tcb@breakable

#  LLM-Enhanced Bayesian Optimization for Efficient Analog Layout Constraint
Generation

Guojin Chen1,2, Keren Zhu1, Seunggeun Kim2, Hanqing Zhu2, Yao Lai3, Bei Yu1,
David Z. Pan2
1Chinese University of Hong Kong 2University of Texas at Austin 3The
University of Hong Kong

###### Abstract

Analog layout synthesis faces significant challenges due to its dependence on
manual processes, considerable time requirements, and performance instability.
Current Bayesian Optimization (BO)-based techniques for analog layout
synthesis, despite their potential for automation, suffer from slow
convergence and extensive data needs, limiting their practical application.
This paper presents the LLANA framework, a novel approach that leverages Large
Language Models (LLMs) to enhance BO by exploiting the few-shot learning
abilities of LLMs for more efficient generation of analog design-dependent
parameter constraints. Experimental results demonstrate that LLANA not only
achieves performance comparable to state-of-the-art (SOTA) BO methods but also
enables a more effective exploration of the analog circuit design space,
thanks to LLM’s superior contextual understanding and learning efficiency. The
code is available at https://github.com/dekura/LLANA.

##  I Introduction

The increasing demand for advanced analog and mixed-signal (AMS) integrated
circuits (ICs) in sectors such as automotive and the Internet of Things (IoT)
necessitates faster design processes and quicker time-to-market. However, the
creation of analog layouts remains a predominantly manual, time-consuming, and
error-prone task. Engineers rely on established practices and insights from
experienced designers, incurring substantial costs and prolonging the design
cycle. The growing complexities of layout-dependent effects in newer
technology nodes further complicate the accurate prediction and assessment of
post-layout performance. Although attempts have been made to automate the
analog layout process, their integration into real-world design practices
remains limited.

Existing tools for analog layout synthesis often come with significant design
complexities or fail to ensure desired post-layout outcomes. Optimization-
driven tools [1] require circuit designers to manually input specific layout
constraints, which are then followed during component placement and routing
(P&R). Methods relying on heuristic constraints face challenges in real-world
applications due to their design-specific nature, lacking adaptability and
universality across different projects. Performance-driven approaches [2]
attempt to account for layout effects by deriving equations, either
analytically or through sensitivity analyses. However, device miniaturization
makes analytical estimations of post-layout effects increasingly inaccurate.
Accurately predicting the impact of design and layout-dependent phenomena,
such as mismatches and parasitics, requires extensive empirical simulations
due to their increased complexity in scaled-down technologies.

Constraint generation plays a crucial role in automatic analog synthesizers
[1, 3], tasked with extracting physical constraints to inform and optimize the
subsequent placement and routing flow. These constraints aim to minimize
discrepancies between pre-layout and post-layout simulations, considering
factors such as device matching, electrical current paths, and thermal
management [4]. While recent advancements [5] have focused on identifying and
enforcing matching constraints, such as symmetry, an exploration into design-
dependent P&R hyperparameters, such as net weighting, remains absent.

Recent progress in analog circuit dimensioning has significantly improved the
effectiveness of leveraging simulations for performance enhancement [3, 6, 7,
8, 9]. These simulation-driven methods approach analog dimensioning as an
optimization problem that does not reveal its internal workings, relying on
circuit simulators to evaluate performance metrics. This approach requires
minimal preliminary knowledge about circuit configurations, contrasting with
model-dependent techniques that necessitate a deep understanding of complex
design and performance interactions. The integration of Gaussian Process
(GP)-based Bayesian Optimization (BO) for automatic transistor sizing [10, 11]
combines the strengths of both paradigms, achieving significant reductions in
simulation requirements while maintaining adaptability across various circuit
designs. This advancement highlights the importance of incorporating iterative
simulations into the design feedback loop, inspiring the development of an
integrated, feedback-oriented analog layout synthesizer.

Figure 1: Overview of the LLANA framework.

Despite advances in BO-based automated analog synthesis [6], BO faces
challenges in high-dimensional tasks due to its computational demands and the
complexity of probabilistic model evaluation. Its effectiveness is further
limited in environments with numerous local optima, owing to its dependence on
prior knowledge and the need to balance exploration and exploitation. The core
of these challenges lies in accurately learning objective functions and
generating solutions with minimal data, a scenario often associated with the
few-shot setting. Large Language Models (LLMs), with their extensive training
on internet data, excel in few-shot learning, demonstrating remarkable
abilities in prediction, generation [12], and contextual understanding [13].
This success is partially attributed to their utilization of encoded priors.

As depicted in Fig. 1, this paper investigates the feasibility of leveraging
LLMs to improve model-based BO for analog design-dependent parameter
generation and fine-tuning, extending LLM applications beyond traditional
natural language tasks. We propose a novel approach that utilizes natural
language representations for BO components, aiming to harness the unique
advantages of LLMs. The central inquiry of our research is whether LLMs’
inherent knowledge and few-shot learning capabilities can enhance crucial
aspects of BO, particularly in generating analog layout constraints, and
assess the efficiency of an LLM-augmented BO pipeline functioning seamlessly
from start to finish. Our main contributions are as follows:

  1. 1.

We formulated the analog layout design-dependent parameter space and provided
a benchmark for BO based on Gaussian processes.

  2. 2.

For the first time, we applied LLM-enhanced BO to analog design-dependent
constraint generation, exploring the capabilities of LLM-based design space
exploration.

  3. 3.

All the algorithms, experiments and benchmarks are open-sourced at
https://github.com/dekura/LLANA.

##  II Preliminaries and Background

###  II-A Analog placement

Analog circuit layout synthesis involves placement and routing for both
building and macro blocks. Placement optimizes component locations within a
predefined bounding box to minimize wire lengths, while routing finalizes the
layout using established component positions, pin locations, and connection
data. Symmetry constraints are respected to minimize mismatches. The placement
and routing engine is design-independent, ensuring efficiency and coherence.
The objective function $f$ for analytical global placement is defined as [14]:

$f=\sum_{i}\alpha_{i}\cdot f_{WL_{i}}+\beta\cdot f_{AREA}+f_{other},$ (1)

where $f_{WL_{i}}$ is the wirelength of net $i$, $f_{AREA}$ is the total
layout area, $\alpha_{i}$ and $\beta$ are the net weightings and area
minimization factor, and $f_{other}$ represents other necessary objectives.

###  II-B Analog design-dependent parameter constraint:

Analog design-dependent parameter constraints, along with symmetric
constraints, define the layout and efficiency of integrated circuits. Net
weightings ($\alpha_{i}$ in eq. 1) direct global placement strategies,
affecting the floorplan and performance. Higher weights reduce parasitics by
aligning connected components more closely. Wire widths and routing sequences
manage the trade-off between parasitic resistance and capacitance, determining
layout complexity. Hyperparameters, such as the area minimization factor
($\beta$ in eq. 1) and net spacing, impact placement and routing optimization,
reducing coupling interference. Design-dependent parameters and
hyperparameters are crucial for optimizing analog circuit performance and
layout. This paper focuses on the impact of net weightings $x_{i}$ on layout
performance.

##  III Algorithm and Framework

###  III-A BO-based methods as baselines

Prior works of analog constraint generation have primarily focused on multi-
objective Bayesian optimization(MOBO) and using Gaussian processes(GP) as
surrogate model [6], formulated as:

$\min(f_{1}(x),f_{2}(x),\ldots,f_{m}(x)),$ (2)

where $f_{i}(x)$ represents the post layout performance metric obtained
through simulation after automatic layout generation using Magical [14], and
$x\in\mathbb{R}^{d}$ are the optimal design specific layout parameters
introduced in Section II-B that minimize the performance metrics.

Algorithm 1 MOBO [6]

1: Sampled data points $x^{t}$, $\\{f_{i}(x^{t})\\}_{i=1}^{m}$

2: Next net weighting $x_{t+1}$

3: Initialize Pareto set $P$ to $\emptyset$;

4: Initialize Pareto set $P$ to $\emptyset$; function MOBO($x^{t}$,
$\\{f_{i}(x^{t})\\}_{i=1}^{m}$)

5: function MOBO($x^{t}$, $\\{f_{i}(x^{t})\\}_{i=1}^{m}$) if
$t<N_{\text{random}}$ then

6: Random sample $x_{t+1}$ from the design space;

7: Random sample $x_{t+1}$ from the design space; Random sample $x_{t+1}$ from
the design space; else

8: Update $P$ with $x^{t}$ and reference point $r$;

9: Update GP models with $f_{i}(x)$;

10: Update GP model with $x^{t}$, $\\{f_{i}(x^{t})\\}_{i=1}^{m}$;

11: Optimize acquisition function to obtain $x_{t+1}$; Optimize acquisition
function to obtain $x_{t+1}$;

12: return $x_{t+1}$; return $x_{t+1}$;

As presented in Algorithm 1, MOBO iteratively samples data points $x^{t}$ and
their corresponding performance metrics $\\{f_{i}(x^{t})\\}_{i=1}^{m}$ to
update the GP models and optimize the acquisition function to obtain the next
net weighting $x_{t+1}$.

###  III-B Large Language Models (LLMs)-enhanced BO

BO’s effectiveness depends on the quality of surrogate models and sampling
strategies, challenged by limited data, sensitivity to inaccuracies, and the
difficulty of incorporating prior knowledge into new tasks. Recently, there
has been a growing trend of utilizing AI and LLMs to address challenges within
the EDA domain [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. [43] explained LLM in-
context learning (ICL) as performing implicit Bayesian inference [12]. LLMs
can enhance BO by leveraging: (1) prior knowledge through ICL for tapping into
pre-trained insights [44], (2) the ability to generalize from limited
examples, aiding in efficient exploration [45], and (3) processing contextual
information to enrich optimization tasks and search strategies [46].

###  III-C LLM-enhanced BO Framework

LLM-enhanced Initial Design: As illustrated in Fig. 2, by leveraging LLM’s
‘zero-shot’ capabilities, we can generate better initial designs through
prompt engineering.

Assist me with automated machine learning using {model}. Explore these
hyperparameters: {configurations, type, ranges}. Suggest {number of
recommendations} diverse, effective configs for BO hyperparameter tuning,
without “None”. Respond with an un-enumerated list of dictionaries, each
describing a recommended config. Figure 2: Prompt for initial design
generation

LLM-enhanced surrogate modeling: BO constructs a surrogate model $p(f|x)$
using $m$ observed input-output pairs
$\mathcal{X}_{m}:=\\{(x_{i},f_{i})\\}_{i=1}^{m}$. Common models include the GP
[6] and random forests (SMAC) [47]. The LLANA framework serializes the
optimization trajectory into natural text, e.g., for an RF model, ‘[max_depth
is 15, min_samples_split is 0.5, …, performance is 0.9]’. These text
representations, denoted as $\mathcal{D}_{m}$, along with the problem
description and queried few-shot examples $x_{k}^{m1}$, are input to the LLM,
denoted as $\mathcal{D}_{m}^{m1}$. The LLM outputs a predicted score and
associated probability: $(\hat{f}_{k},p(\hat{f}_{k}))=$ LLANA
$(x_{k}^{m1},\mathcal{D}_{m}^{m1})$. A shuffling mechanism randomly permutes
the few-shot examples within $\mathcal{D}_{m}^{m1}$ to enhance robustness. The
in-context learning prompts for the LLM-enhanced surrogate model are shown in
Fig. 3.

The following are examples of the performance of a {model} measured in
{metric} and the corresponding model hyperparameter configurations. The model
is evaluated on a tabular {task} containing {number of classes}. The tabular
dataset contains {number of samples} samples and {number of features} features
({number of categorical features} categorical, {number of continuous features}
numerical).
Your response should only contain the predicted accuracy in the format ##
performance##.
Hyperparameter configuration: {C1}. Performance: {P1}.
…
Hyperparameter configuration: {Cn}. Performance: {Pn}.
Hyperparameter configuration: {configuration to predict}. Performance:{}.
Figure 3: Prompt for discriminative surrogate model.

Acquisition strategies: Sampling candidate points is crucial in BO, as high-
potential points can accelerate convergence to the optimum. LLANA introduces a
novel mechanism to conditionally generate candidate points based on desired
objective values through ICL. The samples are generated from high-potential
regions by conditioning on a desired objective value
$x^{\prime}:\tilde{h}_{m}\sim p(f|x^{\prime};\mathcal{D}_{m})$, leveraging the
few-shot generation capabilities of LLMs. The desired objective value is
defined as $x^{\prime}\\!=\\!x_{\min}-\alpha\\!\times\\!(x_{\max}-x_{\min})$,
where $x_{\max}$ and $x_{\min}$ are the worst and best observed points at
related objective values, and $\alpha$ is an exploration hyperparameter.
Positive $\alpha$ sets $x^{\prime}$ to improve over $x_{\min}$, while negative
$\alpha$ results in a more conservative target value within the observed
range. We implement $p(f|x^{\prime};\mathcal{D}_{m})$ through ICL, generating
$M$ candidate points independently, i.e.,
$\tilde{h}_{k}\sim\operatorname{\texttt{LLANA}}(x^{\prime},\mathcal{D}_{m}^{m1})$,
and select the point that maximizes the acquisition function as the next point
to evaluate, using a sampling-based approximation to optimize the acquisition
function.

The following are examples of the performance of a {model} in {metric} and
corresponding hyperparameter configs, evaluated on a tabular {task} task with
{number of classes} classes, {number of samples} samples, {number of features}
features ({number of categorical features} categorical, {number of continuous
features} numerical). Hyperparameter ranges: {configuration and type}.
Recommend a config to achieve {target score}, avoiding min/max/rounded values,
using highest precision. Respond with only the predicted config, as ##
configuration ##.
Performance: {P1}. Hyperparameter config: {C1}
…
Performance: {Pn}. Hyperparameter config: {Cn}
Performance: {performance used to sample configuration}
Hyperparameter config:{}. Figure 4: Prompt for candidate sampling.

End-to-end LLANA framework: The end-to-end procedure iteratively performs
three steps, as depicted in Algorithm 2. (1) sample $M$ candidate points
$\\{\tilde{h}_{m}\\}_{m=1}^{M}$ through ICL. (2) evaluate $M$ points using the
ICL surrogate model, i.e. $p(f\mid\tilde{h}_{m})$ to obtain scores
$\\{a(\tilde{h}_{m})\\}_{m=1}^{M}$ according to an acquisition function. (3)
select point with the highest score to evaluate next,
$h=\operatorname*{argmax}_{\tilde{h}\in\\{\tilde{h}_{m}\\}_{m=1}^{M}}a(\tilde{h})$.
In LLANA , we use expected improvement (EI),
$a(\tilde{h}_{m})=\mathbb{E}[\max(p(f\mid\tilde{h}_{m})-f(h_{\text{best
}}),0)]$.

Algorithm 2 End-to-end LLANA-BO with ICL

1: Initial best point $h_{\text{best}}$

2: Optimal point $h^{\star}$

3: Optimal point $h^{\star}$ function LLANA($h_{m}$)

4: Sample $M$ candidate points $\\{\tilde{h}_{m}\\}_{m=1}^{M}$ through ICL;

5: Evaluate $p(f\mid\tilde{h}_{m})$ using the ICL surrogate model;

6: EI score
$a(\tilde{h}_{m})=\mathbb{E}[\max(p(f\mid\tilde{h}_{m})-f(h_{\text{best}}),0)]$;

7:
$h=\operatorname*{argmax}_{\tilde{h}\in\\{\tilde{h}_{m}\\}_{m=1}^{M}}a(\tilde{h})$;

8:
$h=\operatorname*{argmax}_{\tilde{h}\in\\{\tilde{h}_{m}\\}_{m=1}^{M}}a(\tilde{h})$;
if $f(h)>f(h_{\text{best}})$ then

9: $h_{\text{best}}=h$; $h_{\text{best}}=h$;

10: return $h_{\text{best}}$ as $h$; return $h_{\text{best}}$ as $h$;

##  IV Experiments

The LLANA framework is implemented in Python, utilizing the ‘gpt-3.5-turbo’
model from OpenAI with hyperparameters set to $\alpha=-0.1$ and $M=20$. The
designs used in the experiments are two-stage operational amplifiers from [6],
and the performance benchmark is evaluated by Cadence Spectre after layout
generation using Magical [14]. The design contains a total of 36 devices with
14 design-dependent parameters of the critical net weighting selected by the
MOBO algorithm introduced in Algorithm 1. A dataset of 500 design-performance
pairs is prepared, with 400 pairs used for training and 100 pairs for testing.
The optimization objectives chosen are the common-mode rejection ratio (CMRR)
and absolute input-referred offset (Offset) voltage. The experiments include
three ML models: RandomForest (RF), AdaBoost, and DecisionTree. The scoring
function is the mean squared error (MSE), and the acquisition function is
expected improvement (EI).

###  IV-A Evaluation of the surrogate model

The performance of the LLANA framework is first evaluated against single-
objective GP and SMAC [47] on both CMRR and Offset datasets. For GP and SMAC,
the average result of three ML models is used. The performance of the
surrogate model is assessed using prediction performance and uncertainty
calibration metrics. The Normalized Root Mean Square Error (NRMSE) and the
coefficient of determination ($R^{2}$ score) of the prediction on the test set
are used as performance metrics. Calibration is evaluated using the log
predictive density (LPD) and the normalized regret. The regret metric is
defined as
$\min_{h\in\mathcal{H}_{t}}(f(h)-f_{\text{min}}^{*})/(f_{\text{max}}^{*}-f_{\text{min
}}^{*})$, where $\mathcal{H}_{t}$ denotes the points chosen up to trail $t$,
and $f_{\text{max}}^{*}$,$f_{\text{min }}^{*}$ represent the best and worst
scores [47].

Figure 5: Comparison of LLANA , GP, and SMAC [47] on CMRR dataset. The x-axis
is the number of observed points.

Fig. 5 plots the performance of LLANA against GP and SMAC on the CMRR dataset.
LLANA outperforms GP and SMAC in terms of NRMSE and $R^{2}$ score. Moreover,
this trend becomes more apparent when there are more observed points. As for
normalized regret, LLANA attains lower regret than GP and SMAC when $n<15$.
For uncertainty calibration, LLANA achieves lower LPD than GP and SMAC,
producing the best uncertainty quantification.

Fig. 6 shows the performance of LLANA against GP and SMAC on the Offset
dataset. This time, as the number of observed points increases, LLANA can
achieve NRMSE and $R^{2}$ scores close to those of GP and SMAC when $n=30$.
However, the regret and LPD of LLANA consistently remain far ahead of GP and
SMAC. Additionally, we observe that, consistent with prior findings, LLANA
excels in earlier stages of the search, when fewer observations are available.
As such, empirical evidence supports that permuting few-shot examples, while
straightforward in implementation, improves both uncertainty quantification
and prediction performance, both critical aspects of balancing exploration and
exploitation. LLANA performs effectively as an end-to-end pipeline, exhibiting
sample-efficient search. Its modularity further enables individual components
to be integrated into existing frameworks. Surrogate models implemented
through ICL can produce effective regression estimates with uncertainty,
although there is a tradeoff of stronger prediction performance with worse
calibration than probabilistic methods. The LLM’s encoded prior is crucial to
improving the efficacy of such surrogate models.

Figure 6: Comparison of LLANA , GP, and SMAC [47] on Offset dataset. The
x-axis is the number of observed points.

##  V Conclusion and Discussion

This work introduces LLANA , a novel framework that integrates LLMs with BO to
address the challenge of generating analog net weighting constraints. The
approach incorporates two key enhancements: surrogate models of the objective
function through ICL and a candidate point sampler capable of conditional
generation for specific target values. The study on the analog constraint
problem reveals performance improvements across the three integrations,
particularly when working with limited sample sizes. Moreover, LLANA
demonstrates potential as a stand-alone BO method, exhibiting slightly better
results on CMRR and Offset benchmarks compared to existing techniques.
However, further research is necessary to fully assess the extent of LLANA ’s
capabilities and its potential for generalization across a broader range of
optimization tasks. Codes and experiment results are available at
https://github.com/dekura/LLANA.

Limitations and future work. Despite LLANA ’s higher computational cost due to
LLM inference, the findings suggest that it trades off computational
complexity for improved sample efficiency, which is essential in black-box
optimization. This indicates the potential for combining LLANA with more
computationally efficient methods to achieve better solutions. Furthermore,
unlike the multi-objective approach in [6], this work focuses solely on
single-objective BO. A promising avenue for future research is extending LLANA
to handle multi-objective and higher-dimensional BO tasks with more complex
search spaces, enhancing its applicability and impact in optimization.

## References

  * [1] K. Kunal, M. Madhusudan, A. K. Sharma, W. Xu, S. M. Burns, R. Harjani, J. Hu, D. A. Kirkpatrick, and S. S. Sapatnekar, “INVITED: ALIGN – Open-Source Analog Layout Automation from the Ground Up,” in _ACM/IEEE Design Automation Conference (DAC)_ , 2019.
  * [2] H.-C. Ou, K.-H. Tseng, J.-Y. Liu, I.-P. Wu, and Y.-W. Chang, “Layout-dependent-effects-aware analytical analog placement,” in _ACM/IEEE Design Automation Conference (DAC)_ , 2015.
  * [3] H. Chen, M. Liu, X. Tang, K. Zhu, A. Mukherjee, N. Sun, and D. Z. Pan, “MAGICAL 1.0: An open-source fully-automated ams layout synthesis framework verified with a 40-nm 1GS/s $\Delta$$\sum$ ADC,” in _IEEE Custom Integrated Circuits Conference (CICC)_ , 2021.
  * [4] M. P.-H. Lin, Y.-W. Chang, and C.-M. Hung, “Recent research development and new challenges in analog layout synthesis,” in _IEEE/ACM Asia and South Pacific Design Automation Conference (ASPDAC)_ , 2016.
  * [5] K. Zhu, H. Chen, M. Liu, and D. Z. Pan, “Automating analog constraint extraction: From heuristics to learning: (invited paper),” in _IEEE/ACM Asia and South Pacific Design Automation Conference (ASPDAC)_ , 2022.
  * [6] M. Liu, K. Zhu, J. Gu, L. Shen, X. Tang, N. Sun, and D. Z. Pan, “Closing the design loop: Bayesian optimization assisted hierarchical analog layout synthesis,” in _ACM/IEEE Design Automation Conference (DAC)_ , 2020.
  * [7] K. Zhu, H. Chen, M. Liu, X. Tang, N. Sun, and D. Z. Pan, “Effective analog/mixed-signal circuit placement considering system signal flow,” in _IEEE/ACM International Conference on Computer-Aided Design (ICCAD)_ , 2020\.
  * [8] P. Xu, G. Chen, K. Zhu, T. Chen, T.-Y. Ho, and B. Yu, “Performance-driven analog routing via heterogeneous 3dgnn and potential relaxation,” in _ACM/IEEE Design Automation Conference (DAC)_ , 2024.
  * [9] H. Zhu, K. Zhu, J. Gu, H. Jin, R. T. Chen, J. A. Incorvia, and D. Z. Pan, “Fuse and mix: Macam-enabled analog activation for energy-efficient neural acceleration,” in _Proceedings of the 41st IEEE/ACM International Conference on Computer-Aided Design_ , ser. ICCAD ’22. New York, NY, USA: Association for Computing Machinery, 2022\.
  * [10] W. Lyu, F. Yang, C. Yan, D. Zhou, and X. Zeng, “Multi-objective bayesian optimization for analog/RF circuit synthesis,” in _ACM/IEEE Design Automation Conference (DAC)_ , 2018.
  * [11] S. Zhang, W. Lyu, F. Yang, C. Yan, D. Zhou, X. Zeng, and X. Hu, “An efficient multi-fidelity bayesian optimization approach for analog circuit synthesis,” in _ACM/IEEE Design Automation Conference (DAC)_ , 2019.
  * [12] T. Brown, B. Mann, N. Ryder, M. Subbiah, J. D. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell _et al._ , “Language models are few-shot learners,” _Annual Conference on Neural Information Processing Systems (NeurIPS)_ , 2020.
  * [13] J. Wei, Y. Tay, R. Bommasani, C. Raffel, B. Zoph, S. Borgeaud, D. Yogatama, M. Bosma, D. Zhou, D. Metzler _et al._ , “Emergent abilities of large language models,” _arXiv preprint arXiv:2206.07682_ , 2022.
  * [14] B. Xu, K. Zhu, M. Liu, Y. Lin, S. Li, X. Tang, N. Sun, and D. Z. Pan, “MAGICAL: Toward Fully Automated Analog IC Layout Leveraging Human and Machine Intelligence: Invited Paper,” in _IEEE/ACM International Conference on Computer-Aided Design (ICCAD)_ , 2019.
  * [15] Y. Lai, S. Lee, G. Chen, S. Poddar, M. Hu, D. Z. Pan, and P. Luo, “AnalogCoder: Analog circuit design via training-free code generation,” _arXiv preprint arXiv:2405.14918_ , 2024.
  * [16] Y. Lai, J. Liu, D. Z. Pan, and P. Luo, “Scalable and effective arithmetic tree generation for adder and multiplier designs,” _arXiv preprint arXiv:2405.06758_ , 2024.
  * [17] H. Ren and J. Hu, _Machine Learning Applications in Electronic Design Automation_. Springer, 2022.
  * [18] S. Pujar, L. Buratti, X. Guo, N. Dupuis, B. Lewis, S. Suneja, A. Sood, G. Nalawade, M. Jones, A. Morari, and R. Puri, “Invited: Automated code generation for information technology tasks in yaml through large language models,” in _ACM/IEEE Design Automation Conference (DAC)_ , 2023.
  * [19] Z. He, H. Wu, X. Zhang, X. Yao, S. Zheng, H. Zheng, and B. Yu, “ChatEDA: A Large Language Model Powered Autonomous Agent for EDA,” in _ACM/IEEE Workshop on Machine Learning CAD (MLCAD)_ , 2023.
  * [20] G. Chen, W. Chen, Y. Ma, H. Yang, and B. Yu, “DAMO: Deep agile mask optimization for full chip scale,” in _IEEE/ACM International Conference on Computer-Aided Design (ICCAD)_ , 2020.
  * [21] J. Blocklove, S. Garg, R. Karri, and H. Pearce, “Chip-Chat: Challenges and Opportunities in Conversational Hardware Design,” in _ACM/IEEE Workshop on Machine Learning CAD (MLCAD)_ , 2023.
  * [22] L. Wen, Y. Zhu, L. Ye, G. Chen, B. Yu, J. Liu, and C. Xu, “LayouTransformer: Generating layout patterns with transformer via sequential pattern modeling,” in _IEEE/ACM International Conference on Computer-Aided Design (ICCAD)_ , 2022.
  * [23] W. Zhao, X. Yao, Z. Yu, G. Chen, Y. Ma, B. Yu, and M. D. F. Wong, “AdaOPC: A self-adaptive mask optimization framework for real design patterns,” in _IEEE/ACM International Conference on Computer-Aided Design (ICCAD)_ , 2022\.
  * [24] S. Thakur, B. Ahmad, H. Pearce, B. Tan, B. Dolan-Gavitt, R. Karri, and S. Garg, “VeriGen: A Large Language Model for Verilog Code Generation,” _ACM Transactions on Design Automation of Electronic Systems (TODAES)_ , 2023.
  * [25] Z. Wang, Y. Shen, W. Zhao, Y. Bai, G. Chen, F. Farnia, and B. Yu, “DiffPattern: Layout pattern generation via discrete diffusion,” in _ACM/IEEE Design Automation Conference (DAC)_ , 2023.
  * [26] S. Liu, W. Fang, Y. Lu, Q. Zhang, H. Zhang, and Z. Xie, “Rtlcoder: Outperforming gpt-3.5 in design rtl generation with our open-source dataset and lightweight solution,” _arXiv preprint arXiv:2312.08617_ , 2023.
  * [27] G. Chen, Z. Yu, H. Liu, Y. Ma, and B. Yu, “DevelSet: Deep neural level set for instant mask optimization,” in _IEEE/ACM International Conference on Computer-Aided Design (ICCAD)_ , 2021.
  * [28] Z. He and B. Yu, “Large Language Models for EDA: Future or Mirage?” in _ACM International Symposium on Physical Design (ISPD)_ , 2024.
  * [29] G. Chen, Z. Pei, H. Yang, Y. Ma, B. Yu, and M. Wong, “Physics-informed optical kernel regression using complex-valued neural fields,” in _ACM/IEEE Design Automation Conference (DAC)_ , 2023.
  * [30] M. Liu, T.-D. Ene, R. Kirby, C. Cheng, N. Pinckney, R. Liang, J. Alben, H. Anand, S. Banerjee, I. Bayraktaroglu _et al._ , “ChipNeMo: Domain-Adapted LLMs for Chip Design,” _arXiv preprint arXiv:2311.00176_ , 2023.
  * [31] Z. Pei, F. Liu, Z. He, G. Chen, H. Zheng, K. Zhu, and B. Yu, “AlphaSyn: Logic synthesis optimization with efficient monte carlo tree search,” in _IEEE/ACM International Conference on Computer-Aided Design (ICCAD)_ , 2023\.
  * [32] M. Liu, N. Pinckney, B. Khailany, and H. Ren, “Verilogeval: Evaluating large language models for verilog code generation,” in _IEEE/ACM International Conference on Computer-Aided Design (ICCAD)_ , 2023.
  * [33] G. Chen, Z. Wang, B. Yu, D. Z. Pan, and M. D. Wong, “Ultra-Fast Source Mask Optimization via Conditional Discrete Diffusion,” _IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems (TCAD)_ , 2024.
  * [34] Y. Lu, S. Liu, Q. Zhang, and Z. Xie, “Rtllm: An open-source benchmark for design rtl generation with large language model,” _arXiv preprint arXiv:2308.05345_ , 2023.
  * [35] G. Chen, H. He, P. Xu, H. Geng, and B. Yu, “Efficient bilevel source mask optimization,” in _ACM/IEEE Design Automation Conference (DAC)_ , 7 2024\.
  * [36] L. Chen, Y. Chen, Z. Chu, W. Fang, T.-Y. Ho, Y. Huang, S. Khan, M. Li, X. Li, Y. Liang _et al._ , “The Dawn of AI-Native EDA: Promises and Challenges of Large Circuit Models,” _arXiv preprint arXiv:2403.07257_ , 2024.
  * [37] B. Zhu, S. Zheng, Z. Yu, G. Chen, Y. Ma, F. Yang, B. Yu, and M. Wong, “L2O-ILT: Learning to Optimize Inverse Lithography Techniques,” _IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems (TCAD)_ , 2023.
  * [38] R. Liang, A. Agnesina, G. Pradipta, V. A. Chhabria, and H. Ren, “Invited Paper: CircuitOps: An ML Infrastructure Enabling Generative AI for VLSI Circuit Optimization,” in _IEEE/ACM International Conference on Computer-Aided Design (ICCAD)_ , 2023.
  * [39] W. Li, G. Chen, H. Yang, R. Chen, and B. Yu, “Learning Point Clouds in EDA,” in _ACM International Symposium on Physical Design (ISPD)_ , 2021\.
  * [40] M. DeLorenzo, A. B. Chowdhury, V. Gohil, S. Thakur, R. Karri, S. Garg, and J. Rajendran, “Make Every Move Count: LLM-based High-Quality RTL Code Generation Using MCTS,” _arXiv preprint arXiv:2402.03289_ , 2024.
  * [41] Y. Lai, J. Liu, Z. Tang, B. Wang, J. Hao, and P. Luo, “ChiPFormer: transferable chip placement via offline decision transformer,” in _Proceedings of the 40th International Conference on Machine Learning_ , ser. ICML’23. JMLR.org, 2023.
  * [42] Y. Lai, Y. Mu, and P. Luo, “MaskPlace: Fast chip placement via reinforced visual representation learning,” in _Advances in Neural Information Processing Systems_ , vol. 35, 2022.
  * [43] S. M. Xie, A. Raghunathan, P. Liang, and T. Ma, “An explanation of in-context learning as implicit bayesian inference,” _arXiv preprint arXiv:2111.02080_ , 2021.
  * [44] S. Mirchandani, F. Xia, P. Florence, B. Ichter, D. Driess, M. G. Arenas, K. Rao, D. Sadigh, and A. Zeng, “Large language models as general pattern machines,” _arXiv preprint arXiv:2307.04721_ , 2023.
  * [45] T. Liu, N. Astorga, N. Seedat, and M. van der Schaar, “Large Language Models to Enhance Bayesian Optimization,” _arXiv preprint arXiv:2402.03921_ , 2024\.
  * [46] C. Yang, X. Wang, Y. Lu, H. Liu, Q. V. Le, D. Zhou, and X. Chen, “Large language models as optimizers,” _arXiv preprint arXiv:2309.03409_ , 2023\.
  * [47] M. Lindauer, K. Eggensperger, M. Feurer, A. Biedenkapp, D. Deng, C. Benjamins, T. Ruhkopf, R. Sass, and F. Hutter, “SMAC3: A versatile bayesian optimization package for hyperparameter optimization,” _Journal of Machine Learning Research (JMLR)_ , 2022.


# Reconstruction of the Electron–Phonon Interaction Function for
Superconductors Using Inhomogeneous Point-Contacts and Background Correction
in Yanson Spectra

N. L. Bobrov  B. Verkin Institute for Low Temperature Physics and Engineering,
National Academy of Sciences of Ukraine
prosp. Nauki 47, 61103 Kharkov, Ukraine<EMAIL_ADDRESS>
(_JETPr_ , 160(1), 73, (2021); [_JETPe_ , 133(1), ??, (2021)]; August 27,
2024)

###### Abstract

_The application of inhomogeneous niobium point-contacts in the
superconducting state for reconstructing the electron–phonon interaction
function is considered. The method is based on the use of the nonlinearity of
current–voltage curve, which is due to the inelastic processes of suppressing
excess point-contact current when nonequilibrium phonons are scattered from
the electrons undergoing Andreev reflection. A new model of background
occurrence in point-contact Yanson spectra and some ways to correct this
background are proposed_.

Keywords: Yanson point-contact spectroscopy, electron-phonon coupling,
superconductivity, energy gap, excess current.

###### pacs:

73.40.Jn, 74.25.Kc, 74.45.+c

###### Contents

  1. 1 INTRODUCTION
  2. 2 EXPERIMENTAL
  3. 3 EXPERIMENTAL RESULTS AND THEIR PROCESSING
  4. 4 BACKGROUND CORRECTION IN POINT-CONTACT YANSON SPECTRA
  5. 5 RESULTS AND DISCUSSION
  6. 6 BRIEF CONCLUSIONS

##  1 INTRODUCTION

The spectroscopy of electron–phonon interaction (EPI) of point-contacts with
direct conductivity is based on duplication of carriers: in the current-
conducting state electrons are divided into two groups, for which the energy
difference between the occupied and free electrons states on the Fermi surface
is several millielectronvolts, i.e., is equivalent to the applied voltage [1,
2]. The number of studies devoted to the determination of the EPI function
using point-contacts in the normal state exceeds several hundred; two
monographs generalizing the results obtained in this field have been published
[3, 4].

For the superconducting state, the use of the nonlinearities of the excess
current of ballistic point contacts in the phonon energy range for
reconstructing the EPI function was justified theoretically in the studies by
Khlus and Omel’yanchuk in 1983 [5, 6]. However, their results found
experimental confirmation only almost 30 years after [7, 8]. This time delay
was caused by the existence of superconducting background (disregarded by the
theory), which led to the necessity of using the difference in the second
derivatives of current–voltage ($I\textendash V$) curves in the
superconducting and normal states for this procedure, whereas the theory
predicted that the EPI function is proportional to the first derivative of
excess current or the difference in the first derivatives for the
superconducting and normal states. The mechanism of the formation of excess-
current nonlinearity in the phonon energy range is related to the scattering
of nonequilibrium phonons, generated by electrons with excess energy
(equivalent to the voltage applied to the contact), from the Andreev
electrons. The nonlinearity of the normal-state $I\textendash V$ curve is
formed in a volume with a characteristic size of the same order of magnitude
as the contact diameter. This is related to the geometric limitations of the
backscattering processes responsible for this nonlinearity. The backscattering
processes are considered to be the processes of electron scattering from
nonequilibrium phonons, in which an electron returns as a result of scattering
to the electrode it was emitted from.

The processes of scattering of nonequilibrium phonons from Andreev electrons
are not restricted by these limitations and, in principle, may occur at any
distance from the short-circuit in which they coexist. Nevertheless, the
theory takes into account only the scattering processes occurring in the
region of high current density, where the concentrations of both components
are high. On the whole, the conclusions of the theory were confirmed for
superconductors with a large coherence length, such as, e.g., tin or aluminum
[8]. At the same time, it was found for ballistic tantalum point-contacts that
the region near a shortcircuit with a characteristic size on the order of
coherence length is also involved in the formation of excesscurrent
nonlinearity [7, 9]. The contribution of this region turned out to be much
larger than that from the region with a size on the order of contact diameter.
The significant difference between the superconducting additive to the
spectrum for tantalum and that for superconductors with a larger coherence
length calls for application of peculiar procedures when reconstructing the
EPI function using this additive. These procedures were found useful when
studying the inhomogeneous superconducting point-contacts. The experimental
data on the reconstruction of EPI functions for different superconductors from
the additional nonlinearity arising as a result of the transition of point-
contact with direct conductivity to the superconducting state were reviewed in
[10].

For ballistic point-contacts the nonlinearity of the $I\textendash V$ curve in
the normal state is at least of the same order of magnitude or even larger
than the additional excess-current nonlinearity, arising when a contact passes
to the superconducting state. The necessity of selecting this nonlinearity in
the pure form for reconstructing the EPI function reduces to a great extent
the practical validity of this procedure, because the transfer of many
superconductors to the normal state at low temperatures is highly hindered or
even impossible for technical reasons.

One could get out of this dead-end situation if it were possible to cancel of
the nonlinearity related to the backscattering processes responsible for the
nonlinearity in the normal state, having retained at the same time the
additional excess-current nonlinearity caused by the scattering of
nonequilibrium phonons from Andreev electrons. Since this situation cannot be
implemented in ballistic contacts, we will consider possible alternatives.

The spectrum intensity in the normal state decreases as a result of the
contamination of material in the narrowing region, which is accompanied by the
decrease in the elastic mean free path of electrons. A limiting case of this
decrease is observed in diffusion point contacts, i.e., the contacts
satisfying the condition ${{l}_{i}}\ll d\ll{{\Lambda}_{\varepsilon}}$, where
$d$ is the contact diameter, $l_{i}$ is the elastic relaxation length,
$l_{\varepsilon}$ is the energy relaxation diffusion length,
${{\Lambda}_{\varepsilon}}=\sqrt{{{l}_{i}}{{l}_{\varepsilon}}/3}$, and
$l_{\varepsilon}$ is the energy relaxation length. The spectrum intensity in
the diffusion mode is lower than that in the ballistic mode by a factor of
$\sim l_{i}/d$ in order of magnitude.

In accordance with the theoretical model [11], the second derivative of the
$I\textendash V$ curve of diffusion contact differs from that for a ballistic
contact in only lower intensity. In addition, due to the isotropization of
electron-momentum distribution, one can observe almost complete isotropization
of the EPI spectrum. This may lead to only a small broadening of spectrum and
insignificant variation in its shape. The theory does not predict any other
consequences of the reduction of electron elastic relaxation length.

As was mentioned above, the experimentally observed decrease in the elastic
scattering length is due contamination of the point-contact, i.e., the
increase in the concentration of impurities and lattice defects. Figures 1 and
2 from [12] demonstrate an increase in the background and degradation of high-
energy phonon peaks in copper point-contact spectra with an increase in the
concentration of manganese and iron impurities. A method for direct
determination of the electron elastic scattering length $l_{i}$ in a point-
contact was reported in [13]. For relatively small contacts, the transition
from the ballistic to diffusion mode, when the elastic relaxation length
becomes comparable with the characteristic lattice parameters, is accompanied
by significant lattice distortions. Along with the diffusion of spectra,
predicted by the theory, these distortions lead to strong suppression of high-
energy phonon features, up to their complete disappearance, and to a high
background level [12, 14]. Note that the suppression of high-energy phonons is
also characteristic of the EPI functions reconstructed in tunnel experiments
for films with a distorted lattice [15].

As was noted above, in view of the geometric limitations, backscattering
processes are efficient in a contact volume with a characteristic size on the
same order of magnitude as the contact diameter. However, for diffusion
contacts under certain conditions, these processes can be efficient only near
the contact center. If the maximum concentration of defects and impurities is
obtained at the boundary between the electrodes and rapidly decreases at the
contact periphery, the electrons scattered from nonequilibrium phonons will
diffuse with high probability along the gradient of decreasing impurity
concentration. In other words, the backscattering processes, determining the
spectrum in the normal state, will be concentrated mainly near the electrode
contact interface. This may lead to an increase in the surface-phonon
contribution to the EPI spectrum and, correspondingly, to additional smearing
of the spectrum. Obviously, if only one of the bank of the point contact is in
the diffusion mode, then in the resulting spectrum the intensity of each
partial contribution will be determined by the dirty bank.

Nevertheless, all phonon features remain at their positions in the normal
state in these contacts with a very high background level, diffuse spectra,
and strongly suppressed high-energy phonons [12, 14]. This is an unambiguous
evidence for preservation of electron duplication in these contacts [1, 2];
i.e., they are spectroscopic.

Let us now return to consideration of the excesscurrent nonlinearity caused
by the scattering of nonequilibrium phonons from Andreev electrons. As was
noted above, in tantalum ballistic point contacts [7, 9], the contact banks
make a significant contribution to the additional nonlinearity of
$I\textendash V$ curve during the transition to the superconducting state
because of the smaller coherence length. The formation of excess-current
nonlinearity in these contacts occurs in a volume with a characteristic size
on the order of coherence length. Obviously, the value of this nonlinearity in
point-contacts from similar superconductors (with a comparable coherence
length) will depend on both the excess current value and on the volume within
which this nonlinearity is formed. Excess current for point contact in
diffusion mode is 55% of its value in ballistic contact, and the corresponding
nonlinearity will be formed approximately in the same volume with a
characteristic size on the order of coherence length; therefore, one might
expect the additional nonlinearity of the $I\textendash V$ curve of
superconducting state in the diffusion mode to be only a half of that in the
ballistic mode.

Thus, is appears that the transfer of contact to the diffusion mode may solve
the above-stated problem: one has a significant decrease in the contribution
of backscattering processes to the total nonlinearity and a relatively small
decrease in the contribution of the scattering of nonequilibrium phonons from
Andreev electrons. However, it is pertinent to remind here about the influence
of defects and impurities on the shape of EPI spectra. As was noted in [12,
14], the presence of strong lattice distortions leads to significant
suppression of high-frequency modes and general blur of EPI spectra, which is
confirmed by independent data of tunnel experiments [15]. Since nonequilibrium
phonons reflect the vibrational structure of material in the vicinity of their
generation, to obtain the EPI spectrum of unperturbed material, the impurities
and defects should be concentrated (in the ideal case) only near the contact
center, and the energy relaxation diffusion length should exceed the size of
the region within which the nonlinearity of $I\textendash V$ curve is formed
in the phonon frequency range in the superconducting state. Thus, the final
solution of the above-stated problem is a contact with a nonuniform impurity
distribution, consisting of a dirty nucleus (the point-contact region where
backscattering is efficient) and pure banks; hereinafter, this contact will
be referred to as inhomogeneous. No additional limitations are imposed on the
contact.

Let us outline at once the range of possible candidates applicable for studies
with an inhomogeneous contacts. Superconductors with an extremely large
coherence length can be rejected at once, because banks are not involved in
the formation of nonlinearity in this case, and the volume in which lattice
distortions are concentrated will be close to or coincide with the current
concentration domain. For the same reason, study of high-temperature
superconductors (HTSCs) is unlikely, but now because of the extremely small
coherence length. All other superconductors lying in the intermediate range
can be considered as possible candidates for testing the method.

Pressure point-contacts are the most appropriate objects for verifying the
above suggestions, because surface is always more contaminated as compared
with volume, and additional distortions are introduced at the inter electrode
boundary during the point-contact formation.

##  2 EXPERIMENTAL

The object of investigation was the well-studied superconductor with a known
EPI function: niobium. It is a fairly complex object from the point of view of
both point-contact and tunnel spectroscopies. The oxides on its surface may
vary in composition in wide limits and change their properties in dependence
of the oxidation state. In addition, the oxide layer contains many depairing
centers, which suppress superconductivity.

Point-contacts were formed using pure niobium with a resistance ratio of
$\rho_{300}/\rho_{res}\sim 100$, where $\rho_{300}$ and $\rho_{res}$ are,
respectively, the resistivity at room temperature and the residual resistivity
[16, 17]. The elastic relaxation length in niobium turned out to be $l_{i}\sim
220$ nm, and the coherence length was $\xi_{0}\sim 44$ nm; with allowance for
the elastic scattering, the reduced coherence length was $\zeta\sim 36$ nm.
Here, $\frac{1}{\zeta}=\frac{1}{\xi_{0}}+\frac{1}{l_{i}}$. Contacts were
prepared according to the shear technique [18, 19]. Taking into account the
importance of the role played by oxide on the electrode surface in providing
the high quality and mechanical and electrical stability of point-contacts,
two approaches were applied in the experiments. Native oxides were used in the
first approach: before installing in the device for forming point-contacts,
the electrodes were etched in a mixture of acids, washed, and then dried. In
the second approach, electrodes processed in a similar way were placed in a
sputtering plant and, after heating in vacuum to a premelting temperature, a
thin aluminum layer was deposited on their surface, which was then oxidized in
a boiling solution of hydrogen peroxide. The prepared electrodes were
installed in the device for forming pointcontacts [20]. Hereinafter, the
contacts formed between electrodes coated by native oxides will be referred to
as first-type contacts and the contacts formed using the second approach as
the second-type contacts.

##  3 EXPERIMENTAL RESULTS AND THEIR PROCESSING

Figure 1: EPI spectra for niobium point-contacts. Superconductivity is
suppressed by magnetic field, $H\sim 4\div 5$ T, $T=4.2$ K. The experimental
curves are smoothed by polynomial approximation.

Despite the large statistics (more than a hundred of point-contacts of each
type), even the best spectra in the normal state do not describe quite
adequately the ballistic flight of electrons. The phonon features in them are
significantly diffuse, the high-frequency phonons are strongly suppressed, and
there is a high background level. In addition to the aforesaid, the high-
frequency peak is blue-shifted in a number of spectra. Some spectra have a
form of a wide diffuse maximum in the vicinity of the first peak, with a weak
shoulder near the second peak. Typical representatives of these spectra are
shown in Fig.1. However, the overwhelming majority of the second derivatives
of $I-V$ curves in the normal state for contacts of both types are curves of
very low intensity without any pronounced phonon features. Unfortunately, the
sparse contacts whose spectra in the normal state were similar to those
presented in Fig.1 could not be transferred to the superconducting state,
because only few contacts could withstand the magnetic field switch off.

At the same time, a complete set of characteristics could be recorded for some
contacts from the much more numerous group, with spectra free of pronounced
phonon features in the normal state, because of their much larger number. When
these contacts passed to the superconducting state (at a constant level of
modulating signal), the second-derivative intensity increased approximately by
an order of magnitude, and the spectral shape changed radically.

Figure 2: Procedure of reconstruction of the EPI function for niobium using
superconducting point-contacts: (a) $S$ are the second derivatives of
$I\textendash V$ curves of niobium point-contacts in the superconducting state
and $B$ is the background curve; (b) $S\textendash B$ is the second derivative
with subtracted background, $(S\textendash B)K$ is the second derivative with
subtracted background after the correction, and $K$ is the correction curve;
(c) point-contact EPI functions $g_{PC}$ (the abbreviation PC stands for point
contact) obtained by integrating the second derivative of $I\textendash V$
curve with subtracted background (panel (b)): (1) before the correction and
(2) after the correction. For the 17.5 $\Omega$ contact: (a) $N$ is the second
derivative of $I\textendash V$ curve in the normal state, recorded with the
same modulation current as the curve $S$; $N\times 10$ is the same curve on
the larger scale. For the 50 $\Omega$ contact: (c) curve 3 is obtained from
curve 2 by multiplying the $x$ coordinates of points by 0.8 to compensate for
the series resistance. Contacts 1 and 2 are of the second type, and contact 3
is of the first type.

Figure 2.1a shows an example of this transformation of the second derivative
of $I\textendash V$ curve for one of the second-type contacts. The shape of
the second derivative in the superconducting state is similar to the
difference between the second derivatives of $I\textendash V$ curves in the
superconducting and normal states of the tantalum point-contacts studied in
[9]. There are also some differences: the niobium spectrum does not exhibit
the nonequilibrium feature and transformation of the soft phonon mode in
spectrum from a shoulder in the normal state (at about 10 meV) into a peak
when passing to the superconducting state. In tantalum point-contacts this
transformation is due to the selection of phonons with small group velocities.
The absence of this selection in niobium is explained by the much smaller (as
compared with tantalum) volume in which nonequilibrium phonons interact
efficiently with Andreev electrons and (or) insufficiently long elastic mean
free path of nonequilibrium phonons. Tantalum is closer to the boundary of
coherence-length range in comparison with other superconductors: the coherence
length is sufficiently small to involve the near-contact region in the
formation of superconducting-state nonlinearity and, at the same time,
sufficiently large to implement efficient selection of phonons in this region.

Figure 3: 1(a) Derivatives of the correction curves used as “preforms”. The
ascending portions are identical for all preforms; they are segments of
exponentials. The descending portions for preforms 1 and 2 are also segments
of exponentials, while in "preform" 3 they consist of two segments of
exponentials conjugated by a polynomial. 1(b) Correction curves 1-3, obtained
from the "preforms" presented in panel 1(a). The position of the curves on the
energy axis corresponds to the position of maximum in the preform. Using
different correction curves, one can estimate the degree of influence of their
forms and the position on the energy axis on the result of EPI function
reconstruction. 2(a) Change in the scale of correction curve $K1$ in
dependence of its position on the energy axis (numbers near the curves) for
the point-contact shown in Fig. 2.1. 2(b) Form of the correction curve at the
points of magnitude minima (see formula(1) and Fig.4) for the point-contact in
Fig.2.1. The numbers near the curves indicate their position on the energy
axis.

Despite the fact that the nonlinearity of $I\textendash V$ curve in the normal
state can be neglected for such contacts, one still must subtract the
superconducting background from the second derivative of $I\textendash V$
curve. When plotting the background curve, it is necessary to nullify the
second derivative at voltages above the boundary of the phonon spectrum. At
$eV\gg\Delta$, where $\Delta$ is the superconducting energy gap, the
background is described by a smooth curve without any features, and the main
requirement to the background curve near the gap feature is to exclude the
occurrence of artifacts on the difference curve $S\textendash B$. This also
holds true for the missing portion (starting from zero) in the difference
curve. In Figs.2.1a and 2.1b one can see, respectively, the background curve
satisfying these criteria and the difference curve. The missing portion for
the difference curve at displacements from 0 to 5 mV is plotted by hand. Since
the EPI function is identically zero beyond the phonon spectrum, to
reconstruct correctly the EPI function, the difference curve must satisfy the
geometric rule referred to as the _sum rule_ : _the total areas under the
integrable curve above and below the abscissa axis should be identical_.

Obviously, this rule is not satisfied for the curves presented in Fig.2,
because the high-energy part of the spectrum is significantly suppressed and
must be corrected. As well as in the case of tantalum point contacts [9], the
correction curve is unity in the lowenergy range and exceeds unity in the
region where the spectrum is suppressed. Multiplication of the difference
curve by the correction curve yields a curve for which the sum rule is
satisfied. The EPI functions reconstructed from the curves before and after
the correction are shown in Fig.2c.

The spectrum of the second-type contact (Fig.2.2) is fairly interesting. Its
shape is similar to that of the previous spectrum; however, all features in it
are blueshifted. After subtracting the background $B_{S}$, correction of the
obtained curve, and subsequent integration, we obtain an EPI function with
peaks located near 20 and 28 meV. This pattern corresponds to the EPI function
of niobium, extended by 20% along the x axis. Having multiplied the $x$
coordinates of the curve by 0.8, we obtain an EPI function fairly close to
tunnel. The blue shift of the phonon features may be caused by the complex
contact structure. Since the thickness of the deposited aluminum layer was
monitored with insufficient accuracy, it likely exceeded the conventional
value. During subsequent processing of electrodes in hydrogen peroxide the
aluminum film was not oxidized throughout the entire thickness and played the
role of a series resistance, making the phonon peaks blue-shifted; this shift
can be taken into account if the true position of phonon features is known.

An inverted $S$-shaped curve without any additional structure (Fig.2.3) is
most typical of the first-type contacts. As well as for the previously
considered contacts, the difference curve must be corrected after subtracting
the background. The panel (c) shows the EPI functions reconstructed from the
difference curve before and after the correction. Since the superconducting
additive to the spectrum is formed in a volume with a characteristic size of
the length of conversion of Andreev electrons into Cooper pairs, based on the
shape of the EPI function reconstructed from this additive, one can draw an
indirect conclusion about the degree of lattice perfection in this volume. As
follows from Fig.2.3, the EPI function reconstructed for this contact is a
diffuse bell-shaped curve. This shape is characteristic of a highly deformed
metal. A corresponding example can be observed in Fig. 1 in [21] for the
spectrum of a zirconium break junction. At the same time, a more sparing
technique of shear pressure contacts provided zirconium spectra with a much
better resolution, even with electrodes of obviously worse quality (see Fig. 9
in [22]). Thus, the thickness of the defect layer in the first-type niobium
contacts is most often comparable with the conversion length of Andreev
electrons in Cooper pairs or even exceeds it. Therefore, when using
inhomogeneous point-contacts for reconstructing the EPI function, the key
condition is the minimally possible defect-layer thickness. It must be at
least obviously smaller than the size of the region where the nonlinearity of
I–V curve is formed in the superconducting state. This condition turned out to
be much more easily satisfied for the second-type contacts.

Obviously, when plotting the background curve sticking to the above rules,
possible variations in the same time, the influence of the shape of the
correction curve and its position on the energy axis has been poorly studied.
For tantalum point-contacts, as can be seen in Fig. 15 from [9], there is a
spread in these parameters for different contacts. The correction curves were
plotted in [9] by hand, and their amplitude and position on the energy axis
were chosen empirically. In this study, we attempted to reveal the general
regularities for correction curves of different shapes.

Figure 4: Dependences of the magnitudes of correction curves on their position
on the energy axis for the point-contact presented in Fig.2.1

.

Figure 5: Variations in the form of the EPI function reconstructed for the
contact presented in Fig.2.1, in dependence of the form of correction curve
and its position on the energy axis. The central part of the figure
corresponds to the minima of correction curve magnitudes. Figure 6: Procedure
of background correction in the spectra of point-contacts presented in Fig.1.
(a) The second derivatives of $I\textendash V$ curves: (1) initial and (2)
after the correction. (b) The third derivatives of the $I\textendash V$ curve,
obtained by numerical differentiation of the curves in panels (a) (curves 1);
curves after the correction (2); and correction curves $K$ (3).

Since correction curves always have a step-rise portion, it is convenient to
simulate their shape using a preform consisting of segments of exponentials.
Figure3.1(a) shows the preforms that were integrated to obtain correction
curves of different forms in Fig.3.1(b). The initial portions of the curves
are identical, whereas the descending portions in Fig.3.1(a) differ. The
initial portion for curve 1 is similar to that for curve 2, reflected and
compressed along the $y$ axis; in addition, this portion is extended along the
$x$ axis. For curve 3, the initial portion is a combination of two segments of
exponentials, matched by a polynomial. When carrying out calculations, the
position of the correction curve was identified with the position of the
maximum in the preform.

All three correction curves solved successfully the above-stated correction
problem in a fairly wide range of their positions on the energy axis for the
point-contacts under study. As an example, Fig.3.2(a) shows a change in the
amplitude of the correction curve K1 (see the contact in Fig.2.1 in dependence
of its position on the energy axis, as well as the form of all three
correction curves at the same displacement, approximately at the center of the
interval (Fig.3.2(b)). To describe quantitatively the correction curve,
independent of its shape, it is convenient to use its magnitude, i.e., the
effective area under the curve from zero to the boundary of the phonon
spectrum, normalized by the value Vmax, corresponding to the spectrum
boundary:

${m=\frac{1}{V_{max}}\int\limits_{0}^{V_{max}}{\left[K(\omega)-1\right]d\omega}}$
(1)

Figure4 shows the dependences of correction curve magnitudes on displacement,
and Fig.5 demonstrates variations in the shape of EPI function for different
correction curves and their positions on the energy axis for the same point
contact. As can be seen, the smallest spread in the shape of EPI function
corresponds to the minima of correction curve magnitudes.

##  4 BACKGROUND CORRECTION IN POINT-CONTACT YANSON SPECTRA

Almost all second derivatives of $I\textendash V$ curves of point-contacts at
displacements above the phonon spectrum boundary do not turn to zero, whereas
the EPI functions at these displacements are identically zero. According to
the existing theories, the background is due to the reabsorption of
nonequilibrium phonons by electrons in the point-contact [23, 24, 25]. A
quantitative characteristic of the background level $\gamma$ in Yanson spectra
is the ratio of the magnitude of the second derivative of $I\textendash V$
curve at the phonon spectrum boundary to its maximum value. However, even
under the assumption that nonequilibrium phonons are completely blocked in the
contact, this theory cannot explain the existence of spectra with the
background level $\gamma\geq 0.5$ [26]. To date, the existence of spectra
with a large background level has not been physically justified; therefore,
some empirical selfconsistent iterative procedures are generally applied in
practice, which satisfy the requirements of coincidence (within experimental
error) for EPI functions of different contacts made of identical materials
[3]. This is generally valid for the background level $\gamma\leq 0.3$. For
the spectra with a large background level, the final result of its subtraction
will be determined by the procedure chosen.

Figure 7: Dependences of the magnitudes of correction curves on their position
on the energy axis for the point-contact shown in Fig.6.1. Figure 8:
Variations in the forms of the EPI function reconstructed for the contact
presented in Fig.6.1 (a) in dependence of the position of correction curve
$K1$ on the energy axis and (b) for different correction curves at the points
of minima of their magnitudes. Figure 9: Comparison of the procedures of
background subtraction and correction: (1) initial experimental curve with a
background, (2) curve after the correction, (3) traditionally used background
curve, and (4) curve after background subtraction. The EPI parameter $\lambda$
for curve 2 is larger by a factor of 1.5 than that for curve (4), and the
maximum of curve (4) is about 0.6 of the maximum of curve (2). Figure 10:
Comparison of the EPI functions reconstructed from the spectra of contacts in
the normal and superconducting states and from the tunnel data. All curves are
normalized to unity in the vicinity of the first peak. (a) (1) Amorphous
material in the normal state (Fig.6.3), (2) amorphous material in the
superconducting state (Fig.2.3), and (3) reconstruction from the tunnel data
for amorphous films [15]. (b) (1) Defect material in the normal state
(Fig.6.2), (2) defect material in the superconducting state (Fig.2.1), and (3)
reconstruction from the tunnel data for amorphous films [15]. (c) (1)
Distorted material in the normal state (Fig.6.1), (2) defect-free material in
the superconducting state (Fig.2.2), and (3) reconstruction from the tunnel
data for crystalline films [27].

At the same time, having compared the curves in Figs.1 and 2с, one can suggest
by analogy that the background in Yanson spectra may arise as a result of the
suppression of high-frequency phonons, which should manifest itself in
violation of the sum rule already for the third derivative of $I\textendash V$
curve. The suppression of high-frequency phonons and, correspondingly,
occurrence of background are caused by impurities and lattice defects. Phonons
reflect the vibrational structure of material in their generation region. For
low-energy phonons with a large wavelength, all these local lattice
distortions are averaged on the wavelength and do not affect much the
generation conditions. At the same time, generation of high-energy phonons
(having a wavelength comparable with the lattice period) near lattice
distortions will be hindered, because they are diverse and randomly located.
Thus, it follows from this suggestion that a correlation should be observed
between the degree of suppression of high-frequency phonons and the background
level, which corresponds in principle to the experiment [12, 14].

Derivatives of $I\textendash V$ curves are generally recorded using the
standard modulation technique, which is generally accepted in the point-
contact and tunnel spectroscopy. Since the modulation voltage makes spectra
diffuse, it is taken as low as possible, in order to keep a reasonable
compromise between the resolution and noise level. Therefore, the use of the
initial experimental curve describing the second derivative of $I\textendash
V$ curve to obtain the _third derivative_ by numerical differentiation leads
in most cases to unacceptably high noise level in the curve obtained. The
experimental curves were smoothed to suppress noise: they were initially
divided into overlapping segments, which were then approximated by
polynomials. The initial experimental points and results of their polynomial
approximation are presented in Fig.1.

Since the procedure of background correction in Yanson spectra by
reconstructing the sum rule on the third derivative of $I\textendash V$ curve
in no way differs technically from the similar procedure performed on the
second derivative of $I\textendash V$ curve of superconducting contacts, we
applied the same algorithm.

Figure6 shows the results of application of this algorithm to the contacts
presented in Fig.1. The dependences of the magnitudes of correction curves on
their position on the energy axis (Fig.7) and the variations in the shape of
derivatives after the background correction (Fig.8) are given for different
types of correction curves and their position on the energy axis for one of
the contacts. As can be seen in Fig.8, the smallest variations in the form of
EPI functions correspond to the minima of correction curve magnitudes. Here,
the much smaller magnitude of correction curves (in comparison with the
superconducting spectra; compare Figs.4 and 7) leads to almost complete
identity of curves after the background correction; i.e., the final result is
independent of the form of correction curve.

For Yanson spectra, which are characterized by a high background level, its
subtraction by conventional methods [3] leads, first, to a significant
distortion of the reconstructed EPI function and, second, to underestimated
values of the EPI function $g_{PC}$ and the EPI parameter $\lambda$ (Fig.9).
For the spectrum presented in Fig.9, the parameter $\lambda$ for the
background correction is larger by a factor of 1.5 than for the background
subtraction, and the value of the EPI function at the point of maximum for
background subtraction is about 60% of that in the case of correction. Thus,
the use of the background correction method makes it possible to refine the
form of EPI functions and numerical parameters for the materials whose spectra
are difficult to obtain with a low background level.

It is of interest to compare the results of reconstructing EPI functions from
the spectra of point-contacts in the superconducting and normal states, as
well as to consider them in comparison with the tunnel data [15, 27] (Fig.10).

As follows from the comparison, the obtained EPI functions can arbitrarily be
separated into three groups. The EPI function in Fig.10(a) corresponds to
amorphous niobium. Proceeding from the diffusion of curves, the largest degree
of amorphization is observed for function 2, reconstructed from a
superconducting spectrum, whereas significant recrystallization occurs for the
tunnel EPI function (curve 3). In Fig.10(b), the degrees of recrystallization
are similar, the second peak is blue-shifted in all curves, and their strong
broadening is observed. Finally, Fig.10(c) presents the curves corresponding
to the most perfect lattice. The peak positions coincide for all curves, and
their diffusion is also much lower than in the previous cases. At the same
time, the EPI function reconstructed from the spectrum in the normal state is
significantly broadened, and the second peak is suppressed; the second peak in
the EPI function reconstructed from tunnel measurements is significantly
suppressed. Curve 2, reconstructed from the superconducting spectrum, appears
to be most appropriate.

##  5 RESULTS AND DISCUSSION

The existing theories of the spectroscopy of EPI in superconducting point-
contacts with direct conductivity consider the ballistic flight of carriers
through the constriction and are valid for only superconductors with a large
coherence length. The diffusion limit is considered for only a long channel
with dirty edges, whereas no theoretical model has been developed yet for
inhomogeneous point-contacts. Nevertheless, studies [5, 6] form a solid
theoretical foundation, the quintessence of which is the proof that the EPI
function can be reconstructed from the excesscurrent nonlinearity related to
the scattering of Andreev electrons from nonequilibrium phonons. In the
experimental verification of the results of [5, 6] it was found that the
superconducting background (which was not predicted by the theory) must be
taken into account to reconstruct the EPI function. Of much greater importance
is the fact that the range of application of the theory was found to be wider
than the authors suggested initially: the reconstruction of the EPI function
from excess-current nonlinearity turned out to be possible not only for
ballistic contacts made of superconductors with a very large coherence length
but also for other superconductors with a smaller coherence length, as well as
for inhomogeneous contacts. By analogy with the conventional point-contact
spectroscopy, on the history scale, we are in the interval by which the
Kulik–Omel’yanchuk–Shekhter (KOSh) theory had been developed [1] but the
theory explaining the observation of EPI spectra in diffusion point-contacts
[11] had not been developed yet. The theory explaining the occurrence of
background in Yanson spectra was also absent then. Incidentally, the
theoretical explanation of the possibility of occurrence of background at the
level above 50% is still absent. Thus, the experimental results presented here
may yield useful information for further development of the theory.

Point-contact spectroscopy is based on carrier duplication: in the current-
conducting state electrons are divided into two groups, for which the energy
difference between the occupied and free electron states on the Fermi surface
is $eV$, i.e., corresponds to the applied voltage [1, 2]. If inhomogeneous
contacts in the superconducting state provide non-dissipative transmission of
electrons through the defect region (i.e., electron duplication is preserved),
the second derivative of the $I\textendash V$ curves of these structures is
similar to the difference of the $I\textendash V$ curves of the normal and
superconducting states of ballistic point-contacts. With allowance for the
experimental results for niobium point-contacts, even partially dissipative
mode of transmission through the defect region may be applicable for
reconstructing the EPI function. On the one hand, this circumstance opens
additional possibilities, increasing the number of objects making it possible
to gain in some cases information of better quality about the EPI from an
unperturbed material in the vicinity of generation of nonequilibrium phonons,
as was demonstrated above. At the same time, one must take into account
possible presence of a series resistance. In the case of a superconductor with
a known position of phonon features, this requirement is easy to satisfy, but
a material with unknown phonon spectrum calls for measuring a fairly large
number of contacts with different resistances.

Both the superconducting spectra of inhomogeneous contacts and Yanson spectra
demonstrate suppression of high-energy phonons, although the physical reasons
for this suppression are fundamentally different.

In inhomogeneous superconducting contacts, a nonlinearity is formed due to the
scattering of nonequilibrium phonons from the electrons undergoing Andreev
reflection, in a volume with a characteristic size on the order of reduced
coherence length. The probability of this scattering depends strongly on the
concentration of both components, and there are some threshold concentrations
of both Andreev electrons and nonequilibrium phonons, below which the
probability of their interaction sharply decreases. These threshold
concentrations are interrelated: an increase in the concentration of one
component decreases the threshold concentration of the other. The
concentration of Andreev electrons is determined by the excess current, while
the concentration of nonequilibrium phonons is controlled by the density of
phonon states. With an increase in the voltage across the contact, the value
excess current gradually decreases in the entire energy range. In regards to
the density of phonon states, it first increases and, having reached the
energy of the first phonon peak, starts rapidly decreasing, thus causing a
sharp drop of scattering probability. Another extremely important factor is
the dependence of the effective scattering cross section of nonequilibrium
phonons from Andreev electrons on the energy of these phonons. Nonequilibrium
phonons are not point objects, and the scattering efficiency decreases with an
increase in their energy.

Thus, specifically the combination of all these factors leads to the
suppression of high-energy phonons. As was shown in the study of
superconducting tantalum point-contacts [5], the degree of this suppression
depends very strongly on the rate with which the excess current decreases,
whereas the suppression onset is always related to the energy corresponding to
the sharp decrease in the density of phonon states. For the contacts
considered here, the minima of correction curve magnitudes lie in the energy
ranges of $15.8\div 16$ meV for the contacts with an ordered lattice (with
allowance for the voltage correction for the $50\leavevmode\nobreak\ \Omega$
contact) and $16.5\div 16.8$ meV for the amorphous contact.

The suppression of high-energy phonons in Yanson spectra is due to the lattice
distortions (induced by impurities and defects). A number of conditions must
be satisfied to allow an electron with an excess energy corresponding to the
applied voltage to loose this energy via emission of a nonequilibrium phonon.
First, the energy of this phonon should coincide with the excess electron
energy, i.e., the vibrational lattice parameters in the vicinity of the phonon
generation point should allow for this process. Second, the phonon should be
able to propagate throughout the crystal; i.e., the vibrational lattice
parameters in neighboring volumes should not differ significantly. When the
voltage applied to the contact is low, the emitted phonons have a large
wavelength, and the defects do not affect much the possibility of phonon
generation, because many atoms can be placed within a distance equal to the
wavelength, and the lattice parameters are averaged. At the same time, if the
phonon wavelength is comparable with lattice parameters, its distortions may
hinder phonon generation. Indeed, there will be individual vibrational lattice
parameters in the vicinity of each defect, which will differ from those of un
perturbed lattice and, possibly, other defects. In addition, random
arrangement of defects hinders the propagation of the phonons generated in
their vicinity. Thus, the effective generation volume for short-wavelength
phonons is smaller than that for long-wavelength phonons, due to which their
intensity decreases and general spectral broadening occurs.

The minima of correction curve magnitudes in Yanson spectra are located at
energies of $16.6\div 17.2$ meV for contacts with an ordered lattice and at
$17.6$ meV for amorphous contacts. Thus, the minima of correction curve
magnitudes in Yanson spectra are blue-shifted by approximately 1 meV in
comparison with the spectra of the superconducting contacts. This pattern
correlates with the observed increase in the degree of suppression of high-
frequency phonons with a decrease in their wavelength.

So different physical reasons responsible for the suppression of high-energy
phonons in the spectra of inhomogeneous superconducting contacts and the
spectra of Yanson point-contacts obviously lead to different results when
reconstructing the form of EPI functions. The sum-rule reconstruction on the
third derivatives of I–V curves of Yanson spectra removes the background on
the second derivatives but, however, does not eliminate the reason for its
occurrence. Therefore, the form of the EPI function corresponds to a
perturbed material, which is especially pronounced for spectra with a large
background and, correspondingly, large magnitude of correction curves: high-
frequency peaks are suppressed, and phonon features are broadened. Since the
nonlinearity formation domain is removed from the surface (where the lattice
defect concentration is maximum) in inhomogeneous superconducting point-
contacts, the probability of obtaining spectra corresponding to unperturbed
material is higher for these contacts than for Yanson contacts. This remark is
especially true for materials with a problematic surface, which are
technologically complex for point-contact spectroscopy.

Despite the fact that the derivation of EPI functions from the spectra of
inhomogeneous superconducting contacts involves double transformation
(subtraction of superconducting background and subsequent correction of the
high-energy part of the curves to make the latter satisfy the sum rule), the
variance of the obtained results is not of crucial importance. The
superconducting background corresponds at large displacements to the general
shape of the spectrum, and its small variation barely leads to any significant
changes in the form and position of phonon features on the reconstructed EPI
function. A change in the form of the correction curve also affects little the
form of the reconstructed EPI function if its position on the energy axis
corresponds to the magnitude minimum. Nevertheless, there is a strong
relationship between the variance in the form of reconstructed EPI function
and the correction curve magnitude. There are some small variations in form
for curves in Fig. 5 at the points of minima of correction-curve magnitudes (m
= 8.6–9.5), whereas for the contact with a resis-tance of 110 . (see Fig. 1)
with a background level . 0.7 the variations in the forms of EPI function
after the reconstruction (see Fig. 8, m 1) are negligible.

A possible factor hindering curve processing is the occurrence of nonspectral
features (related to the destruction of superconductivity in the near-contact
region) in derivatives of I–V curves. These features may be of different
nature, for example, thermal or related to nonequilibrium processes. They are
not reproducible, and their position depends on temperature and point-contact
resistance. Therefore, a rather large volume of experimental data must be
gained to exclude these features from consideration.

##  6 BRIEF CONCLUSIONS

  1. 1.

Inhomogeneous microcontacts exhibit a radical difference between the second
derivatives of $I\textendash V$ curves in the normal and superconducting
states: the phonon-related features on the derivatives are either absent or
significantly weakened in the normal state. Specific features arise on the
second derivatives of $I\textendash V$ curves in the superconducting state,
which make it possible to reconstruct the EPI function.

  2. 2.

A necessary and sufficient condition for reconstructing the EPI function with
the aid of inhomogeneous superconducting contacts is to use the second
derivative of $I\textendash V$ curve in the superconducting state.

  3. 3.

The background in the second derivatives of $I\textendash V$ curves of point-
contacts in the normal state is caused by lattice defects, which reduce the
contribution of high-energy phonons to the spectrum and, correspondingly,
violate the sum rule even in the third derivatives of $I\textendash V$ curve.
Background correction is performed by reconstructing the broken sum rule on
this derivative.

This study was supported by the National Academy of Sciences of Ukraine,
project FTs 4-19.

## References

  * [1] I.O. Kulik, A.N. Omel’yanchuk, R.I. Shekhter, Sov.J. Low Temp.Phys, 3, 740 (1977); _Fiz. Niz. Temp._ 3 1543 (1977).
  * [2] Yanson I.K., Kulik I.O., Omel’yanchuk A.N, Shekhter R.I., Sharvin Yu.V. Yavleniya pereraspredeleniya energii nositelei zaryada v metallicheskikh mikrokontaktakh pri nizkikh temperaturakh"" ( Phenomena of redistribution of charge carrier energy in metal point contacts at low energies""), Diploma 328 (Moscow: AllRussian Scientific Research Institute of Patent Information, 1986); _Otkrytiya Izobret._ (40) 3 (1987)[_Fiz. Niz. Temp._ , 13(9), 1005 (1987)] [_Sov. J. Low Temp. Phys._ , 13(9), 574 (1987)]
  * [3] Khotkevich A.V., Yanson I.K. _Atlas of Point-Contact Spectra of Electron-Phonon Interaction in Metals_ (Dordrecht: Kluwer Acad. Publ., 1995); Translated from Russian: Yanson I.K., Khotkevich A.V. _Atlas Mikrokontaktnykh Spektrov Electron-Fononnogo Vzaimodeistviya v Metallakh_ (Kyiv: Naukova Dumka, 1986)
  * [4] Yu.G. Naidyuk, I.K. Yanson, _Point-Contact Spectroscopy_ (New York: Springer, 2005)
  * [5] V.A. Khlus, A.N. Omel’yanchuk, Sov.J. Low Temp.Phys, 9, 189, (1983) [_Fiz. Niz. Temp._ 9 373 (1983)]
  * [6] A.V. Khlus, Sov.J. Low Temp.Phys, 9, 510, (1983) [_Fiz. Niz. Temp._ 9 985 (1983)]
  * [7] N.L. Bobrov, V.V. Fisun, O.E. Kvitnitskaya, V.N. Chernobay, I.K. Yanson, Low Temp. Phys. 38, 373 (2012), [_Fiz. Niz. Temp._ 38, 480 (2012)] [arXiv:1207.6486]
  * [8] N.L. Bobrov, A.V. Khotkevich, G.V. Komarchuk, P.N. Chubov, Low Temp. Phys. 40, 215 (2014) [_Fiz. Niz. Temp._ 40, 280 (2014)], arXiv:1405.6869
  * [9] N.L. Bobrov, Low Temp. Phys., 45, 482, (2019) [_Fiz. Niz. Temp._ 45, 562, (2019)], arXiv:1906.04380
  * [10] N.L. Bobrov, UFNe, 63(11), 1072, (2020) [UFNr, 190(11), 1143, (2020)], arXiv:2109.00806
  * [11] I.O. Kulik, R.I. Shekhter, A.G. Shkorbatov, JETP, 54(6), 1130, (1981) [Sov. Phys. JETP, 81, 2126, (1981)]
  * [12] Yu.G. Naidyuk, O.I. Shklyarevskii, I.K. Yanson, Sov. J. Low Temp. Phys., 8(7), 362, (1982) [_Fiz. Niz. Temp._ 8(7), 725 (1982)]
  * [13] A.I. Akimenko, A.B. Verkin, N.M. Ponomarenko, I.K. Yanson, Sov. J. Low Temp. Phys., 8(3), 130, (1982) [_Fiz. Niz. Temp._ , 8(3), 260 (1982)]
  * [14] I.O. Kulik, I.K. Yanson, Sov. J. Low Temp. Phys., 4(10), 596, (1978) [_Fiz. Niz. Temp._ , 4(10), 1267 (1978)]
  * [15] B.D. Kumhi, T.H. Geballe, Phys.Rev. Lett., 45, 1039, (1980)
  * [16] I.K. Yanson, N.L. Bobrov, L.F. Rybal’chenko, V.V. Fisun, Sov. J. Low Temp. Phys., 9, 596, (1983) [_Fiz. Niz. Temp._ , 9, 1155 (1983)] arXiv:1604.07067
  * [17] I.K. Yanson, N.L. Bobrov, L.F. Rybal’chenko, V.V. Fisun, Solid State Communications, 50, 515, (1984), arXiv:1605.09022
  * [18] P.N Chubov, I.K. Yanson, A.I. Akimenko Sov. J. Low Temp. Phys. 8, 32 (1982) [_Fiz. Niz. Temp._ , 8, 64 (1982)]
  * [19] P.N. Chubov, A.I. Akimenko, and I.K. Yanson, USSR Patent No. 834803, Byull. Izobret., No. 20 (1981), p. 232.
  * [20] N.L. Bobrov, L.F. Rybal’chenko, A.V. Khotkevich, and I.K. Yanson, USSR Patent No. 1631626, Byull. Izobret., No. 8 (1991).
  * [21] V.V. Khotkevich, A.V. Khotkevich, A.P. Zhernov, T.N. Kulagina, and E.K. Fol’k, Visn. KhNU, Ser. Fiz., No. 476 (4), 96 (2000).
  * [22] N.L. Bobrov, L.F. Rybal’chenko, V.V. Fisun, A.V. Khotkevich, Low Temp. Phys., 42, 811, (2016) [_Fiz. Niz. Temp._ , 42, 1035 (2016)], arXiv:1612.03396
  * [23] A.P. Van Gelder, Solid State Communs., 35(1) 19 (1980)
  * [24] I.O. Kulik, A.N. Omel’yanchuk, I.K. Yanson, Sov.J. Low Temp. Phys., 7(2), 129, (1981) [_Fiz. Niz. Temp._ , 7(2), 263 (1981)]
  * [25] I.O. Kulik, Sov. J. Low Temp. Phys., 11(9), 516, (1985) [_Fiz. Niz. Temp._ , 11(9), 937 (1985)]
  * [26] G.R. Shustov, Sov.J. Low Temp.Phys., 15(4), 240, (1989) [_Fiz. Niz. Temp._ , 15(4), 426, (1989)]
  * [27] E.L. Wolf, J. Zasadzinski, J.W. Osmun, J. Low Temp. Phys. 40(1/2) 19 (1980)