative and has an 1/r12 dependence for r12 λh , where λh = 32.1˚ A = 0.97a0 for h ¯ ωl = 16.5 meV. 27 For r12 < λh , however, Vs (r12 ) stops increasing as r12 further decreases, indicating that the screening becomes ineffective. The obtained 1s exciton, with all parameters from literature, has Eb = 14.9 meV, in excellent agreement with the experiment value of 16 meV. 15 The wave functions of 1s and 2s excitons, (0)

(0)

ψ1 and ψ2 , are more delocalized as compared to ψ1 and ψ2 with the unscreened Coulomb −1 = hψ1 |r−1 |ψ1 i, is rav = 40.0 ˚ A, potential. The average size of the 1s exciton, defined as rav

which is only slightly larger than λe(h) . This explains the incomplete screening of excitons in the OP. The energy En and oscillator strength fn of the first five ns excitons, shown in (0)

(0)

Fig. 1e, are found to deviate from the hydrogenic spectrum, En(0) /E1 = n−2 and fn(0) /f1 = (0)

|ψn(0) (0)|2 /|ψ1 (0)|2 = n−3 . The deviation is caused by the mixing among ψn(0) in the presence of Vs . In particular, the relative energy difference between the 1s and 2s excitons increases from 3/4 to 0.84, indicating that it would be inaccurate to estimate Eb from the energy splitting between the exciton states by assuming the hydrogenic spectrum. The POP-induced screening can be explicated further by systematically varying the exciton size, or ∞ , since a0 ∝ ∞ . We see in Figs. 2a and 2b that for small exciton sizes, or large binding energies, with rav ≤ λh , Eb is close to Eb0 , indicting a minimal screening effect. By contrast, for large exciton sizes with rav > λh , the screening gains strength and Eb becomes considerably smaller than Eb0 . The effects of polar coupling strength αh and phonon frequency h ¯ ωl on the 1s exciton energy are summarized in Figs. 2c and 2d. We see that the exciton binding decreases with both αh and h ¯ ωl . A large αh means that polarization caused by the POP mode is strong and consequently, the screening effect more pronounced. The frequency dependence can be understood in terms of the ratio rav /λh : a lower h ¯ ωl would give a larger λh , and when it encompasses the entire exciton, the electron and hole sense no screening. In other words, the atoms move so slowly that they cannot follow electrons and holes to adequately screen the e-h Coulomb attraction. These results also justify our neglect



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of the Pb-X bending mode, which has a much lower h ¯ ωl (5 meV) and smaller αh (0.6) 27 than the Pb-X stretching mode. Now we study the TP of HOIPs, in which the MA dipoles are randomly oriented. 24,25 These dipoles are responsible for the Curie-Weiss type temperature dependence in 0 . 19 Since the rotational excitation of these dipoles has an energy of 10−4 eV, 30 orders-of-magnitude smaller than Eb , the electron and hole in an exciton see a snapshot of “frozen” fluctuations of the MA dipoles. Using the free energy of an MA ion in an electric field E, F = −p · E − E · γ · E/2, 31 with p and γ being the dipole moment and polarizability of the MA ion, we express the free-energy change of all dipoles due to the electric field created by an electron at re and a hole at rh as

F =−

X

p i · Ei −

i

where Ei =

e ( ri −rh ∞ |ri −rh |3

−

ri −re ), |ri −re |3

1X Ei · γ · Ei ≡ V1 + V2 , 2 i

(9)

and ri is the individual dipole location. (0)

To the ground-state exciton, ψ1 , the free energy (9) can be treated as a perturbation. (0)

(0)

Because of the random orientations of MA dipoles, hψ1 |V1 |ψ1 i = 0, so one must go to the second order to capture the dipoles’ effect: (0)

(0)

∆H1

(0)

(0)

hψ1 |V1 |ψi ihψi |V1 |ψ1 i 1 (0) (0) (0) = ' − 0 [hψ1 |V12 |ψ1 i − |hψ1 |V1 |ψ (0) i|2 ] 0 0 E1 − Ei Eb i6=1 X

= −

N p2 Z 3 d r|E(r)|2 , 3Eb0

(10) (0)

(0)

where we have used an approximation Ei −E1 ' Eb0 for all exciton excited states (although (0)

strictly 0.75Eb0 ≤ Ei

(0)

− E1

≤ Eb0 ) and hp(r)p(r 0 )i = 31 N p2 δ(r − r 0 )I for uncorrelated

random dipoles with N being the density of MA dipoles and I the unit tensor. Combining the contributions from V1 and V2 , we obtain an extra e-h coupling due to the MA dipoles


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(see Supporting Information),

Vd (r12 ) =

e2 4πN 2p2 + γ , 2∞ 3Eb0 r12

(11)

where γ = γI is assumed to be isotropic. The 1/r12 dependence in Eq. (11) suggests that excitons in the TP, with the POP-induced screening included, can still be described by −1 ˜−1 Hamiltonian (7), if −1 ∞ ≡ ∞ [1 − ∞ is replaced by

4πN 2p2 ( ∞ 3Eb0

+ γ)].

From the measured N and p, as well as the calculated γ from first principles in Table I, we have ˜∞ = 7.18. Moreover, the elevated temperatures in the TP indicate that the phonon number nph is significant and that Vs has a prominent oscillatory tail, as shown in Fig. 1c. Using the variational basis functions in Eq. (8), we obtain the exciton binding energy Eb = 12.6 meV at T = 161 K, agreeing with the experiment value of 12 meV. 16 Because of the dipole-induced screening Vd , the exciton wave functions become more delocalized in Fig. 1d than in the OP, with rav = 51.9 ˚ A for the 1s exciton. The exciton energies and oscillator strengths of the ns exciton, displayed in Fig. 1f, also exhibit a more pronounced deviation from the hydrogenic spectrum than in the OP. The larger exciton wave functions in the TP further enhance the screening due to POPs, as illustrated in Figs. 2a and 2b. The screening effect, as shown in Figs. 2c and 2d, generally becomes stronger with increases of the phonon frequency h ¯ ωl and polar coupling strength αh , similar to that at low temperatures. The initial decrease of screening at small phonon frequencies in Fig. 2c can be traced to the large nph for a small h ¯ ωl , and accordingly, a severely oscillatory potential Vs . The phonon number nph in Eq. (6) implies that even within the OP and TP, the screening potential Vs , exciton size rav , and exciton binding energy Eb depend on temperature, as shown in Figs. 3a, 3b, and 3c. With increase of temperature, Vs becomes more oscillatory, which enlarges the exciton and actually slightly increases Eb in the OP of MAPbI3 . This can be understood from the expectation value of a potential VT (r) = nph [cos(r/λ) − exp(−r/λ)] in



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(0)

the hydrogen’s 1s wave function ψ1 , (0)

(0)

hψ1 |VT |ψ1 i = 4nph f (η) ≡ 4nph

12 + 16η − η 2 − 4η 3 − η 4 , (4 + η 2 )(2 + η)2

(12)

where η = a0 /λ. Since function f (η) becomes negative when η ≥ 1.5, for a relatively large exciton, the oscillatory component makes the screening out of phase at different e-h distances and weakens the screening effect. In the TP, because of the larger exciton size, Eb increases with temperature more rapidly than in the OP, consistent with experiment. 23 It is also observed experimentally that MAPbBr3 has a much larger Eb , but weaker temperature dependence, than MAPbI3 . 22,23 Moreover, Eb in MAPbBr3 changes little from the OP to the TP. 22,23 This stark contrast between MAPbI3 and MAPbBr3 is in fact consistent with our microscopic theory. The larger Eb in MAPbBr3 , which has a similar µ as MAPbI3 , 16 means that the exciton is much more compact in size and likely smaller than λe(h) . Consequently, the phonon screening is largely disabled, leading to the weak temperature dependence of excitons in MAPbBr3 . The dipole-induced screening Vd , according to Eq. (11), is also less effective in MAPbBr3 because it is proportional to 1/Eb0 . In summary, we have developed a microscopic theory of excitons in the OP and TP of HOIPs, which systematically incorporates the screening due to the POPs and MA dipoles as well as its temperature dependence. The POPs can screen the Coulomb interaction between the electron and hole in an exciton, which, however, is incomplete when the e-h distance is comparable to the virtual radius of electron (hole). In the high-temperature TP, randomly oriented MA ions mediate an extra e-h coupling that modifies ∞ . The model quantitatively accounts for the measured Eb in MAPbI3 and consistently explains the much weaker screening effects of the POPs and MA dipoles in MAPbBr3 . We emphasize that the screening effects due to the MA dipoles and POPs are distinct: the former has a classical electrostatic origin, whereas the latter is quantum-mechanical in nature and frequently reveals itself in many important optical and transport properties, such as excitons and carrier mobility, 27 in


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2 a

e

c

0

-1

0

10

-2

10 -2

-3

b

ψ1,ψ2

2

f

d

10

0

10

-1

10

1

Oscillator strength

Potential

10

-2

10

0

-3

-2

10

-1

0

10

10

1

-2

10 10

-1

10

r12/a0

0

10

1

-2

-1

10 10

0

10

r12/ã0

10

10

En/E1

Figure 1: e-h potentials and exciton wave functions in MAPbI3 . (a) and (c) illustrate the unscreened Coulomb potential Vc (red), POP-induced screening Vs (green), dipole-induced screening Vd (blue), and the total potential (black) in the the OP at T = 2 K and in the TP at T = 161 K, respectively. The energy units in (a) and (b) are Eb0 and E˜b0 = Eb0 (∞ /˜∞ )2 . (b) and (d) plot the 1s (magenta) and 2s (organge) exciton wave functions at T = 2 K and T = 161 K, where dash lines are for the corresponding excitons with potential −e2 /(∞ r12 ) in (b) and −e2 /(˜∞ r12 ) in (d). a ˜0 = a0 ˜∞ /∞ . (e) and (f) show energies and oscillator strengths for the first five ns excitons at T = 2 K and T = 161 K with red bars representing values from the hydrogenic spectrum.

0.00

4

a

c

d

rav/λh

3 2

-0.01

-0.02

b

0 b

1

E1 (eV)

1 0

Eb/E

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-0.03

0 0.1

0.2

1/ε∞

0.3

0

0.005_ 0.01 0.015 0

hωl (eV)

0.5

αh

1

1.5

-0.04

Figure 2: Effects of exciton size, phonon frequency, and polar coupling on the screening. (a) and (b) delineate rav and Eb as a function of −1 ∞ . (c) and (d) plot energy of the 1s exciton versus phonon frequency h ¯ ωl and polar-coupling strength αh . Black and red lines in (a) and (b) are for T = 2 and 161 K, respectively, with fixed αh = 1.1 and h ¯ ωl = 16.5 meV. Black, red, and blue lines correspond to αh = 0.1, 0.5, and 1.1 in (c) and h ¯ ωl = 4.5, 10.5, and 16.5 meV in (d). Solid and dashed lines in (c) and (d) are for T = 2 and 161 K, respectively.



2.0

b

a

0 b

f(η)

1.5 0

1

rav/λh

1

1.0 -1 0

1

2

0.02

c

η

0

0.01

0

8

4

12 0

r12/a0

100

200

Eb (eV)

2

Vs/E

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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300

Temperature (K)

Figure 3: Temperature dependence of excitons in MAPbI3 . (a) illustrates the POP-induced screening Vs for T = 2 (black), 161 (red), and 302 K (orange). The inset draws f (η) in Eq. (12). (b) and (c) describe exciton size rav and binding energy Eb as a function of temperature, where the dot-dashed line marks the transition temperature between the OP and the TP. the exciting platform – HOIPs.

Acknowledgement This work was partly supported by the US Army Research Office under Contract No. W911NF-15-1-0117. Competing financial interest: The author declares no competing financial interest.

Supporting Information Available • si.pdf: Methods, derivations, and additional results.

References (1) Kojima, A.; Teshima, K.; Shirai, Y.; Miyasaka, T. Organometal Halide Perovskites as Visible-Light Sensitizers for Photovoltaic Cells. J. Am. Chem. Soc. 2009, 131, 60506051. 12 ACS Paragon Plus Environment

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Graphical TOC Entry 0.02

2

T=2K T = 161 K

e

Eb (eV)

1

Vs(r)

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h

0

0.01

0

1

r

10

0

100

200

Temperature (K)


300

Excitons in Orthorhombic and Tetragonal Hybrid OrganicâInorganic

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Excitons in Orthorhombic and Tetragonal Hybrid OrganicâInorganic