Carrier Dynamics and Evaluation of Lasing
Actions in Halide Perovskites
Jiajun Qin,
1Xiao-Ke Liu,
1Chunyang Yin,
1and Feng Gao
1,*
Metal halide perovskites have shown rapid development in variousfields such as photovoltaics, photodetectors, light-emitting diodes (LEDs), and optically pumped lasers owing to their superior optoelectronic properties. Here, we review the basic optoelectronic properties of halide perovskites from a photophysical perspective. We highlight that halide perovskites are promising in various opto-electronic devices functioning at a wide range of carrier densities. We discuss optically and electrically generated carrier density under two different excitation modes [continuous wave (CW) and pulsed] as well as the impact of carrier density on the optoelectronic behavior of perovskites. Moreover, we discuss lasing actions at high carrier densities and summarize key rules to evaluate the lasing actions. Last, we provide an outlook on perovskite-based electrically pumped lasers.
Halide Perovskites: High-Performing Materials Over a Wide Range of Carrier Densities
Metal halide perovskites, which can be expressed by the formula ABX3, comprise corner-sharing [BX6]4–octahedral frameworks embedding the A cations [1]. These materials have excellent optoelectronic properties such as a large absorption coefficient (>104cm–1), high charge carrier mobility, readily tunable bandgaps, and excellent defect tolerance [2–4]. As a result, perovskite materials perform well in various optoelectronic devices working at a wide range of carrier densi-ties (Figure 1). Specifically, perovskite photodetectors have been demonstrated with detectable light intensities ranging from subpicowatts per cm2to ~100 mW/cm2[5–8], which corresponds to carrier densities ranging from ~104to ~1015cm–3. In the carrier density region from ~1010to ~1016cm–3(usually corresponding to a light intensity of ~1μW/cm2to 1000 mW/cm2) where pe-rovskite solar cells are usually evaluated, a variety of high-efficiency perovskite solar cells have been reported [9–12]. For instance, at 100 mW/cm2(the intensity equivalent to one sun, with a carrier density usually located at ~1015to ~1016cm–3for most perovskitefilms), perovskite solar cells reach high power conversion efficiencies of >25% [13,14] (NREL efficiency chart: https://www.nrel.gov/pv/cell-efficiency.html). Perovskite LEDs can even work well at current densities over 500 mA/cm2and can further extend the working range of halide perovskites (from ~1μW/cm2to ~10 W/cm2, and in terms of carrier density ranging from ~1010to ~1017cm–3) [15–17]. When the excitation intensity goes even higher, to above 100 W/cm2[sometimes intense femtosecond laser (seeGlossary) excitation is required], lasing actions will emerge when the carrier density reaches ~1018cm–3, resulting in the realization of optically pumped perovskite lasers [18–21]. Considering the many kinds of functional devices that can work at a wide range of excitation in-tensities, halide perovskites appear to be incredibly promising materials that bring about a broad research interest in revealing and understanding the underlying mechanisms [11–26]. In this review, we focus on the carrier dynamics of halide perovskites centering around carrier density (which is positively correlated with excitation intensity) and discuss the optoelectronic behaviors of these materials under various carrier densities.
Highlights
Carrier density and carrier lifetime play important roles in determining recombi-nation dynamics, which are useful for un-derstanding the excellent optoelectrical properties and the rich applications of halide perovskites.
Guidance is provided on the calcula-tion of carrier density under both continuous-wave and pulsed excita-tions. The carrier density and carrier lifetime are correlated, which deter-mines the competition between the monomolecular, bimolecular, and Auger recombination processes. Four key factors are discussed to evalu-ate lasing actions: optical gain, spectral narrowing, carrier density, and largely re-duced carrier lifetime.
The threshold current densities required for perovskite electrically pumped lasers are potentially within reach based on the recent development by introducing pulsed voltages in perovskite light-emit-ting diodes (LEDs).
1
Department of Physics, Chemistry and Biology (IFM), Linköping University, Linköping SE-581 83, Sweden
*Correspondence: feng.gao@liu.se(F. Gao).
Carrier-Density Rate Equation and Recombination in Halide Perovskites
The change of carrier density n in halide perovskites can be described by the following rate equation [25,27]:
−dn
dt¼ −G þ an þ bn
2þ cn3; ½1
where t is the time, G refers to the carrier generation rate, and a, b, and c correspond to first-order, second-first-order, and third-order recombination rate coefficients, respectively. It should be noted that this equation considers only an ideal case in which a, b, and c are constants while the densities of electrons and holes are the same. The factors that may influence the validity of Equation1are discussed inBox 1. In this equation, thefirst-order term contains radiative recom-bination initiated by excitonic emission and/or nonradiative recomrecom-bination mainly caused by traps, quenchers, or interfacial dissociation [28,29]. In general, the value of a ranges from ~105 to ~109s–1, mainly extracted from the photoluminescence (PL) lifetimeτ at low carrier densities (where a1τ) [30–32]. The second-order term, which is also called the bimolecular recombina-tion, contains radiative free electron–hole recombination in 3D perovskites due to their small ex-citon binding energies [33,34]. Note that nonradiative bimolecular recombination also contributes to the second-order term, the mechanism of which remains unclear and requires further investi-gations [35]. Here, the value of b varies from ~10–12to ~10–7cm3s–1[35–38]. The third-order term refers to Auger recombination, which is a three-particle process that involves one electron and two holes (ehh) or two electrons and one hole (eeh) [39]. The Auger recombination rate coefficient c of most perovskites usually ranges from 10–26to 10–29cm6s–1[35,40,41], which is close to that of some inorganic semiconductors such as Si and GaAs [39].
The values a, b, and c determine that the carrier dynamics are dominated by differing processes at different carrier densities. As exampled inFigure 2A (where we set a = 106s−1, b = 10−10cm3s−1,
Glossary
Auger recombination: a three-particle nonradiative process comprising two electrons and one hole (eeh) or two holes and one electron (ehh). During the process, one electron and one hole recombine to transfer the excess energy to another particle (electron or hole) without giving off photons. Photoluminescence quantum efficiency (PLQE): refers to the optically induced photon emission event per photon absorbed by the system. Population inversion: statistical view of the states within one system, such as a group of atoms or molecules. Normally in thermal equilibrium, the population of excited states is lower than that of ground states. When the population of excited states exceeds the population of ground states, we call it population inversion.
Stimulated emission (SE): light emission occurs when an incoming photon (the stimulus) of a specific frequency interacts with an excited electron. The emitted photons share the same information in phase, frequency, polarization, and propagation direction as the incoming photons.
Trap filling: a process related to free carriers and trap states. When a trap state is unoccupied, it can become a nonradiative recombination center. The filling process occurs when the trap is filled with a charge carrier; thus, the trap becomes inactive.
Perovskite photodetectors
Perovskite solar cells Perovskite LEDs Perovskite lasing Excitation intensity ~1010 Carrier density (cm–3) ~104 ~1013 ~1016 ~1019 1 pW/cm2 1 µW/cm2 1 mW/cm2 1 W/cm2 1 kW/cm2 Trends Trends inin ChemistryChemistry
Figure 1. Schematic Diagram Showing the Applications of Halide Perovskites and the Related Excitation Intensity as well as Carrier Density Regions.Specifically, perovskite photodetectors (red area) work at carrier densities from ~104
cm–3to ~1015
cm–3(corresponding to photoexcitation intensities from sub-pW/cm2
to ~100 mW/cm2 ). The characterizations of perovskite solar cells (orange area) locate at carrier densities from ~1010
cm–3to ~1016 cm–3 (corresponding to illumination intensities from ~1μW/cm2
to 100s of mW/cm2
). Furthermore, the emission of perovskite light-emitting diodes (LEDs) (green area) ranges from ~1010cm–3to ~1017cm–3in terms of carrier densities (corresponding to emission power intensities between ~1μW/cm2
and ~1 W/cm2
). For perovskite lasing actions (purple area), much higher carrier densities are required; typically, >1018
and c = 10−28cm6s−1), all of the individual recombination rates (first, second, and third order) in-crease with increasing carrier density. At low carrier density (<1014cm–3), thefirst-order recombi-nation possesses nearly 100% of the total recombirecombi-nation. As the carrier density increases, the ratio of the bimolecular recombination (second order) grows and becomes dominant at a carrier density of ~1017cm–3, where the ratio of the bimolecular recombination is larger than that of the other two. Further increase of carrier density leads to dominant Auger recombination (third order). Therefore, the carrier density plays an important role in regulating the carrier dynamics of perovskite materials. To have a clearer picture of how carrier density influences carrier recombination, we present a cartoon inFigure 2B to summarize the rate equation by an analogy of‘water’ in a ‘bucket’. The water canfill and leak, which are analogous to the carrier injection process through optical or elec-trical excitation and the carrier recombination process, respectively. The carriers may be lost
Box 2. Commonly Used Transient Techniques
Transient absorption (TA) measurement refers to the characterization of absorption changes in the absorbance/transmittance mode of the sample after pumping with a femtosecond laser pulse. In a typical experiment, two pulses are required: one for excitation (called the‘pump’, exerted at time 0) and one for absorption measurement (called the ‘probe’, exerted at controllable delay time t after the pump). The probe beam is the white light produced from supercontinuum generation through a nonlinear optical process, to distinguish it from typical incoherent light sources. Carriers are optically injected through the pump beam. After controllable delay time t, the probe beam is applied to characterize the absorption change. This change signal correlates with the density of states change at the corresponding wavelength, which reflects the carrier density n. By tracing the dynamic TA signal at wavelengths corresponding to carrier recombination, the n-t curve can be plotted [35,91,92].
Ultrafast optical-pump–THz-probe spectroscopy is also a kind of TA measurement with short pulses of terahertz radiation as the probe, which can be used to track the concentration of free carriers. The more free carriers that are induced with the pump pulse, the more the THz probe is absorbed by the free-carrier absorption mechanism. It is also possible to track the exciton dynamics due to differing THz absorption for bound excitons. Therefore, optical-pump–THz-probe spectroscopy can be applied to reveal the dynamics of free carriers or bound excitons. Photocurrent inside the semiconductor will lead to electromagnetic radiation at THz frequencies after femtosecond pulse excitation, which is measured through the THz emission technique. The photocurrent can arise due to drift (built-infield necessary), diffusion (high carrier concentration gradients are necessary), or the ballistic movement of charge carriers [93–96]. Notably, the THz emission technique is used to track the n-t curve if photocurrent can be detected.
trPL measurements use only one pump beam without the probe beam, and PL signal is detected after a variable time delay from the pump. The PL intensity has contributions from the radiative recombination of total carriers, which can be either monomolecular (excitonic recombination) or bimolecular (free electron–hole recombination). Therefore, the IPL-t curve can be used to reflect the carrier dynamics (n-t curve). As in the example inFigure 4C in main text at low pumpfluence (<1015
cm–3, n ¼ n0e−tτ), the trPL is single exponential decay both for radiative monomolecular recombination (IPL∝ n ¼ n0e−tτ) and radiative bi-molecular recombination (IPL∝ n2¼ n02e−2tτ).
Box 1. Factors Not Included in the Carrier-Density Rate Equation (Equation1)
In Equation1, the electron density is assumed to be equal to the hole density (n = ne= nh), which is usually realized through optical excitation or balanced electrical injection. However, exceptions exist when we consider the nonnegligible intrinsic or doping concentration terms [77]. Moreover, electrons and holes can be unbalanced when the majority of electrons or holes are trapped due to trap filling processes. Notably, in the low carrier density region, PL lifetime increases with increasing ex-citation intensity by considering trapfilling processes [23,78,79]. As the optical excitation increases, electrons and holes tend to be equal due to limited carriers for trapfilling. For the electrical excitation case, it is more complicated even if the trap filling process is not considered. For instance, balanced electron and hole injection can be easily obtained at low operational bias through device engineering, but it is difficult to maintain the balanced injection when the operational bias is very high [80,81]. Therefore, Equation1is just a simplified form of rate equation by neglecting the intrinsic carrier densities and doping densities and assuming equal injection of electrons and holes [25]. In addition, the major discussions in this review present an ideal case when the three parameters a, b, and c are constant with respect to carrier density. However, in practical cases these parameters could vary with carrier density. Specifically, the bimolecular recombination term (bn2
) can be viewed as a pseudo-first-order process when the doping density or intrinsic carrier density is much larger than the injected carrier density [77]. In addition, in perovskite quantum-confined systems such as nanowires or quantum dots [82–87], the large exciton binding energy leads to dominant bound excitons rather than free carriers. Thus, Equation1must be rewritten due to the existence of other complex processes such as exciton–exciton annihilation and biexciton emission [88–90].
through both radiative and nonradiative paths. Specifically, at low carrier density (usually below 1015cm–3), the main leakage of carriers comes from excitonic radiative recombination and trap-assisted Shockley–Read–Hall (SRH) nonradiative recombination [42–44]. At medium carrier density (usually ~1015to ~1017cm–3), bimolecular recombination appears to be the dominant ‘water leakage’ mechanism. At high carrier density (usually >1017cm–3for optical excitation and maybe even lower for electrical excitation), Auger recombination grows, serving as the main nonradiative loss [45–49]. Interestingly, many perovskites are good gain media, which can show stimulated emission (SE) at carrier densities over ~1018cm–3[18].
Calculating Carrier Density under CW and Pulsed Excitation
Carrier Lifetime versus Carrier Density
To determine how fast the recombination rate is, the carrier lifetime,τcarrier, is introduced through nondimensionalization of Equation1. It can be expressed as
τcarrier¼
1
aþ bn þ cn2: ½2
In Equation2, the carrier lifetime is inversely proportional to the carrier recombination rate coefficient, which is determined by three effective lifetimes: the monomolecular recombination lifetime (τa=1a); the bimolecular recombination lifetime (τb=bn1); and the Auger recombination lifetime (τc=cn12) [50]. It should
be noted that the total carrier lifetime is no longer than any of the three individual lifetimes. As shown in
Figure 3A, the monomolecular recombination lifetime is constant while the other two decrease with increasing carrier density. Especially, the Auger recombination lifetime decreases more significantly with increasing n than that of the bimolecular recombination lifetime. By comparing the three lifetimes, the dominant recombination process at a given carrier density can be identified. For example, at a car-rier density as low as 1013cm–3, the shortest lifetime is the monomolecular recombination lifetime, while the bimolecular recombination and Auger recombination lifetimes are several orders of magni-tude larger. This implies that the recombination incidence for monomolecular recombination is much larger than that of the other two terms, and thus nearly all recombination will be monomolecular. Similarly, when the carrier density is ~1017cm–3, the three lifetimes are of similar magnitude. In this
1013 1013 1010 1012 1014 1016 1018 0 50 100 ) %( oit a R Carrier density (cm–3) c=10-28 b=10-10cm3s-1 a=106s-1 1010 1012 1014 1016 1018 1015 1018 1021 1024 1027 m c( t d/n d-–3 s –1) Carrier density n (cm–3) cn3 bn2 an c = 10–28 cm6s–1 b = 10–10 cm3s–1 a = 106s–1 SRH SE Auger Exciton Carrier injection 1015 1017 1019 Radiative Nonradiative
(A)
(B)
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Figure 2. Recombination Rate Contributions from First-, Second-, and Third-Order Terms.(A) Ratios of the three recombination terms as a function of the carrier density. Inset shows the three recombination rates as a function of the carrier density. The ratio refers to the proportion of the recombination rate in the total recombination rate. Here, we set a = 106
s−1, b = 10−10cm3
s−1, and c = 10−28cm6
s−1. (B) Cartoon showing the recombination dynamics. The carrier density is analogous to the water inside the bucket and the water leakage is analogous to carrier recombination. Different recombination processes dominate different carrier densities. Leakages on the left of the bucket are nonradiative losses, which do not contribute to light emission, while leakages on the right are spontaneous emission and stimulated emission (SE).
case, all three recombination processes play key roles. If the carrier density is even higher (usually >1018cm–3), the Auger recombination lifetime is the shortest, leading to dominant recombination from the Auger term. The above analysis in view of the recombination lifetime is consistent with
Figure 2. Moreover, the changes of the lifetime with carrier density have been verified by various ex-perimental techniques, such as transient PL in quasi-2D perovskites [37].
Carrier Density under CW Excitation
Under CW excitation, equilibrium is realized when the carrier generation rate is equal to the carrier recombination rate (−dn
dt¼ 0). Thus, the carrier-density rate equation can be expressed as
G¼ an þ bn2þ cn3¼ n τcarrier: ½3 10–4 10–2 100 102 104 106 108
PL
P
(mW/cm
2)
Monomolecular recombine slope = 1 Bimolecular recombine slope = 2 10–4 10–2 100 102 104 106 108 10–7 10–6 10–5 10–4 10–3 10–2 10–1 100 101E
Q
L
P
.
mr
o
N
P
(mW/cm
2)
1012 1015 1018 10–510–310–1101103105107 1091011 1012 1015 1018m
c(
n
–3)
b = 10–10cm3s–1 c = 10–28 cm6s–1 a (s–1) 105 106 107 c (cm6s–1) 10–28 10–27 10–26P (mW/cm
2)
a = 106 s–1 b =10–10 cm3s–1 1013 1015 1017 1019 1 ps 1 ns 1 µs 1 ms 1 s 1 kse
mit
efi
L
n (cm
–3)
(A)
Monomolecular(C)
(D)
(B)
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Figure 3. Relationship between Carrier Lifetime and Carrier Density.(A) Lifetime estimations of three recombination terms (green, monomolecular recombination; red, bimolecular recombination; purple, Auger recombination) and the total carrier recombination (yellowish green) as a function of the carrier density. Here, a ranges from ~105to ~109s–1, b ranges from ~10–12to ~10–7cm3
s–1, and c ranges from 10–26to 10–29cm6
s–1. Parameters a, b, and c taken from [30–32,35– 38,40,41]. (B) Estimated carrier density (n) as a function of continuous-wave (CW) excitation power. Top-panel and bottom-panelfigures show the influence of a and c on the carrier density, respectively. (C) Logarithmic power-dependent photoluminescence (PL) intensity under CW laser excitation. (D) Normalized PL quantum efficiency (PLQE) as a function of CW power intensity. (C,D) Black-square curves (red-circle curves) refer to the case where the radiative recombination is monomolecular (bimolecular). Here, we set a = 106
s−1, b = 10−10cm3
s−1, and c = 10−28cm6
s−1. (B–D) When calculating the carrier density of a perovskitefilm, we set the following film parameters: excitation wavelength, λ = 405 nm; reflectance, R = 0.2; absorption coefficient, α = 2 × 104
The average carrier generation rate produced through CW optical excitation, G, can be expressed as G¼ 1−Rð Þ 1−e−αd P d∙hc λ ; ½4
where R is the reflectance, α is the absorption coefficient, d is the thickness of the perovskite thinfilm, P is the power intensity of the excitation at wavelength λ, h is Plank’s constant, and c is the speed of light [35]. Notably, owing to the Beer–Lambert law, the excitation light decays exponentially through the perovskitefilm along its direction of travel, yielding photogenerated carriers distributed mainly within the distance below the penetration depth L¼α1. Equation4
is applicable when thefilm is thin enough (i.e., d < L). Otherwise, G must be modified by replacing d with L, where we consider that the majority of the carriers are located on top of thefilm or crystal [51,52].
Figure 3B shows an example of the calculated carrier density at different CW excitation power intensities based on the different recombination rate constants (we setλ = 405 nm, R = 0.2,α = 2 × 104cm–1, d = 100 nm). We should point out that decreasing the value of a (usually realized by decreasing defect densities in experimental conditions) can lead to increased carrier density when P is very low. At higher ranges for lasing (usually n ~ 1018cm–3 for 3D perovskites [18]), the key factor that limits the carrier density under CW excitation originates from the Auger recombination coefficient c. It is also clear that the carrier density within perovskite films is proportional to P at low CW excitation intensities (usually lower than ~1 to ~1000 mW/cm2
). In this region, the radiative recombination type can be identified through the logarithmic slope of the CW power-dependent PL intensity (Figure 3C), whereas slopes of 1 and 2 refer to excitonic (monomolecular) and free electron–hole (bimolecular) recombination, respectively [37,53–56]. However, when P increases, the second-order term (bn2) becomes comparable with the first-order term (an), resulting in the nonlinear relationship between P and n (Figure 3B). Therefore, the linear nature of logarithmic power-dependent PL plots is typically lost at higher CW excitation intensities [57]. Since monomolecular and bimolecular recombination processes dominate at different carrier density regions, the PL quantum efficiency (PLQE) of perovskites with monomolecular radiative recombination behaves differently compared with those with bimolecular radiative recombination. The maximal PLQE of the former appears at very low CW excitation intensities and relatively high inten-sities for the latter (Figure 3D), which is consistent with reported power-dependent PLQE results [17,37,58]. Thus, two practical characterization methods are proposed to deter-mine the radiative recombination type (monomolecular or bimolecular): power-dependent PL intensity measurements at low CW excitation intensity and power-dependent PLQE measurements.
Carrier Density under Pulsed Excitation
In addition to CW excitation at equilibrium, pulsed excitation (where the carrier density varies with time) is also frequently used. When the time evolution of the excitation power intensity is tracked, it appears to be in pulses of some durationτpulseat some repetition frequency f for pulsed excita-tion. Thus, the carrier generation rate G is nonzero only when t is within the region of the pulse. If the pulse duration is much longer than the carrier lifetime (τpulse>>τcarrier), equilibrium can be established in the duration. Therefore, long-pulse-duration excitations can be regarded as quasi-CW excitations due to the same carrier dynamics as in CW cases [21]. Ifτpulse<<τcarrier, carrier recombination is negligible compared with the carrier generation rate, G, during this
pulse duration. Therefore, the carrier density right after the pulse is attributed only to the carrier generation under short-duration pulsed excitation:
n¼ 1−Rð Þ 1−e−αd Ips d∙hc
λ
; ½5
where Ipsis the pumpfluence of a single pulse (usually in units of J/cm2) [35]. Similar to Equation4, thefilm thickness d must be replaced by the penetration depth, L, when d > L. It is worth noting that the initial generation rate varies with the penetration length of the light according to the Beer– Lambert law. Here, the carrier density is regarded as a uniform distribution when thefilm is thin enough by considering the carrier diffusion process. Considering that the carrier lifetime is usually longer than subnanoseconds, the carrier density calculation of Equation5is usually applicable for femtosecond (τpulse~ 150 fs) and picosecond (τpulse~ 100 ps) lasers. For pulsed excitation of a duration comparable with the carrier lifetime, the carrier density should be derived from Equation1
by considering both the carrier injection and the carrier recombination simultaneously within the duration.
Transient Techniques to Reveal Carrier Dynamics
To reveal carrier dynamics, several transient techniques have been applied wherein the perov-skites are excited with a short pulse, followed by characterization of the sequent optoelectronic properties (seeBox 2).Figure 4A provides an example of temporal carrier density change (n-t curve) after initial carrier injection at t = 0. During the carrier decay process, only carrier recombi-nation rates are considered (i.e., G = 0). If the initial carrier injection is high enough to render Auger recombination dominant, the n-t curve will undergo three processes with time: Auger recombina-tion dominant, followed by bimolecular recombinarecombina-tion dominant, followed by monomolecular re-combination dominant. Notably, perovskites differ significantly on the three recombination rate coefficients; thus, the corresponding optoelectronic properties are quite different. As shown in
Figure 4B, perovskite dimensionality can be tuned from 3D to 2D or 0D by controlling the fabrica-tion condifabrica-tions. As the dimensionality decreases, the exciton binding energy increases, so that the contribution of excitonic emission will increase [36].
Figure 4C shows an example of how initial carrier density, which is tuned by changing the pumpfluence, can affect transient PL (trPL) results. When the initial carrier density is low (usually <1015cm–3), the normalized trPL curve remains unchanged because monomolecular recombina-tion withfixed carrier lifetime is dominant at the low carrier density region, as shown inFigure 3A. When the initial carrier density increases, the carrier lifetime decreases due to growing contributions from bimolecular or Auger recombination, showing faster PL decay at initial time. After the carrier density decays to the low-density region with monomolecular recombination being dominant, the line shape of PL decay will be the same as the case with very low initial carrier density. Therefore, we can expect parallel curves for different initial carrier densities at longer time (e.g., the region with t > 2μs inFigure 4C). If the recombination rate coefficients a, b, and c are the same, free electron–hole recombination leads to faster PL decay than excitonic recombination. Note that trPL results are usually applied to compare defect densities of different perovskitefilms, where longer PL lifetime is used to indicate lower defect density [59–61]. Essentially, this argument is valid only when we compare PL lifetimes at long times or take measurements with reasonably low excitation power densities to enable monomolecular recombination dominant processes. Evaluation of Lasing Actions
Halide perovskites have emerged as promising gain media for solution-processed semiconduc-tor lasers owing to their large absorption coefficient (>104
loss (c usually in the range of 10–26to 10–29cm6s–1) [35,40,41]. Lasing physics centering around materials engineering, structure modulation, and light–matter interaction has been systematically reviewed elsewhere [62,63]. In this section, we aim to provide a general view to identify lasing ac-tions through carrier dynamics analysis.
For lasing actions, population inversion should be reached, which usually requires a carrier den-sity as high as 1018cm–3in most halide perovskites [18]. Under these conditions, the major nonradiative recombination loss becomes the Auger recombination, with the Auger lifetime as fast as several nanoseconds or even below. From the materials-design point of view, the Auger re-combination rate should be reduced to reach a low lasing threshold. Regarding the lasing action, the SE lifetime should be faster than the Auger lifetime based on the analysis inFigure 3A. There-fore, the dominant recombination process becomes the SE process, with which the recombination rate increases much faster with increasing carrier density than the other three recombination processes (monomolecular, bimolecular, and Auger recombination). As a result, the power-dependent PL slope will increase above the lasing threshold. Consistent with our previous calculation, the threshold carrier density for lasing can be achieved either through pulsed excitation with an excitation intensity of a fewμJ/cm2to a few tens ofμJ/cm2[64] or through CW excitation with an excitation intensity of a few kW/cm2at room temperature [65,66] for typical perovskitefilms.
10–5 10–4 10–3 10–2 10–1 100 0 1 2 3 4 5 10–5 10–4 10–3 10–2 10–1 100 yti s n et ni L P . mr o N 1019 10 18 1017 1016 1015 1014 1013
I
PL=
a′n
Increase initialcarrier densitiy (unit: cm–3)
Increase initial
carrier densitiy (unit: cm–3)
I
PL=
b′n
2 Delay (μs) 1013 1014 1015 1016 1017 1018 10193D
2D
0D
Higher binding energy
Free carriers Excitons
0 1 2 3 n (cm –3 ) Delay (μs)
(A)
(B)
(C)
n n2 n3 n > ~1017cm–3 n n2 ~1013cm–3<n < ~1018cm–3 n n < ~1016 cm–3 − = + + TrendsTrends inin ChemistryChemistry
Figure 4. Carrier Dynamics at Different Initial Carrier Densities.(A) Schematic diagram to show that the carrier density varies with time with an initial carrier density of 1019
cm−3. It undergoes three processes with time: Auger recombination dominant, followed by bimolecular recombination dominant, followed by monomolecular recombination dominant. (B) Schematic diagram to show dimensionality tuning of perovskites from 3D to 2D and 0D. With the dimensionality decreasing, the exciton binding energy increases. (C) Normalized transient photoluminescence results at different initial carrier densities ranging from 1019
cm−3to 1013
cm−3. Here, we set a = 106
s−1, b = 10−10cm3
s−1, and c = 10−28cm6 s−1. For thefigure in the top panel, the radiative recombination is set to be monomolecular recombination (IPL∝ n, with a′ as a constant) only; for thefigure in the bottom panel, the radiative recombination is set to be bimolecular recombination (IPL∝ n2, with b′ as a constant) only.
Figure 5A shows an example to identify the lasing actions of perovskite nanowires through steady-state PL results. Two features are observed: (i) the increase of the excitation power-dependent PL slope after a threshold to show optical gain properties; and (ii) spectrum narrowing to show coherent emission. Strictly speaking, very narrow spectral width (<1 nm) and cavity-related emission are required for a real laser [67]. The laser cavity is an optical resonator that forms a stand-ing wave for a specific emission wavelength. The vertical-cavity surface-emitting laser (VCSEL) and the distributed feedback laser (DFB) are commonly used feedback structures for perovskite laser devices [68,69]. However, the gain property is not obvious through power-dependent PL results in some situations, which usually causes discussion on whether lasing occurs under certain exci-tation powers [70,71]. Here, we strongly suggest adding excitation-intensity-dependent trPL re-sults together with the respective values of carrier density to confirm lasing actions. According to the carrier lifetime analysis, by including SE in the carrier-density rate equation, the lifetime for SE (τSE) must be faster than the other three recombination lifetimes, including the Auger lifetime (τAuger), when the carrier density is above the lasing threshold. Therefore, carrier lifetime will de-crease significantly when SE appears. As in the example of perovskite nanowire inFigure 5B, the PL lifetime is∼150 ns at very low excitation power. When the excitation power increases to 15% below the lasing threshold, the PL lifetime decreases to∼5.5 ns, which may be caused by an increasing contribution from the second- or third-order recombination and consistent with our analysis inFigure 3. When the lasing action is realized (the excitation power is only 10% higher
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Figure 5. Lasing Actions and Related Lifetimes.(A) An example to show lasing characteristics based on hybrid perovskites: narrow spectral width and optical gain. Inset shows the full width at half maximum (FWHM) of the photoluminescence (PL) spectra and PL intensity as a function of the excitation intensity. (B) Normalized transient PL results after photoexcitation withfluence far below the lasing threshold PTh(<<PTh, black) and below (P∼ 0.85PTh, blue) and above (P∼ 1.1PTh, red) the threshold. With increasing photoexcitationfluence, the PL lifetime decreases. When the lasing process occurs, the decrement is more significant and goes to ≤20 ps. (C) Amplified spontaneous emission (ASE) spectra for perovskites can be obtained from 400 nm to 800 nm. The peak position is tuned through halide composition engineering. (D) Schematic diagram to characterize carrier dynamics under intense electrical excitation. (A,B) adapted from [19]; (C) adapted from [18]; (D) adapted from [70].
than the threshold), the PL lifetime decreases sharply to below 20 ps [19]. This severely shortened lifetime, together with spectral narrowing and the optical gain character (Figure 4A), provide strong evidence for lasing.
Perspectives for Electrically Pumped Perovskite Lasers
Due to the high tunability of halide hybrid perovskites, SE at various wavelengths covering a wide range (from 390 to 800 nm) can be realized (Figure 5C) [18]. Especially, the success of CW opti-cally pumped lasers [21] has triggered great research interest in the development of perovskite-based electrically pumped lasers [15,72,73]. Here, we provide some perspectives on salient op-portunities and challenges.
First, as indicated by the low lasing threshold (~1018cm–3) [18], the minimum threshold current density for a typical perovskite electrical pumped laser with an active layer thickness of 100 nm is estimated to be as low as ~1.6 kA/cm2from
n¼jτcarrier
qd ; ½6
where j is the current density and q is the elementary charge. It should be noted that the estimation is an ideal case when balanced charge carrier injection is achieved with a carrier lifetime of ~1 ns. Based on our previous discussion, the lifetime will decrease very quickly with increasing carrier den-sity when lasing action occurs, making it difficult to increase carrier density through electrical injec-tion. Moreover, in real devices, Joule heating is more serious at high current densities and the rate of nonradiative recombination loss will also be enhanced, resulting in a further increase of the threshold current density. In addition, the continuous Joule heating under direct circuit (DC) bias causes serious degradation, making it difficult to maintain high balanced carrier injection for lasing actions. To reduce heat accumulation with time, pulsed voltages have been applied, which enables sufficient time for heat dissipation during the OFF period (voltage = 0). Similar to optically the pumped case, the carrier density under pulsed voltage is determined by
n¼jτpulse
qd ; ½7
where this equation is valid only whenτpulse<<τcarrier. Ifτpulse>>τcarrier, which means that the pulsed voltage is close to the CW case, the carrier density should still be derived through Equation6.
Second, severe current efficiency roll-off is usually observed at high current densities, owing to unbalanced charge carrier injection and Joule heating, which always lead to dominant unwanted nonradiative recombination loss. Therefore, the poor efficiency at high injection current densities provides additional challenges for electrically pumped lasers. To solve this issue, a good device structure with balanced injection through both anode and cathode at extremely high current den-sity is required. For instance, combined measures of careful addressing of proper transport layers, the use of thermally conductive sapphire substrates, decreased cell size, and the applica-tion of pulsed operaapplica-tions have led to decent external quantum efficiencies (EQEs) at ultrahigh cur-rent densities of near 1 kA/cm2in both iodide [74] and bromide [15] -based perovskite LEDs.
Third, carrier lifetime characterization under electrical operation is strongly suggested to evaluate the possibility of electrically pumped lasing actions.Figure 5D shows an example of such a measurement, where the trPL curve is tracked by applying an optical pulse during the electrical operation [72]. When the carrier density at electrical injection exceeds the lasing threshold, the PL lifetime will decrease significantly with increasing carrier density, owing to the dominant
recombination process of SE. Otherwise, the decrease with increasing carrier density will follow the lifetime trend of Auger or bimolecular recombination processes.
The earlier discussion mainly focuses on continuous 3D polycrystalline perovskitefilms, where the injected carrier density is regarded as uniformly dispersed. When the dimensionality is decreased to 2D or quasi-2D, carriers tend to be cascaded into small-gap phases within the film, leading to carrier accumulation inside [75]. It provides the opportunity to realize population inversion at a relatively low optical excitationfluence toward lasing actions. However, under electrical operation the current density could be limited in these low-dimensional perovskites owing to the insulating long-chain organic cations [76]. For 1D perovskite nanowires, the main challenge is to achieve an efficient LED device; the device in this case is often limited by poor coverage [19,64]. For 0D perovskite quantum-dot-based LEDs, although we can achieve high-performance devices working at a relatively low carrier density region, serious Auger loss or exciton–exciton quenching currently limits their application toward electrically pumped lasers [47].
Concluding Remarks
Metal halide perovskites have shown excellent performance in various optoelectronic devices that work under different excitation intensities. This feature highlights the importance of under-standing the carrier dynamics of these materials (see Outunder-standing Questions). For perovskites, monomolecular, bimolecular, and Auger recombination processes are usually considered, whose rates are strongly dependent on the carrier density. We highlight that the carrier density plays an important role in regulating these three processes in perovskite materials. The monomolecular, bimolecular, and Auger recombination processes dominate in the carrier density ranges of <1014, ~1015to ~1017, and >1017cm–3, respectively. Moreover, we have provided guidance on the calculation of carrier density under both CW and pulsed excitations and have discussed the relationship between the carrier density and carrier lifetime. We also have shown that transient techniques with ultrafast pulsed excitation present direct observations of carrier densities as well as carrier lifetimes to reveal photophysical processes such as Auger recombina-tion and SE. The characterizarecombina-tion of carrier density and carrier lifetime is strongly encouraged when reporting the lasing actions of perovskites since they provide solid evidence of SE. In addition, we have evaluated the current density required for perovskite electrically pumped lasers, which is potentially within reach by combining a suitable cavity, optimal device structure, a low-threshold perovskite layer, and pulsed voltage operation, although challenges remain regarding the charac-terization and evaluation of electrically pumped lasers.
Acknowledgments
We thank Tiankai Zhang and Ignas Nevinskas for helpful suggestions and valuable discussions. We acknowledge the support from the ERC Starting Grant (No. 717026), the Swedish Energy Agency Energimyndigheten (No. 48758-1 and 44651-1), the Swedish Foundation for International Cooperation in Research and Higher Education (No. CH2018-7736), and the Swedish Government Strategic Research Area in Materials Science on Functional Materials at Linköping University (Faculty Grant SFO-Mat-LiU No. 2009-00971). F.G. is a Wallenberg Academy Fellow.
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To reduce the threshold for perovskite lasing, it is important to reduce the Auger recombination loss. What is the most effective way to measure the Auger recombination lifetime? What are the factors that can influence Auger recombination from the materials- design point of view? What are the origins of the current efficiency roll-off in perovskite LEDs? If we attribute the efficiency roll-off to Auger recombination, what is the Auger lifetime or recombination rate coefficient?
Can the carrier lifetime be prolonged to 1 ns or even longer when lasing actions appear? The prolonged carrier lifetime will benefit the decrease of lasing threshold under CW excitation. What are the carrier dynamics under electrical excitation? Will the external electricfield affect carrier recombination? Will there be a difference between the lasing thresholds for optical pumping and electrical pumping?
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