# Dynamic nsNet2: Efficient Deep Noise Suppression with Early Exiting

###### Abstract

Although deep learning has made strides in the field of deep noise
suppression, leveraging deep architectures on resource-constrained devices
still proved challenging. Therefore, we present an early-exiting model based
on nsNet2 that provides several levels of accuracy and resource savings by
halting computations at different stages. Moreover, we adapt the original
architecture by splitting the information flow to take into account the
injected dynamism. We show the trade-offs between performance and
computational complexity based on established metrics.

Index Terms—  Deep Noise Suppression, Dynamic Neural Networks, Early-exiting

††This research has received funding from the European Union’s Horizon
research and innovation programme under grant agreement No 101070374.

IEEEexample:BSTcontrol

##  1 Introduction

In recent years, audio products such as earbuds, headsets, and hearing aids
have become increasingly popular and are driving the demand for real-time
solutions capable of improving speech quality. Noise suppression techniques
based on deep learning — also referred to as Deep Noise Suppression (DNS) —
are superseding conventional speech enhancement techniques based on digital
signal processing. This is thanks to their ability to effectively reduce non-
stationary noise and deal with diverse background sounds. Several DNS models,
in particular those capable of real-time causal inference, rely on Recurrent
Neural Networks (RNNs) [1, 2, 3]. Such approaches usually involve feeding a
frequency-domain signal — in the form of Short-time Fourier Transform (STFT)
or Mel spectrograms — one frame at a time, into an encoding layer and one or
more recurrent layers such as Gated Recurrent Units (GRU), thereby providing
temporal sequence modelling abilities. This internal representation is then
decoded to form a time-varying filter, which is applied to the noisy input to
obtain a clean speech estimate.

Unfortunately, such architectures are not particularly suited for resource-
constrained devices due to the large amount of computation required by the
recurrent units. On that note, several efforts have been put in trying to
optimise and deploy recurrent models on embedded hardware [4, 5] with
techniques such as quantization. Nevertheless, one fundamental aspect remains
unaddressed, i.e., the model requires the same amount of computation
irrespective of energy/power requirements, initial speech quality of the input
signal, and desired output speech quality.

To address this issue, Dynamic Neural Networks (DyNNs), also known as
conditional neural networks, are one path to undergo [6]. DyNNs can adaptively
adjust their parameters as well as their computational graph, based on the
input they receive. This dynamism grants the network more flexibility and
efficiency in handling various resource budgets, real-time requirements, and
device capacities while maintaining a good performance trade-off. Amongst the
most promising techniques for DyNNs that appear suitable for addressing
limited hardware resources, we find early exiting [7, 8, 9]. Early exiting was
introduced in the context of image classification [10, 11]. It appends the
architecture with decision blocks (e.g., internal classifiers) that decide
when to halt the computations based on the output of each exit stage (e.g.,
image class probability). This paradigm helps avoid using the full-fledged
network, but various challenging aspects emerge from these new types of
architectures. For instance, the absence of feature sharing between internal
decision blocks can lead to computational waste [12, 13]. Another challenge
lies in the optimal placement of the decision blocks in the architecture based
on either performance gain or loss minimisation [14, 15]. Moreover,
convergence is also subject to instability due to the accumulation of
gradients coming from the different decision blocks, which can be mitigated by
gradient re-scaling [16] or a mixture of training strategies [17].

In this work, we introduce a novel application of DyNNs to noise suppression.
We adapt nsNet2 [1], a popular architecture for noise suppression, into a
model capable of early exiting given user-predefined constraints. We then
demonstrate that the resulting architecture achieves a monotonic increase in
its denoising capabilities with each consecutive exit stage. Our architecture
allows the user to choose the denoising/computational cost trade-off that best
suits their needs. Due to the additional challenges associated with
unintrusively modelling speech quality in real-time, automatic exiting is
beyond the scope of this work. In this paper:

  * •

We convert nsNet2 into an early-exiting model and explore several
architectural adaptations aimed at maintaining its denoising capabilities at
each stage;

  * •

We investigate the impact of different early exit training strategies (layer-
wise or joint) on the denoising performance of the models;

  * •

We evaluate the speech quality and computational efficiency of our models for
each exit stage and show that our architectural adaptations decrease
computational cost without degrading baseline performances.

##  2 Deep noise suppression

We assume that the observed signal $X(k,n)$ — defined in the STFT-domain where
$k$ and $n$ are the frequency and time frame indices, respectively — is
modelled as an additive mixture of the desired speech $S(k,n)$ and interfering
noise $N(k,n)$, expressed as:

$X(k,n)=S(k,n)+N(k,n)$ (1)

To obtain the clean speech estimate $\widehat{S}(k,n)$, we compute a real-
valued suppression gain spectral mask $\widehat{M}$ such that:

$\widehat{S}(k,n)=X(k,n)\cdot\widehat{M}(k,n)$ (2)

This can be formulated as a supervised learning task, where a mask
$\widehat{M}(k,n)$ is learned to minimise the distance between $S$ and
$\widehat{S}$.

###  2.1 Dynamic Architecture

$\mathbf{X}$log-
powFC1GRU1GRU2FC2FC3FC4$\mathbf{\widehat{M}_{0}}$$\mathbf{\widehat{M}_{1}}$$\mathbf{\widehat{M}_{2}}$$\mathbf{\widehat{M}_{3}}$$\mathbf{\widehat{M}_{4}}$$\mathbf{\widehat{M}_{5}}$$\times$$\mathbf{\widehat{S}_{0}}$

Fig. 1: nsNet2 architecture with exit stages (dotted lines show an example of
full inference path)

As our baseline, we use nsNet2 [1], an established DNS model with the
following architecture comprising recurrent and fully-connected (FC) layers:
FC-GRU-GRU-FC-FC-FC. Each fully-connected layer is followed by a ReLU
activation, except for the last one, which features a sigmoid nonlinearity.
The model operates on real-valued log-power spectrograms, computed as
$\log({|X|^{2}+\varepsilon})$, for a small $\varepsilon$.

We introduce exit stages after each layer, as shown in Fig. 1. Each exit
outputs a mask $\widehat{M}_{i}$, which is a subset of that layer’s
activations with size given by the number of frequency bins, to be applied to
the noisy input (chosen, in our case, as the first 257 features). Since the
suppression gains are bound between 0 and 1, we use the sigmoid function to
clamp our FC activations — i.e., the outputs of layers FC1, FC2, FC3 —
similarly to how the baseline model implements its last activation (i.e., the
output of layer FC4). For the GRU layer activations, which are inherently
bound between $-1$ and $1$ due to the final $\tanh$ activation, we employ the
simple scaling function $0.5\cdot(1+\text{GRU}_{i}(x))$. Note that these extra
steps are only performed when there is a need to extract the mask at the early
stage, otherwise, we use the activations mentioned earlier. This results in an
architecture able to recover the signal with up to 6 different denoising
abilities.

Given the significant differences between the masks and the model’s internal
representation, forcing our model to derive the former at each layer may
degrade its performance. We address this by reducing the number of
intermediate exit stages to promote the emergence of richer internal feature
representations in non-exiting layers. Thus, we introduced a version of our
dynamic model with only the 4 exits marked with $\mathbf{\widehat{M}_{0}}$,
$\mathbf{\widehat{M}_{1}}$, $\mathbf{\widehat{M}_{3}}$,
$\mathbf{\widehat{M}_{5}}$ in Fig. 1. We decided to remove later-occurring
exits because, during our experimental observations, they were more prone to
performance degradation.

###  2.2 Split Layers

FC1FC${}^{*}_{1}$concatGRU1GRU${}^{*}_{1}$$\mathbf{\widehat{M}_{0}}$$\mathbf{\widehat{M}_{1}}$

(a) Regular split layer

FC1FC${}^{*}_{1}$concatGRU1GRU${}^{*}_{1}$$\mathbf{\widehat{M}_{0}}$$\mathbf{\widehat{M}_{1}}$

(b) With concatenated inputs

Fig. 2: Different styles of split layer adaptations

As mentioned earlier, when dealing with early exiting, a degradation in
performance in the deepest exit stages is often observed, when comparing the
model against its non-dynamic analogous. This is due to the emergence of task-
specialized features that prevent useful information to flow further [6].
Therefore, we attempt to mitigate the issue by introducing additional data
paths in the form of duplicate layers. These act as ancillary feature
extractors, tasked with deriving an increasingly refined internal
representation that proves useful for the downstream layers. This
configuration also avoids subsetting the layers’ activations to yield a mask.

As shown in Fig. 2, two alternative split-layer topologies have been
formulated. In both cases, the layers $\Phi_{i}$ generate mask estimates,
while layers $\Phi_{i}^{*}$ propagate features. The main difference between
the two variants is whether a given $\Phi_{i}^{*}$ layer receives only the
output from $\Phi_{i-1}^{*}$ or the concatenated output from both previous
layers, i.e., $concat(\Phi_{i-1},\Phi_{i-1}^{*})$. The former variant assumes
that previous masks do not contain features that are useful for the model’s
internal representation. Conversely, the latter lifts this assumption at the
expense of computational complexity, which is increased as a result of the
larger input size for $\Phi_{i}^{*}$ layers.

##  3 Training

Our model is trained on the loss function shown in Eq. 3 that was initially
proposed in [18] and subsequently adapted in [1]. The loss comprises two
terms: the first term computes the mean-squared error between the clean and
estimated complex spectra, whereas the second term corresponds to the mean-
squared error of the magnitude spectra. The two terms are weighted by a factor
of $\alpha$ and $(1-\alpha)$, respectively, with $\alpha=0.3$. The spectra are
power-law compressed with $c=0.3$:

$\displaystyle\mathcal{L}\left(S,\widehat{S}\right)={}$
$\displaystyle\alpha\sum_{k,n}\left||S|^{c}e^{j\angle
S}-|\widehat{S}|^{c}e^{j\angle\widehat{S}}\right|^{2}+$ (3)
$\displaystyle(1-\alpha)\sum_{k,n}\left||S|^{c}-|\widehat{S}|^{c}\right|^{2}$

To avoid the impact of large signals dominating the loss and creating
unbalanced training in a batch of several samples, we normalise $S$ and
$\widehat{S}$ by the standard deviation of the target signal $\sigma_{S}$
before computing the loss, as per [1].

In general, training early-exit models can fall into two categories [7]:
layer-wise training and joint training. Since each has its advantages and
drawbacks, we have adopted both strategies to challenge them against each
other.

###  3.1 Layer-wise Training

Layer-wise training is a straightforward way to train early exiting models
with or without pre-trained backbones (i.e., non-dynamic architectures). The
idea is to train the first sub-model, from input $X$ and target $S$ to the
first exit stage that outputs an estimate
$\widehat{S}_{0}=X\odot\widehat{M}_{0}$. Once the training reaches an optimum,
the sub-model’s weights are frozen. The subsequent sub-models are afterwards
trained iteratively taking as inputs their previous sub-model’s last feature
vector. This strategy is helpful for mitigating vanishing gradients as it
allows the training of smaller parts of a bigger network. However, its main
drawback is its shortsightedness as early freezing might degrade later feature
representations, and thus, impede the model’s expressivity.

###  3.2 Joint Training

Joint training inherits from multi-objective optimisation the idea of
minimising a weighted sum of the competing objective functions at play (a
practice known as linear scalarisation). For each sub-model, we attribute a
loss function $\mathcal{L}_{i}$ as defined in Eq. 3. Thus, for $N$ exit stages
(i.e., $N$ sub-models), the total loss can be written as a linear combination
of $\mathcal{L}_{i}$:

$\mathcal{L}_{tot}=\sum_{i=0}^{N-1}\alpha_{i}\mathcal{L}_{i}\left(S,X\odot\widehat{M}_{i}\right)$
(4)

where $\alpha_{i}$ are weighting factors applied to each respective sub-model
loss. Since we do not want to prioritise any specific exit stage, we set each
$\alpha_{i}$ to 1.

By definition, joint training establishes information sharing between the
different exit stages. The model is optimised in a multiplayer-game setting
since each exit (player) strives to minimise its loss along with the best
internal representations for its task. Nonetheless, the loss complexity
imposed on the model leads to an accumulation of gradients from the different
sub-models which may result in unstable convergence [16].

##  4 Experimental setup

Our input and target signals are 4 seconds long and pre-processed using an
STFT with a window of $512$ samples ($32$ ms) and 50% overlap, resulting in
$257$ features per time frame. Baseline and joint trainings were allowed to
run for up to $400$ epochs whereas layer-wise training was capped at $50$
epochs per exit stage for pre-trained models or $50\cdot(i+1)$ epochs, where
$i$ is the exit stage, for the split-layer cases. We set the learning rate to
$10^{-4}$ and batch size to $512$. To prevent overfitting, we implemented
early stopping with patience of $25$ epochs and decreased the learning rate by
$0.9$ every $5$ epochs if there was no improvement.

###  4.1 Dataset

We trained and evaluated our models using data from the 2020 DNS Challenge
[19]. This is a synthetic dataset, composed of several other datasets (clean
speech, noises, room impulse responses) that are processed so as to introduce
reverberation and background noises at different SNRs and target levels,
thereby allowing for arbitrary amounts of noisy-clean training examples.
Similarly, the challenge provides evaluation sets — here, we use the synthetic
non-reverberant test set.

###  4.2 Evaluation Metrics

For this study, we are interested in assessing the performance of our models
from two perspectives: speech quality and computational efficiency. To gauge
the denoised speech quality, we utilise two commonly adopted metrics, namely
PESQ and DNSMOS [20]. For evaluating the computational efficiency of the
models, we consider three measures: Floating Point Operations (FLOPs),
Multiply-Accumulate Operations (MACs), and Inference Time.

FLOPs and MACs are computed using DeepSpeed’s built-in
profiler111https://github.com/microsoft/DeepSpeed and the TorchInfo
library222https://github.com/TylerYep/torchinfo, respectively. Inference time
was computed on CPU for a single frame and averaged out over $1000$ samples.
In all cases, we normalise the metrics to 1 second of input data,
corresponding to $63$ STFT frames.

###  4.3 Model Configurations

Model class | Trainable params. | Size (FP32)
---|---|---
pretrain and baseline | 2.78 M | 11.13 MB
split_layers | 1.62 M | 6.48 MB
concat_layers | 1.88 M | 7.54 MB

Table 1: Model complexity for different configurations

We experimented with several configurations, all derived from the baseline.
These comprise the baseline itself, which has been replicated and trained
accordingly, as well as combinations of the techniques and adaptations
mentioned earlier, and can be subdivided into the following three dimensions:

  * •

Exit stages: we trained both the 6-exit and 4-exit variants introduced in
Section 2.1;

  * •

Split layers: we trained both the straightforward early-exiting model
described in Section 2.1, starting from a checkpoint of the pre-trained
baseline, as well as both split-layer variants described in Section 2.2;

  * •

Training strategies: we employ both the layer-wise and joint training schemes
described in Section 3 to determine the most suitable strategy.

To isolate the impact of each adaptation, we fix our architectural
hyperparameters to the following values: for baseline and simple pre-trained
early-exiting models, we adhered to the original feature sizes of $400$ units
for FC1, GRU1, and GRU2, $600$ units for FC2 and FC3, and $257$ units for FC4
— i.e., the number of frequency bins in the suppression mask. In split-layers
models, the number of features in each $\Phi_{i}$ must also match the size of
the suppression mask. To avoid introducing any unfair advantage into our
evaluation, we picked $128$ as the size of $\Phi_{i}^{*}$ layers, so that the
overall number of propagated features per layer is less than in the baseline.
This results in models with sizes described in Table 1.

##  5 Results

| PESQ | DNSMOS
---|---|---
training | model $\downarrow$ exit $\rightarrow$ | 0 | 1 | 2 | 3 | 4 | 5 | 0 | 1 | 2 | 3 | 4 | 5
— | noisy | 1.58 | 3.16
— | baseline | 2.75 | 3.88
layerwise | pretrain_6exits | 1.73 | 2.20 | 2.32 | 2.29 | 2.27 | 2.21 | 3.27 | 3.57 | 3.68 | 3.67 | 3.66 | 3.63
layerwise | pretrain_4exits | 1.73 | 2.17 |  | 2.38 |  | 2.38 | 3.27 | 3.56 |  | 3.72 |  | 3.71
layerwise | split_layers_6exits | 1.71 | 2.08 | 2.36 | 2.39 | 2.37 | 2.37 | 3.26 | 3.48 | 3.65 | 3.69 | 3.68 | 3.67
layerwise | split_layers_4exits | 1.73 | 2.10 |  | 2.51 |  | 2.45 | 3.27 | 3.50 |  | 3.75 |  | 3.75
layerwise | concat_layers_6exits | 1.72 | 2.08 | 2.38 | 2.43 | 2.44 | 2.44 | 3.27 | 3.49 | 3.72 | 3.73 | 3.73 | 3.74
layerwise | concat_layers_4exits | 1.71 | 2.09 |  | 2.54 |  | 2.55 | 3.26 | 3.49 |  | 3.77 |  | 3.78
joint | pretrain_6exits | 1.72 | 2.18 | 2.40 | 2.47 | 2.53 | 2.56 | 3.27 | 3.45 | 3.67 | 3.73 | 3.77 | 3.80
joint | pretrain_4exits | 1.70 | 2.22 |  | 2.51 |  | 2.58 | 3.26 | 3.54 |  | 3.72 |  | 3.78
joint | split_layers_6exits | 1.71 | 2.13 | 2.43 | 2.50 | 2.52 | 2.52 | 3.26 | 3.45 | 3.73 | 3.76 | 3.75 | 3.76
joint | split_layers_4exits | 1.69 | 2.02 |  | 2.41 |  | 2.42 | 3.26 | 3.42 |  | 3.66 |  | 3.68
joint | concat_layers_6exits | 1.71 | 2.14 | 2.44 | 2.53 | 2.55 | 2.54 | 3.26 | 3.45 | 3.74 | 3.77 | 3.78 | 3.78
joint | concat_layers_4exits | 1.71 | 2.13 |  | 2.63 |  | 2.64 | 3.26 | 3.51 |  | 3.80 |  | 3.81

Table 2: Results of all the models at different exit stages (in bold, best
score for each training strategy and exit stage).

A full overview of the different configurations is shown in Table 2. Here, we
observe a monotonous performance increase along the exit stages until
asymptotically approaching the baseline, showing that deeper layers develop
more expressive representations. Predictably, this trend is observed across
all model variants and applies to both PESQ and DNSMOS scores, with the
notable advantage of smaller model size in split-layer configurations.

Split-layer designs trained in a layer-wise fashion were demonstrably
effective at reducing the performance gap with the baseline, despite using
fewer trainable parameters and operations than the straightforward variants.
Indeed, the features encoded by the auxiliary layers are beneficial to the
denoising tasks, especially at later exit stages (see bold figures in Table 2)
where we derive richer features. Layer concatenation (Fig. 2.b) presents the
highest scores, confirming that the auxiliary pipeline benefits from mask-
related features.

Overall, joint training accounts for the most impact on performance. As
presented in Section 3.2, here we allow the model to take full advantage of
the freedom given by all its parameters to find the best representations for
all exit stages. Here we also notice that later exit stages benefit more from
this training strategy, which could be addressed by using different
$\alpha_{i}$ in Eq. 4.

Fig. 3: Boxplot of quality metrics at different exit stages.

In Fig. 3, it is interesting to observe that PESQ scores exhibit
heteroskedasticity with respect to the exit stage; this can be seen as a
generalisation of the baseline behaviour, which shows the largest variance,
hinting that very low-quality input data are harder to recover. Bizarrely
enough, the opposite phenomenon is observed when considering DNSMOS, where
values are more stable around the mean.

Fig. 4: Comparison of performance/efficiency trade-off at different exit
stages, relative to 1 second of data.

Fig. 4 shows the different computational demands imposed by each layer, in
terms of both operations and processing time. Expectedly, the GRU layers
occupy the majority of the computational budget. Although splitting the layers
into main and ancillary paths requires less computation than the baseline, we
notice a slight increase in inference time. This could be caused by how
Pytorch schedules computations across the layers, or by additional retrieval
and copy operations. The GRU layers also take up the majority of the inference
time, due to the sequential nature of their computation. This also causes the
split-layer variation to be moderately slower since they feature more GRU
layers. However, exiting at a given stage $i$ will spare the computational
cost of the following layers as well as that of its respective ancillary layer
$\Phi_{i}^{*}$. Moreover, as mentioned earlier, the layers $\Phi_{i}^{*}$ are
of small dimensions by design, thus presenting negligible overhead (see Table
1).

Fig. 5: Comparison of example noisy input (in color), suppression masks (in
grayscale) at different exits, and baseline.

The incremental improvements provided by each processing block can be fully
appreciated in Fig. 5. Most noticeably, stage 0 extracts a coarse spectral
envelope of the noise, while later stages refine the contour of speech
features such as formats, their harmonics, and high-frequency unvoiced
components. When comparing against the baseline, a small but noticeable
compression in dynamic range is also observed; this could be a symptom of the
conflicting objectives that joint training aims to optimise — indeed, models
trained layer-wise exhibit more contrast.

Fig. 6: Improvement in PESQ caused by each exit stage, over a range of input
SNR values.

Finally, in Fig. 6 we provide an overview of how each model stage improves the
output PESQ over different input SNR ranges. Here, we notice that the model is
most effective at higher input SNRs, and that each stage’s contribution to the
denoising task is almost equal for each SNR bracket.

##  6 Conclusion

In this work, we proposed an efficient and dynamic version of nsNet2, built
upon early-exiting. The models presented herein provide a diverse range of
performance/efficiency trade-offs, that could benefit embedded devices with
computational and power constraints such as headphones or hearing aids. Our
best-performing models can achieve 96% and 98% of baseline performance on PESQ
and DNSMOS metrics, respectively, on the last exit stage. When considering the
second exit stage, we are able to reach 77% and 90% of baseline performances
with 62% savings in multiply-accumulate operations.

Our future work will advance the current proposed model by automatically
selecting the optimal exit stage based on the properties of the current input
signal as well as a non-intrusive quality assessment module.

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# Relatively spectral homomorphisms and $K$–injectivity††thanks: Project
supported by Natural Science Foundation of China (no.10771069) and Shanghai
Leading Academic Discipline Project(no.B407)

Yifeng Xue
Department of Mathematics, East China Normal University
Shanghai 200241, P.R.China email<EMAIL_ADDRESS>
###### Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be unital Banach algebras and
$\phi\colon\mathcal{A}\rightarrow\mathcal{B}$ be a unital continuous
homomorphism. We prove that if $\phi$ is relatively spectral (i.e., there is a
dense subalgebra $X$ of $\mathcal{A}$ such that
$\mathrm{sp}_{\mathcal{B}}(\phi(a))=\mathrm{sp}_{\mathcal{A}}(a)$ for every
$a\in X$) and has dense range, then $\phi$ induces monomorphisms from
$K_{i}(\mathcal{A})$ to $K_{i}(\mathcal{B})$, $i=0,1$.

##  1 Introduction and Preliminaries

Let $\mathcal{A},\,\mathcal{B}$ be unital Banach algebras and
$\phi\colon\mathcal{A}\rightarrow\mathcal{B}$ be a unital homomorphism (i.e.,
$\phi(1)=1$). If
$\mathrm{sp}_{\mathcal{B}}(\phi(a))=\mathrm{sp}_{\mathcal{A}}(a)$ for all
$a\in\mathcal{A}$, we say that $\phi$ is spectral, here
$\mathrm{sp}_{\mathcal{A}}(a)$ denotes the spectrum of $a$ in $\mathcal{A}$.
Recall from [3, Definition 10] that $\phi$ is said to be relatively spectral
(to $X$) if there is a dense subalgebra $X$ of $\mathcal{A}$ such that
$\mathrm{sp}_{\mathcal{B}}(\phi(x))=\mathrm{sp}_{\mathcal{A}}(x)$ for all
$x\in X$. Furthermore, $\phi$ is said to be completely relatively spectral if
$\phi_{n}\colon\text{M}_{n}(\mathcal{A})\rightarrow\text{M}_{n}(\mathcal{B})$
is relatively spectral (to $\text{M}_{n}(X)$) for each $n$, where
$\phi_{n}((a_{ij})_{n\times n})=(\phi(a_{ij}))_{n\times n}$,
$(a_{ij})_{n\times n}\in\text{M}_{n}(\mathcal{A})$.

It is known that if $\phi$ is spectral and has dense range, then $\phi$
induces an isomorphism $K_{*}(\mathcal{A})\cong K_{*}(\mathcal{B})$ (cf. [2]).
Recently, B. Nica shows that if $\phi$ is completely relatively spectral and
$\mathrm{Ran}\,(\phi)=\phi(\mathcal{A})$ is dense in $\mathcal{B}$, then
$\phi$ also induces an isomorphism $K_{*}(\mathcal{A})\cong
K_{*}(\mathcal{B})$ (cf. [3, Theorem 2]).

Since we do not know if a relatively spectral homomorphism is completely
relatively spectral in general, except some special cases listed in [3], it is
significant to investigate if the relatively spectral homomorphism $\phi$ with
dense range could induce an isomorphism between $K_{*}(\mathcal{A})$ and
$K_{*}(\mathcal{B})$.

In this short note, we prove following result which partially generalizes [3,
Theorem 2].

######  Theorem 1.1.

Let $\mathcal{A},\,\mathcal{B}$ be two unital Banach algebras and $\phi$ be a
unital continuous homomorphism from $\mathcal{A}$ to $\mathcal{B}$ with dense
range. If $\phi$ is relatively spectral, then $\phi$ induces a monomorphism
$K_{*}(\mathcal{A})\rightarrow K_{*}(\mathcal{B})$.

We will prove this theorem in §2. Now we introduce some notations used in next
section. For a Banach algebra $\mathcal{A}$ with unit $1$, let
$GL(\mathcal{A})$ (resp. $GL_{0}(\mathcal{A})$) denote the group of invertible
elements in $\mathcal{A}$ (resp. the connected component of $1$ in
$GL(\mathcal{A})$). For a Banach algebra $\mathcal{A}$, we view
$\mathcal{A}^{n}$ and ${\text{M}}_{n}(\mathcal{A})$ as the set of all $n\times
1$ and $n\times n$ matrices over $\mathcal{A}$ respectively. The norm on
$\mathcal{A}^{n}$ (resp. $\text{M}_{n}(\mathcal{A})$) is given by
$\|(a_{1},\cdots,a_{n})^{T}\|=\sum\limits^{n}_{i=1}\|a_{i}\|$ (resp.
$\|(a_{ij})_{n\times n}\|=\sum\limits^{n}_{i,j=1}\|a_{ij}\|$). Set
$GL_{n}(\mathcal{A})=GL({\text{M}}_{n}(\mathcal{A})),\,GL^{0}_{n}(\mathcal{A})=GL_{0}({\text{M}}_{n}(\mathcal{A}))$.

Suppose $\mathcal{A}$ is unital and let $x\in GL(\mathcal{A})$. Denote by
$[x]$ the equivalence class of $x$ in $GL(\mathcal{A})/GL_{0}(\mathcal{A})$.
The $K_{1}$–group of $\mathcal{A}$, denoted $K_{1}(\mathcal{A})$ is defined as
$K_{1}(\mathcal{A})=\bigcup\limits_{n=1}^{\infty}GL_{n}(\mathcal{A})/GL_{n}^{0}(\mathcal{A})$,
where $GL_{n}(\mathcal{A})/GL_{n}^{0}(\mathcal{A})\subset
GL_{n+1}(\mathcal{A})/GL_{n+1}^{0}(\mathcal{A})$ in the sense that
$[x]\mapsto[\mathrm{diag\,}(x,1)]$, $\forall\,x\in
GL_{n}(\mathcal{A})/GL_{n}^{0}(\mathcal{A})$, $n=1,2,\cdots$. We can define
the $K_{0}$–group of $\mathcal{A}$ by
$K_{0}(\mathcal{A})=K_{1}((S\mathcal{A})^{+})$, where

$(S\mathcal{A})^{+}=\\{f\in
C([0,1],\mathcal{A})|\,f(0)=f(1)=\text{constant}\\}.$

More detailed information about $K_{0}(\mathcal{A})$ and $K_{1}(\mathcal{A})$
can be found in [1].

##  2 Proof of main theorem

In this section, we assume that $\mathcal{A}$ and $\mathcal{B}$ are unital
Banach algebras and $\phi$ is a unital continuous homomorphism from
$\mathcal{A}$ to $\mathcal{B}$.

Let $M$ be a compact Hausdorff space and let $C(M,\mathcal{A})$ denote the
Banach algebra consisting of all continuous maps $f\colon
M\rightarrow\mathcal{A}$ with the norm $\|f\|=\sup\limits_{t\in M}\|f(t)\|$.
When $M=\mathbf{S}^{1}$, we set
$\Omega(\mathcal{A})=C(\mathbf{S}^{1},\mathcal{A})$. Let $X$ be a dense
subalgebra of $\mathcal{A}$ and put $M(X)=\\{f\colon M\rightarrow X\
\text{continuous}\\}$. Define a homomorphism $\phi_{M}\colon
C(M,\mathcal{A})\rightarrow C(M,\mathcal{B})$ by $\phi_{M}(f)(t)=\phi(f(t))$,
$\forall\,f\in C(M,\mathcal{A})$ and $t\in M$.

######  Lemma 2.1.

Let $\phi$ be a relatively spectral (to $X$) homomorphism with
$\mathrm{Ran}\,(\phi)$ dense in $\mathcal{B}$. Then $\phi_{M}$ is a relatively
spectral (to $M(X)$) homomorphism with $\mathrm{Ran}\,(\phi_{M})$ dense in
$C(M,\mathcal{B})$.

###### Proof.

Given $g\in C(M,\mathcal{B})$ and $\epsilon>0$. Since $g$ is continuous and
$M$ is compact, it follows that $g(M)$ is also compact in $\mathcal{B}$. Thus,
there are $y_{1},\cdots,y_{n}\in g(M)$ such that
$g(M)\subset\mathop{\bigcup}\limits_{i=1}^{n}O(y_{i},\epsilon)$, where
$O(y_{i},\epsilon)=\\{y\in\mathcal{B}|\,\|y-y_{i}\|<\epsilon\\}$. Set
$U_{i}=g^{-1}(O(y_{i},\epsilon))$, $i=1,\cdots,n$. Then
$\\{U_{1},\cdots,U_{n}\\}$ is an open cover of $M$. Choose a partition of
unity $\\{f_{1},\cdots,f_{n}\\}$ subordinate to this cover, so that each
$f_{i}$ is a continuous function from $M$ to $[0,1]$ with support contained in
$U_{i}$ and $\sum\limits^{n}_{i=1}f_{i}(t)=1$, $\forall\,t\in M$.

From $\overline{\phi(X)}=\mathcal{B}$, we can find $a_{1},\cdots,a_{n}\in X$
such that $\|\phi(a_{i})-y_{i}\|<\epsilon$, $i=1\,\cdots,n$. Set
$f_{\epsilon}(t)=\sum\limits^{n}_{i=1}a_{i}f_{i}(t)$, $\forall\,t\in M$. Then
$f\in M(X)$ and

$\|\phi(f_{\epsilon}(t))-g(t)\|\leq\sum\limits^{n}_{i=1}\|\phi(a_{i})-y_{i}\|f_{i}(t)+\sum\limits^{n}_{i=1}\|y_{i}f_{i}(t)-g(t)f_{i}(t)\|<2\epsilon,$

$\forall\,t\in M$. Thus, $\phi_{M}(M(X))$ is dense in $C(M,\mathcal{B})$.

If we set $\mathcal{B}=\mathcal{A}$ and $\phi=\mathrm{id}$ in above argument,
then we have $\overline{M(X)}=C(M,\mathcal{A})$.

Now we show that $\phi_{M}$ is relatively spectral. But it is enough to prove
that $\phi_{M}(f)$ is invertible in $C(M,\mathcal{B})$ for $f\in M(X)$ implies
that $f$ is invertible in $C(M,\mathcal{A})$. Since $\phi_{M}(f)$ is
invertible in $C(M,\mathcal{B})$, it follows that $\phi(f(t))$ is invertible
in $\mathcal{B}$, $\forall\,t\in M$. Thus, from the relatively spectral
property of $\phi$, we have $f(t)\in GL(\mathcal{A})$, $\forall\,t\in M$. This
means that $f\in GL(C(M,\mathcal{A}))$. ∎

######  Corollary 2.2.

Let $\phi$ be a relatively spectral (to $X$) homomorphism with dense range.
Suppose there is $a\in X$ such that $\|1-\phi(a)\|<1$. Then $a\in
GL_{0}(\mathcal{A})$.

###### Proof.

Choose $x\in X$ such that $\|1-x\|\leq\dfrac{1}{1+\|\phi\|}$. Then $x\in
GL_{0}(\mathcal{A})$ and $\|1-\phi(x)\|<1$. Put $f(t)=(1-t)x+ta$,
$\forall\,t\in\mathrm{I}=[0,1]$. Then $f\in\mathrm{I}(X)$ and
$\|1-\phi_{I}(f)\|<1$. So $f\in GL(C(I,\mathcal{A}))$ with $f_{0}=x$ and
$f_{1}=a$ by Lemma 2.1, which means that $a\in GL_{0}(\mathcal{A})$. ∎

######  Lemma 2.3.

Let $\phi$ be a relatively spectral ( to $X$) homomorphism with dense range
and let $z\in\mathrm{M}_{n}(X)$ with $\|1_{n}-\phi_{n}(z)\|<\dfrac{1}{\,3\,}$,
where $1_{n}$ is the unit of $\mathrm{M}_{n}(\mathcal{A})$. Then for any
$\epsilon>0$, there is $z^{\prime}\in\mathrm{M}_{n}(X)\cap
GL^{0}_{n}(\mathcal{A})$ such that $\|z-z^{\prime}\|<\epsilon$.

###### Proof.

When $n=1$, the statement is true by Corollary 2.2. We assume that the
statement is true for $1\leq n\leq m$. We now prove the argument is also true
for $n=m+1$.

Let $y=\phi_{m+1}(z)\in\phi_{m+1}(\mathrm{M}_{m+1}(X))$ with
$\|1_{m+1}-y\|<\dfrac{1}{\,3\,}$. Write $z=(z_{ij})_{m+1\times m+1}$ (resp.
$y=(y_{ij})_{m+1\times m+1}$) as $z=\begin{pmatrix}z_{1}&z_{2}\\\
z_{3}&z_{4}\end{pmatrix}$ (resp. $y=\begin{pmatrix}y_{1}&y_{2}\\\
y_{3}&y_{4}\end{pmatrix}$), where $y_{ij}=\phi(z_{ij})$, $i,j=1,\cdots,m+1$,
$z_{1}=(z_{ij})_{m\times m}\in\mathrm{M}_{m}(X)$ and

$z_{2}=\begin{pmatrix}z_{1\,m+1}\\\ \vdots\\\ z_{m\,m+1}\end{pmatrix},\quad
z_{3}=\begin{pmatrix}z_{m+1\,1}&\dots&z_{m+1\,m}\end{pmatrix},\quad
z_{4}=z_{m+1\,m+1}.$

Then $\|1_{m}-y_{1}\|<\dfrac{1}{\,3\,}$, $\|y_{2}\|<\dfrac{1}{\,3\,}$,
$\|y_{3}\|<\dfrac{1}{\,3\,}$ and $\|1-y_{4}\|<\dfrac{1}{\,3\,}$. By
assumption, there is $z_{1}^{\prime}\in
GL_{m}^{0}(\mathcal{A})\cap\mathrm{M}_{m}(X)$ such that
$\|z_{1}-z_{1}^{\prime}\|<\dfrac{\epsilon}{3(\|\phi_{m}\|+1)}$. Pick
$x_{1}\in\mathrm{M}_{m}(X)$ such that

$\|(z_{1}^{\prime})^{-1}-x_{1}\|<\dfrac{\epsilon}{3(\|z_{1}^{\prime}\|+1)(\|z_{3}\|+1)(\|\phi_{m}\|+1)}.$
(2.1)

Set

$z^{\prime}=\begin{pmatrix}1_{n}&\ 0\\\ z_{3}x_{1}&\
1\end{pmatrix}\begin{pmatrix}z_{1}^{\prime}&z_{2}\\\ \ 0&\
z_{4}-z_{3}x_{1}z_{2}\end{pmatrix}=\begin{pmatrix}z_{1}^{\prime}&z_{2}\\\
z_{3}x_{1}z_{1}^{\prime}&z_{4}\end{pmatrix}\in\mathrm{M}_{m+1}(X).$

Then
$\|z-z^{\prime}\|\leq\|z-z^{\prime}\|+\|z_{3}\|\|1_{m}-x_{1}z_{1}^{\prime}\|<\epsilon$.

Note that $\|1_{m}-\phi_{m}(z_{1})\|<\dfrac{1}{\,3\,}$ and
$\|\phi_{m}(z_{1})-\phi_{m}(z_{1}^{\prime})\|<\dfrac{1}{\,3\,}$. So
$\|1_{m}-\phi_{m}(z_{1}^{\prime})\|<\dfrac{2}{\,3\,}$ and hence
$\|(\phi_{m}(z_{1}^{\prime}))^{-1}\|<3$. Consequently, $\|\phi_{m}(x_{1})\|<4$
by (2.1). Therefore,

$\|1-\phi(z_{4}-z_{3}x_{1}z_{2})\|<\|1-y_{4}\|+\|y_{3}\|\|\phi_{m}(x_{1})\|\|y_{2}\|<1.$
(2.2)

Applying Corollary 2.2 to (2.2), we have $z_{4}-z_{3}x_{1}z_{2}\in
GL_{0}(\mathcal{A})$. Thus, we can deduce that
$\begin{pmatrix}z_{1}^{\prime}&z_{2}\\\ \ 0&\
z_{4}-z_{3}x_{1}z_{2}\end{pmatrix}\in GL^{0}_{m+1}(\mathcal{A})$. Since
$\begin{pmatrix}1_{n}&\ 0\\\ z_{3}x_{1}&\ 1\end{pmatrix}\in
GL_{m+1}^{0}(\mathcal{A})$, it follows that $z^{\prime}\in
GL_{m+1}^{0}(\mathcal{A})$. This completes the proof. ∎

Now we give the proof of Theorem 1.1 as follows.

###### Proof.

Let $G\in GL_{n}(\mathcal{A})$ such that $G_{0}=\phi_{n}(G)\in
GL_{n}^{0}(\mathcal{B})$ for some $n$. We will prove $G\in
GL_{n}^{0}(\mathcal{A})$. Since $X$ is dense in $\mathcal{A}$ and $\phi$ has
dense range, we can find $A\in\mathrm{M}_{n}(X)$ with $\|A-G\|$ small enough
so that $A\in GL_{n}(\mathcal{A})$ with $[A]=[G]$ in
$GL_{n}(\mathcal{A})/GL_{n}^{0}(\mathcal{A})$ and $\phi_{n}(\mathcal{A})\in
GL_{n}^{0}(\mathcal{B})$. Noting that $\phi_{n}(A)$ can be written as
$\phi_{n}(A)=e^{b_{1}}\cdots e^{b_{s}}$ for some
$b_{1},\cdots,b_{s}\in\text{M}_{n}(\mathcal{B})$, we can find
$a_{1},\cdots,a_{s}\in\text{M}_{n}(\mathcal{A})$ with
$\|\phi_{n}(a_{i})-b_{i}\|$ small enough, $i=1,\cdots,s$, such that

$\|\phi_{n}(A)^{-1}-\phi_{n}(e^{-a_{1}}\cdots
e^{-a_{s}})\|<\dfrac{1}{6(\|\phi_{n}(A)\|+1)}.$

Choose $B_{0}\in\mathrm{M}_{n}(X)$ such that

$\|e^{-a_{1}}\cdots e^{-a_{s}}-B_{0}\|<\dfrac{1}{6(\|e^{a_{1}}\cdots
e^{a_{s}}\|+1)(\|\phi_{n}\|+1)(\|\phi_{n}(A)\|+1)}.$

Then $B_{0}\in GL_{n}^{0}(\mathcal{A})$ and

$\|I_{n}-\phi_{n}(AB_{0})\|\leq\|\phi_{n}(A)\|\|\phi_{n}(A)^{-1}-\phi_{n}(B_{0})\|<\frac{1}{\,3\,}.$

Therefore there exists $Z\in GL_{n}^{0}(\mathcal{A})\cap\mathrm{M}_{n}(X)$
such that $\|AB_{0}-Z\|<\dfrac{1}{\|(AB_{0})^{-1}\|}$ by Lemma 2.3. So,
$AB_{0}\in GL_{n}^{0}(\mathcal{A})$ and hence $G\in GL_{n}^{0}(\mathcal{A})$.

Since $\phi_{\mathbf{S}^{1}}$ is relatively spectral and
$\mathrm{Ran}\,(\phi_{\mathbf{S}^{1}})$ is dense in $\Omega(\mathcal{B})$ by
Lemma 2.1, we get that the induced homomorphism
$(\phi_{\mathbf{S}^{1}})_{*}\colon K_{1}(\Omega(\mathcal{A}))\rightarrow
K_{1}(\Omega(\mathcal{B}))$ is injective by above argument. Thus, from the
commutative diagram of split exact sequences

$\begin{CD}0@>{}>{}>K_{1}(S\mathcal{B})@>{}>{}>K_{1}(\Omega(\mathcal{B}))@>{}>{}>K_{1}(\mathcal{B})@>{}>{}>0\\\
@A{\phi_{*}}A{}A@A{(\phi_{\mathbf{S}^{1}})_{*}}A{}A@A{\phi_{*}}A{}A\\\
0@>{}>{}>K_{1}(S\mathcal{A})@>{}>{}>K_{1}(\Omega(\mathcal{A}))@>{}>{}>K_{1}(\mathcal{A})@>{}>{}>0,\end{CD}$
(2.3)

we get that $\phi_{*}\colon K_{1}(S\mathcal{A})\rightarrow
K_{1}(S\mathcal{B})$ is injective, that is, $\phi_{*}\colon
K_{0}(\mathcal{A})\rightarrow K_{0}(\mathcal{B})$ is injective. ∎

######  Remark 2.4.

If $\mathrm{csr\,}(C(\bf S^{1},\mathcal{B}))\leq 2$, where
$\mathrm{csr\,}(\cdot)$ is the connected stable rank of Banach algebras
introduced by Rieffel in [5], then $\phi_{*}$ is also surjective.

In fact, when $\mathrm{csr\,}(C(\mathbf{S}^{1},\mathcal{B}))\leq 2$, we have
$\mathrm{csr\,}(\mathcal{B})\leq 2$ by [4, Proposition 8.4]. So the natural
homomorphism $i_{\mathcal{B}}\colon
GL(\mathcal{B})/GL_{0}(\mathcal{B})\rightarrow K_{1}(\mathcal{B})$ is
surjective by using the proof of Theorem 10.10 in [5]. On the other hand, if
$\phi$ is relatively spectral and has dense range, then the induced
homomorphism $\phi_{*}\colon GL(\mathcal{A})/GL_{0}(\mathcal{A})\rightarrow
GL(\mathcal{B})/GL_{0}(\mathcal{B})$ is injective by the proof of Theorem 1.1.
To show $\phi_{*}$ is surjective, let $b\in GL(\mathcal{B})$ and pick $a\in X$
such that $\|\phi(a)-b\|<\dfrac{1}{\|b^{-1}\|}$. Then $\phi(a)\in
GL(\mathcal{B})$ with $[b]=[\phi(a)]$ and hence $a\in GL(\mathcal{A})$. Thus,
$\phi_{*}([a])=[b]$ and consequently, $\phi_{*}\colon
K_{1}(\mathcal{A})\rightarrow K_{1}(\mathcal{B})$ is surjective.

By the above argument, we also have $(\phi_{\mathbf{S}^{1}})_{*}$ is
surjective. Thus, using the commutative diagram (2.3), we obtain that
$\phi_{*}\colon K_{1}(S\mathcal{A})\rightarrow K_{1}(S\mathcal{B})$ is
surjective.

Especially, when $\mathcal{B}$ is of topological stable rank one,
$\mathrm{csr\,}(\mathcal{B})\leq 2$ and
$\mathrm{csr\,}(C(\mathbf{S}^{1},\mathcal{B}))\leq 2$. So $\phi_{*}$ is
surjective.

Acknowledgement. The author is grateful to the referee for his (or her) very
helpful comments and suggestions.

## References

  * [1] B. Blackadar, “$K$–Theory for Operator Algebras”, Springer–Verlag, 1986.
  * [2] J.-B. Bost, _Principe d’Oka, $K$–théorie et systémes dynamiques non commutatifs_, Invent. Math., 101(2) (1990), 261–333.
  * [3] B. Nica, _Relatively spectral morphisms and applications to $K$–theory_, J. Funct. Anal., 255 (2008), 3303–3328.
  * [4] B. Nica, _Homotopical stable ranks for Banach algebras_ , http://arxiv.org/ abs/0911.2945v1.
  * [5] M.A. Rieffel, _Dimension and stable rank in the $K$–Theory of $C^{*}$–algebras_, Proc. London Math. Soc., 46(1983), 301–333.


# Charge carriers trapping by the full-configuration defects in metal halide
perovskites quantum dots

Xiao-Yi Liu  [  Yu Cui  [  Jia-Pei Deng  [  Yi-Yan Liu  [  Xu-Fei Ma  [  Yu-
Xuan Hou  [  Jun-Ye Wei  [  Zhi-Qing Li  [  Zi-Wu Wang  [<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Metal halide perovskites quantum dots (MHPQDs) have aroused enormous
interesting in the photovoltaic and photoelectric because of their marvelous
properties and size characteristics. However, one of key problems that how to
systematically analyze charge carriers trapping by different defects is still
a challenge task. Here, we study nonradiation multiphonon processes of the
charge carrier trapping by various defects in MHPQDs based on the well-known
Huang-Rhys model, in which a method of full-configuration defect, including
different defect species with variable depth and lattice relaxation strength,
is developed by introducing a localization parameter in the quantum defect
model. With the help of this method, these fastest trapping channels for
charge carriers transferring from the quantum dot ground state to different
defects are found. Furthermore, the dependences of the trapping time on the
radius of quantum dot, the defect depth and temperature are given. These
results not only enrich the knowledge of charge carrier trapping processes by
defects, but enlighten the designs of MHPQDs-based photovoltaic and
photoelectric devices.

###### keywords:

American Chemical Society, LaTeX

Tianjin University] Tianjin Key Laboratory of Low Dimensional Materials
Physics and Preparing Technology, Department of Physics, School of Science,
Tianjin University, Tianjin 300354, China Tianjin University] Tianjin Key
Laboratory of Low Dimensional Materials Physics and Preparing Technology,
Department of Physics, School of Science, Tianjin University, Tianjin 300354,
China Tianjin University] Tianjin Key Laboratory of Low Dimensional Materials
Physics and Preparing Technology, Department of Physics, School of Science,
Tianjin University, Tianjin 300354, China Tianjin University] Tianjin Key
Laboratory of Low Dimensional Materials Physics and Preparing Technology,
Department of Physics, School of Science, Tianjin University, Tianjin 300354,
China Tianjin University] Tianjin Key Laboratory of Low Dimensional Materials
Physics and Preparing Technology, Department of Physics, School of Science,
Tianjin University, Tianjin 300354, China Tianjin University] Tianjin Key
Laboratory of Low Dimensional Materials Physics and Preparing Technology,
Department of Physics, School of Science, Tianjin University, Tianjin 300354,
China Tianjin University] Tianjin Key Laboratory of Low Dimensional Materials
Physics and Preparing Technology, Department of Physics, School of Science,
Tianjin University, Tianjin 300354, China Tianjin University] Tianjin Key
Laboratory of Low Dimensional Materials Physics and Preparing Technology,
Department of Physics, School of Science, Tianjin University, Tianjin 300354,
China Tianjin University] Tianjin Key Laboratory of Low Dimensional Materials
Physics and Preparing Technology, Department of Physics, School of Science,
Tianjin University, Tianjin 300354, China IR,NMR,UV

TOC Graphic

##  1 Introduction

Metal halide perovskites quantum dots (MHPQDs) have gained extensive attention
in the past few years due to their excellent photoelectric properties such as
tunable bandgap, long carrier diffusion length, narrow emission width, high
photoluminescence quantum yield and high defect tolerance.1, 2, 3, 4 The power
conversion efficiency of MHPQDs-based solar cell has remarkably increased to
16.6% in a relatively short period of time.4 Besides in photovoltaic field,
MHPQDs also exhibit immense potential applications in light-emitting diodes
and lasers.5, 6, 7 MHPQDs have the size-tunability of their bandgap energies
due to quantum confinement effect,8, 9 which enriches the wavelength range of
absorption and emission.10, 6 At the same time, the enhancement of quantum
confinement effect and the weakening of dielectric screening effect can result
in the larger exciton binding energy and enhance the photoluminescent
efficiency.11, 12, 6 These structural features are apparently conducive to
MHPQDs’ applications. It is just the zero-dimensional structure with specific
surface area, giving inevitably rise to various defects in fabrication
processes.4 These defects, especially the deep level defects, mediated as the
non-radiative recombination centers (or called as the trapping centers),13,
14, 15 leading to charge and energy losses.

There were many experimental strategies have been made to reduce or modify the
pernicious influence of defects on material properties.16, 17, 18, 19
Undoubtedly, a series of new experimental method for this issue will be sprang
up rapidly in following years. In the theoretical aspect, the dynamical
processes of different types of defects in MHPQDs have been extensively
studied by first-principal calculation based on the density function
theory.20, 21 However, on the one hand, there are still numbers of pernicious
defects in MHPQD due to its soft lattice structure,22, 23 resulting in the
comprehensively analysis of these defects become almost impossible, even
though costing the huge calculational labor. On the other hand, the simulation
for the trapping probability (or trapping time) of the charge carrier by so
many defects is also a challenge task based on first-principal calculation. In
particular, the disagreement between the experimental measurement and the
theoretical simulation becomes obviously with the increasing of quantum dots
size and temperature. Therefore, an effective method not only classifying so
many defects clearly, but also covering all possible trapping channels of the
charge carrier by defects is needed urgently.

In this paper, we propose a full-configuration defect model by introducing a
localization parameter in the classical quantum defect model,24, 25, 26 which
allows us not only to classify various defects into three types (accepter-like
and donor-like and neutral defects),27 but also to effectively describe the
variational defect depth in bandgap with the appropriate lattice relaxation
strength. By virtue of this model, we present charge carriers are trapped from
the ground state (GS) of MHPQDs to different defect states by non-radiation
multiphonon processes, in which the fastest trapping channel is found. More
important is that the relations of the fastest channel with the radius of
quantum dot, the defect depth and temperature could be obtained, which give
the direct guideline to analyze the influence of the charge carrier trapping
by defects on the performance of the newly devices, such as the open-circuit
voltage of MHPQDs solar cells,28, 29, 30 the instability of bias voltage in
MHP transistors31, 32 and so on.

##  2 Results and discussion

FIG. 1: (a) The schematic diagram for an electron is trapped from the ground
state in the metal halide perovskite quantum dot to various defects in the
bandgap (a hole trapping has the similar process). (b) The coordinate
configurations for before and after the carrier trapping. $Q\left(R\right)$
and $Q\left({\nu,\beta}\right)$ denote the vibrational equilibrium position
for the quantum dot system when an electron is in the ground state (the
initial state) and in the defect state (the final state), respectively, in
which the same depth of diverse defects with different localization strength
are described by $\cdots
Q\left({{\nu_{n}},{\beta_{n-1}}}\right),Q\left({{\nu_{n}},{\beta_{n}}}\right),Q\left({{\nu_{n}},{\beta_{n+1}}}\right)\cdots$
and the variable depth of defects with the same localization strength are
described by $\cdots
Q\left({{\nu_{n-1}},{\beta_{n}}}\right),Q\left({{\nu_{n}},{\beta_{n}}}\right),Q\left({{\nu_{n+1}},{\beta_{n}}}\right)\cdots$.
The variable intersections of two coordinate configurations, such as A, B, C,
stemming from the diversity of defects with different lattice relaxation
strength in bandgap, denote different nonradiation multiphonon processes.

We mainly consider an electron in the ground state in MHP quantum dot are
trapped by different depth defects (the trapping process of a hole is similar
as electron) as shown in FIG. 1 (a). The eigenwave function and eigenenergy
for the quantum dot are given in part A of Supplemental Material. In the
classical quantum defect model,24, 25, 26 various defects are divided into
three types of donor-like, accepter-like and natural defects by the parameter
$\mu=-\nu,\nu,0$ in the defect wave function, respectively.27 Meanwhile, the
variation of defect depth $\Delta E$ and the corresponding lattice relaxation
strength that arising from defects couple with the surrounding lattice filed
(or phonon) can be effectively described by the parameter $\nu$, which is an
advantage of this model. However, for the various defects in the same depth
with the different lattice relaxation strength are failed description.
Therefore, we introduce a localization parameter $\beta$ in the defect wave
function to quantitatively describe the lattice relaxation effect for various
defects in the same energy level (see the part B of Supplemental Material).
Namely, all kinds of defects with varying lattice relaxation strength
(thereafter called as full-configuration defects) in these MHP quantum dots
could be described by this modified quantum defect model, whose practicability
for analyzing the charge carrier trapping will be given further in the
following sections.

In the frame of Huang-Rhys model,33, 34, 35, 36 an electron (or a hole)
trapping by the defects via non-radiation multiphonon processes are
extensively described by the coordinate configuration diagrams shown in FIG. 1
(b), in which the vibrational equilibrium position of system is assumed at
$Q\left(R\right)$ when the electron is in the GS state (the initial state),
depending on the radius of the quantum dot ($R$), and the equilibrium position
shifts to $Q\left({\nu,\beta}\right)$ as the electron trapped by defect (the
final state), depending on the depth of defect and the localization parameter.
The difference between two equilibrium positions can be expressed as:

$\displaystyle{\mathcal{S}_{LO}}{\left({\hbar{\omega_{LO}}}\right)^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum\limits_{\mathbf{q}}{{{\left[{Q\left(R\right)-Q\left({\nu,\beta}\right)}\right]}^{2}}}$
(1) $\displaystyle=$
$\displaystyle\frac{1}{2}\sum\limits_{\mathbf{q}}{{{\left|{\mathcal{M}\left(q\right)\int{\left[{\psi_{100}^{*}\left(r\right){e^{i\mathbf{q}\cdot\mathbf{r}}}{\psi_{100}}\left(r\right)-\varphi_{\nu,\beta}^{*}\left(r\right){e^{i\mathbf{q}\cdot\mathbf{r}}}{\varphi_{\nu,\beta}}\left(r\right)}\right]dr}}\right|}^{2}}},$

where ${\mathcal{S}_{LO}}$ is the well-known Huang-Rhys factor, in which the
longitudinal optical (LO) phonon ($\hbar{\omega_{LO}}$) and electron-LO phonon
coupling ($\mathcal{M}\left(q\right)$) in the Fr$\mathrm{\ddot{o}}$hlich
mechanism are mainly considered, since this coupling plays the dominate rule
in the carrier scattering proved by many experiments.23, 37, 38, 39, 40 The
detailed expression and calculation for Huang-Rhys factors are present in part
C of Supplemental Material. ${\mathcal{S}_{LO}}$ as functions of the depth of
donor-like defect and the radius of CH3NH3PbI3 quantum dot are shown in FIG. 2
(a) (the related parameters for CH3NH3PbI3 material are listed in Table $S$1
in Supplemental Material). One can see that (1) ${\mathcal{S}_{LO}}$ increases
obviously with increasing the defect depth, which is consistent with the
results of the classical quantum defect model as the variational defects
($\cdots
Q\left({{\nu_{n-1}},{\beta_{n}}}\right),Q\left({{\nu_{n}},{\beta_{n}}}\right),Q\left({{\nu_{n+1}},{\beta_{n}}}\right)\cdots$)
schemed in FIG. 1 (b);26, 41 (2) ${\mathcal{S}_{LO}}$ increases slightly as
the radius increases,42 which can be attributed to the quantum confinement
effect become weak with increasing the radius,4, 43, 7 resulting in the
difference of equilibrium positions in coordinate configuration between
quantum dot and defect are enhanced. The influence of the localization
parameter ($\beta$) on ${\mathcal{S}_{LO}}$ are plotted in FIG. 2 (b). From
it, one can see that ${\mathcal{S}_{LO}}$ increases signally as $\beta$ varies
in the range of 0.1$\sim$3.0 at fixed defect depth. Which reflects the
difference strength of lattice relaxation for diverse defects in the same
depth in the bandgap as the varying position ($\cdots
Q\left({{\nu_{n}},{\beta_{n-1}}}\right),Q\left({{\nu_{n}},{\beta_{n}}}\right),Q\left({{\nu_{n}},{\beta_{n+1}}}\right)\cdots$)
schemed in FIG. 1 (b). The similar results for the acceptor-like and neutral
defects are also given in FIG. $S$1 and $S$2 in Supplemental Material.

FIG. 2: (a) Huang-Rhys (HR) factor ${\mathcal{S}_{LO}}$ as functions of the
CH3NH3PbI3 quantum dot radius $R$ and the depth of donor-like defect ${\Delta
E}$ at $T=300$ K. (b) The influence of localization parameter $\beta$ on HR
factor ${\mathcal{S}_{LO}}$ with different defect depths at $T=300$ K and
$R=50$ nm for donor-like defect. (c) Under different quantum dot radii, the
carrier trapping lifetime $\tau$ as a function of the donor-like defect depth
${\Delta E}$ at $T=300$ K and $\beta{\rm{=}}1$.

In fact, this factor presents an appropriate metric for the systematically
energy variation before and after the carrier trapping by different defects.
With the aid of this key factor, the trapping probability ${\tau^{-1}}$
($\tau$ denotes the carrier lifetime or the trapping time) of carrier by
various defects via non-radiation multiphonon processes can be written as

$\displaystyle{\tau^{-1}}=\frac{{2\pi{{\left|{{H_{gd}}}\right|}^{2}}}}{{\hbar\Delta
E}}{\left({\frac{{{{\bar{n}}_{LO}}+1}}{{{{\bar{n}}_{LO}}}}}\right)^{{p\mathord{\left/{\vphantom{p2}}\right.\kern-1.2pt}2}}}\exp\left[{-{\mathcal{S}_{LO}}\left({2{{\bar{n}}_{LO}}+1}\right)}\right]{I_{p}}\left[{2{\mathcal{S}_{LO}}\sqrt{{{\bar{n}}_{LO}}\left({{{\bar{n}}_{LO}}+1}\right)}}\right],$
(2)

where ${{H_{gd}}}$ is the transition matrix between GS of quantum dot and
defect, whose definition and the related parameters in Eq. (2) are given in
part D of Supplemental Material. FIG. 2 (c) shows the carrier lifetime as a
function of the defect depth ${\Delta E}$ at different radii. One can see that
$\tau$ is prolonged by three orders of magnitude as ${\Delta E}$ increases,
which stems from more LO phonons are required to match the increasing of
${\Delta E}$, hindering the probability of occurrence for the carrier trapping
by non-radiation multiphonon processes significantly. Meanwhile, $\tau$ also
has the very large changing magnitude with the quantum dot radius, even though
${\mathcal{S}_{LO}}$ varies slightly with the radius shown in FIG. 2 (a),
which attributes to the strong effect of quantum confinement of the quantum
dot radius on the transition matrix ${{H_{gd}}}$. This implies that the
smaller quantum dot, the easier carrier trapping, which is very agreement with
the experimental measurements.44, 45, 46 These conclusions are also same as
the accepter-like and neutral defects as shown in FIG. $S$3 and $S$4 in
Supplemental Materials.

FIG. 3: (a) The dependence of carrier trapping lifetime $\tau$ on the
localization parameter $\beta$ and the donor-like defect depth ${\Delta E}$,
$T=300$ K and $R=50$ nm. The fastest channel of the carrier trapping is shown
in (b) by the pink hollow circles. (c) The temperature dependence of lifetime
of the carrier trapping by the donor-like defect with different $\beta$ at the
fixed depth $\Delta E=12\hbar{\omega_{LO}}=0.198{\rm{eV}}$ and $R=50$ nm. The
fastest channel of the carrier trapping is given in (d) by the pink hollow
circles.

In addition to the defect depth, the localization parameter $\beta$ should
play a key role to determine the carrier trapping processes, because which
describes the different relaxation strength of the same depth defect with
different origins, resulting in the variation of intersection points in the
coordinate configurations for before and after the carrier trapping shown in
FIG. 1 (b). The influence of $\beta$ and ${\Delta E}$ on the $\tau$ are shown
in FIG. 3 (a) and (b). We can see that all possible lifetime (or all possible
channels) of the carrier trapping by the full-configuration defects can be
obtained effectively, whose values are very closely to the results of both
first-principle calculation and experimental measurements.47, 48, 49, 50 More
important is that the fastest channel (the shortest time) of the carrier
trapping can be given as these hollow circles shown in FIG. 3 (b). With the
increasing of the defect depth, the charge carriers are inclined to trap by
these defects with large localization parameter $\beta>1$, which a complement
for the only case of $\beta=1$ in the classical quantum defect model. Huang
and Rhys proposed nonradiation multiphonon recombination from the quantum
treatment first time and pointed out these quantum transitions are thermal
excited processes, depending on the intersection points between two coordinate
configurations as shown in FIG. 1 (b).33 We illustrate the temperature
dependence of the lifetime of the carrier trapping by the fixed defect depth
with different $\beta$ in FIG. 3 (c) and (d). One can see that the temperature
dependence can be divided into two regions as we proposed in previous
studies.41 Namely, (1) the lattice relaxation region with large $\beta$ where
the trapping processes are hindered by the temperature increasing; (2) the
thermal activated region where $\tau$ increases with temperature. However,
among these trapping channels, the fastest one follows the thermal activated
tendency as listed in FIG. 3 (d), which indicates that the thermal activated
mechanism still plays the dominate role for these nonradiation multiphonon
processes in MHPQDs. FIG. $S$5 and $S$6 present the similar results for the
accepter-like and neutral defects.

Lastly, we must emphasize that (1) CH3NH3PbI3 will undergo the phase
transition from orthohombic to tetragonal and then to cubic phase,51, 52, 23,
53 which alters the extent of tilting of PbI6 octahedra and then influences
the phonon energy and the Fr$\mathrm{\ddot{o}}$hlich coupling strength.54, 55
These effect are, however, not considered in this paper; (2) the present
method is not designed to compete with first-principle calculations in
accuracy,56, 55, 57 it is expected to provide qualitatively prediction for the
carrier trapping by variable defects along with simple explanations for the
underlying physics.

###  2.1 Conclusion

In summary, we theoretically study the carrier trapping by various defects via
nonradiative multiphonon processes in the MHPQDs based on Huang-Rhys model. It
is concluded that (1) all possible channels of the carrier trapping by defects
with different depth and localization strength could be obtained and the
corresponding carrier lifetime are very consistent with the experimental
measurements; (2) the smaller quantum dot, the easier carrier trapping; (3)
the fastest channel of the carrier trapping are found with the variable defect
depth, radius of quantum dot and temperature, which still follow the thermal
excited mechanism.

This work was supported by National Natural Science Foundation of China
(Grants 11674241 and 12174283).

The eigenwave function and eigenenergy of an electron in quantum dot, the
quantum defect model with the localization parameter $\beta$, calculations of
Huang-Rhys factor and the trapping probability of carrier by defects,
parameters adopted in calculation for CH3NH3PbI3 and the results, which are
similar to the conclusions in discussion, for the acceptor-like and neutral
defects.

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