# MT2ST: Adaptive Multi-Task to Single-Task Learning

Dong Liu  Department of Computer Science
University of Wisconsin-Madison
<EMAIL_ADDRESS>Meng Jiang  Department of Computer Science
University of Notre Dame
<EMAIL_ADDRESS>
###### Abstract

The conventional training approaches often face challenges in balancing the
breadth of multi-task learning (MTL) with the depth of single-task learning
(STL). To address this issue, we introduce the Multi-Task to Single-Task
(MT2ST) framework, a groundbreaking approach that can combine the
generalizability of MTL with the precision of STL. Our work include two
strategies: ’Diminish’ and ’Switch’. ’Diminish’ Strategy will gradually reduce
the influence of auxiliary tasks, while the ’Switch’ strategy involves a shift
from multi-tasking to single-tasking at a specific timepoint at the training
process.

In this paper, we propose the Multi-Task to Single-Task (MT2ST) framework, a
novel approach that significantly enhances the efficiency and accuracy of word
embedding training while concurrently addressing prevalent issues such as
overfitting. Our empirical studies demonstrate that MT2ST can reduce training
time by 67% when contrasted with single-task learning approaches, and by 13%
compared to traditional multi-task learning methods. These findings underscore
MT2ST’s potential to be a powerful tools for word embedding training
acceleration.

###### Index Terms:

Efficient ML, ML Acceleration

##  I Introduction

Recent developments in natural language processing have highlighted that word
embeddings are foundational to many NLP tasks,its performance and training
efficiency directly impacting the effectiveness of linguistic tasks.

Traditional word embedding training methods predominantly rely on either MTL,
which offers a wide learning scope at the expense of task-specific detail, or
STL, which excels in focus on specific task but lacks the comprehensiveness of
MTL. Our research introduces a novel paradigm, the MT2ST framework, which
combines comprehensive scope of MTL and task-sepecific precision of STL
together. [CPKC09]

The MT2ST framework incorporates two specific strategies for training word
embeddings: ’Diminish’ and ’Switch’. The Diminish strategy innovatively
reduces the impact of auxiliary tasks over time, ensuring a gradual shift to
STL. In contrast, the ’Switch’ strategy splits training into two stages:
starting with MTL and then making a direct transition to STL. MT2ST framework
has been proven effectiveness in our emprical study. [BOHG19]

Our contributions can be summarized as follows:

  * •

Bridge the gap between MTL and STL with the unique MT2ST framework.

  * •

Introduce two pioneering strategies within this framework. ’Diminish’ and
’Switch’.

  * •

Empirically demonstrate MT2ST’s superiority in training efficiency and model
performance. [SK18]

##  II Motivation

###  II-A Challenges in Single-Task Word Embedding Training

Word embeddings map words to continuous vector spaces, reflecting their
semantic meanings. It has wide applications in text classification, sentiment
analysis, and machine translation. [AX23] However, its training process is
hindered by several constraints, which are as follows:

Data Volume Requirement: The need for large-scale textual datasets to train
word embeddings results in significant time and computational resource
consumption. Processing these large datasets is inherently slow and demands
extensive computational power. [EWVD19]

Computational Complexity: Training involves complex mathematical computations.
These algorithms, while effective, require high computational resources,
leading to increased training duration and processing power needs.

Optimization Difficulties: Optimizing word embedding models for accurate
semantic representation is a meticulous process. It often necessitates
iterative tuning and adjustments, which are resource-intensive and time-
consuming.

###  II-B Optimizing Training Efficiency in NLP Models with Multi-Task
Learning

The training efficiency of word embeddings is hindered by the mismatch between
large training datasets and narrow objective function metrics. This often
leads to underutilization of datasets and focuses on limited semantic or
syntactic aspects, causing inefficiencies. Altough work like attention based
multi-task learning [LJD19] can adjust weights for different task overtim,
models still struggle with polysemous words due to objective functions missing
their subtle meanings, leading to repetitive training and reduced efficiency.

Multi-task learning (MTL) emerges as an effective solution for these
inefficiencies. [ZYY+23] Recent research indicates that training on multiple
related tasks can yield computational benefits. MTL has wide application in
industry, such as building end-to-end dialogue modelling. [SSM+22] MTL enables
models to share core representations across tasks, enhancing word embeddings
with a richer semantic understanding, thus improving efficiency.

###  II-C Potential Inefficiency of Multi-Task Learning in Word Embedding
Training

MTL brings notable benefits but can also introduce inefficiencies, especially
in word embedding training. In MTL, the shared embedding layer across tasks
may receive divergent gradient directions from different tasks, which may
conflict.

This issue arises as tasks emphasize distinct aspects of word meanings. For
example, sentiment analysis prioritizes emotional connotations, while part-of-
speech tagging focuses on grammatical features. Consequently, the embedding
layer faces mixed directions of gradient descent in tuning embeddings, which
can potentially result in less effective task-specific embeddings. [LLJ+21]

Additionally, computational demands rise in MTL. Each task’s will update the
parameters the embedding layer, which leads to time inefficiency and high
resource usage. This issue intensifies with large vocabularies and high-
dimensional embeddings common in NLP.

##  III Model

###  III-A MT2ST Framework

In this research, we propose ”MT2ST” (Multi-Task to Single-Task) model to
improve the efficiency of word embedding training, specifically addressing the
challenges in multi-task learning (MTL) environments.

The process begins with a multi-task learning phase, where a unified model,
featuring a shared embedding layer, is simultaneously trained across various
NLP tasks. This phase utilizes the advantages of MTL, enabling the model to
absorb a wide array of linguistic features and nuances. The shared embedding
layer benefits from varied linguistic inputs, fostering a more comprehensive
word representation.

Once the multi-task phase is established, MT2ST shifts focus to single-task
learning. Here, the pre-trained shared embeddings are fine-tuned in a task-
specific manner. This targeted fine-tuning aligns the embeddings more closely
with the distinct demands and subtleties of each task, ensuring that while
they maintain the general linguistic knowledge from the MTL phase, they are
also tailored for superior performance in specific tasks.

MT2ST employs advanced techniques for a smooth and effective transition from
multi-task to single-task learning, which provides a fine-tuning and
pretraining method for effcient NLP[TLJ+23]. These include adaptive learning
rate mechanisms and selective freezing of certain embedding vector dimensions
based on their relevance to the specific task. These strategies enable a
balance between retaining the broad linguistic understanding gained in the MTL
phase and focusing on task-specific details during the single-task phase.

###  III-B Model Construction

Instead of mono-task fine-tuning used in Multi-Task Supervised Pretraining
[MST+20], our MT2ST model develop an adaptive method to accelerate word
embedding training within a multi-task to single-task learning framework.

####  III-B1 Input Layer

  * •

Processes raw text input.

  * •

Converts text into a numerical format suitable for NLP tasks using
tokenization and encoding techniques.

####  III-B2 Embedding Layer

  * •

Learns dense vector representations of words, capturing semantic and syntactic
information.

  * •

Trained across multiple tasks in the initial phase to obtain generalized word
embeddings.

####  III-B3 Shared Layers

  * •

Extract and learn features common across different tasks.

  * •

Play a crucial role in the multi-task learning phase.

####  III-B4 Dynamic Weighting Mechanism (Dimish Strategy Only)

  * •

Integral to the Diminish strategy, this mechanism adjusts the influence of
each task over time.

  * •

Employs time-dependent weighting functions that reduce the impact of auxiliary
task losses as the training progresses.

  * •

Ensures that the transition from multi-task learning to single-task learning
is smooth and performance-driven.

####  III-B5 Task-Specific Layers (Switch Strategy Only)

  * •

Dedicated layers for each task, activated during the single-task learning
phase in the Switch strategy.

  * •

Fine-tuned to align the model’s output closely with the requirements of the
specific task.

####  III-B6 Feedback Loop

  * •

Continuously evaluates the model’s performance, providing insights for
adjusting the dynamic weighting and learning parameters.

  * •

Facilitates a data-driven approach to optimize the transition phase in the
training process.

###  III-C Model Overview

Figure 1: MT2ST Word Embedding Training Framework Overview222This is a
footnote explaining the MT2ST model. 22footnotetext: Code:
https://github.com/NoakLiu/MT2ST-Word-Embeddings-Acceleration.

###  III-D MT2ST: Diminish Strategy

The Diminish strategy integrates a time-dependent weighting mechanism to
progressively reduce the influence of auxiliary tasks, thereby increasingly
highlight the primary task i the training process.

Figure 2: MT2ST-Dimish: Dimish Strategy for Auxiliary Tasks in Training to
Single Task Learning

Algorithmic Implementation: Given an input sequence $X$, embeddings are
obtained via an encoder network (such as BiLSTM):

$Emb=Encoder(X;\theta_{Encoder})$ (1)

where $\theta_{Encoder}$ represents the parameters of the encoder. The
gradients for both the primary task $\nabla\mathcal{L}_{primary}$ and the
auxiliary tasks $\nabla\mathcal{L}_{aux_{k}}$ are computed. The update of the
model parameters at each epoch $t$ is calculated by a diminishing weight for
each auxiliary task, defined as:

$\gamma_{k}(t)=\gamma_{k,0}\cdot e^{-\eta_{k}t^{\nu_{k}}}$ (2)

$\gamma_{k,0}$ is the initial weight, $\eta_{k}$ is the decay rate, and
$\nu_{k}$ is the curvature parameter of the decay function for the $k$-th
auxiliary task. The parameter update rule is as follow:

$\theta^{(t+1)}=\theta^{(t)}-\eta\left(\nabla\mathcal{L}_{primary}+\sum_{k=1}^{K}\gamma_{k}(t)\nabla\mathcal{L}_{aux_{k}}\right)$
(3)

The MT2ST-Diminish gradient descent iterates until the primary task loss
converges, which indicates the model’s gradually focus from multi-task to
primary-task.

Figure 3: MT2ST to realize quicker convergence [An example of MT2ST Diminish]

input : Initial parameters $\theta^{(0)}$, learning rates
$\\{\eta_{k}\\}_{k=1}^{K}$, decay curvature parameters
$\\{\nu_{k}\\}_{k=1}^{K}$, sequence input $X$

output : Optimized parameters $\theta^{*}$

initialize : $\theta\leftarrow\theta^{(0)}$;
$\gamma_{k}\leftarrow\gamma_{k,0}$ for $k\in\\{1,\ldots,K\\}$

for  _$t\leftarrow 1$ to $T$_ do

$Emb\leftarrow$ Encoder(_$X,\theta$_);

$\nabla\mathcal{L}_{primary}\leftarrow$ BackwardPass(_$Emb,\theta$ ,
primary_);

for  _$k\leftarrow 1$ to $K$_ do

$\nabla\mathcal{L}_{aux_{k}}\leftarrow$ BackwardPass(_$Emb,\theta$ ,
$aux_{k}$_);

$\gamma_{k}(t)\leftarrow\gamma_{k,0}\cdot e^{-\eta_{k}t^{\nu_{k}}}$;

end for

$\Delta\theta\leftarrow\nabla\mathcal{L}_{primary}$;

for  _$k\leftarrow 1$ to $K$_ do

$\Delta\theta\leftarrow\Delta\theta+\gamma_{k}(t)\cdot\nabla\mathcal{L}_{aux_{k}}$;

end for

$\theta\leftarrow\theta-\eta\cdot\Delta\theta$;

if  _CheckConvergence(_$\nabla\mathcal{L}_{primary}$_)_ then

return _$\theta$_ ;

end if

end for

Algorithm 1 Diminish Strategy for MT2ST

###  III-E MT2ST: Switch Strategy

The Switch strategy features a clear transition in its training approach.
Initially, the model simultaneously engages with both primary and auxiliary
tasks. However, upon reaching a predetermined epoch, denoted as
$T_{\text{switch}}$, the model shifts its focus solely to the primary task.

Figure 4: MT2ST-Switch: Optimal Solution Area Searching and Task Switching

Algorithmic Implementation: The key difference between Switch and Diminish
strategy is the parameter update methodology. For epochs before $T_{switch}$,
the parameter update is a combination of the gradients from all tasks:

$\theta^{(t+1)}=\theta^{(t)}-\eta\left(\nabla\mathcal{L}_{primary}+\sum_{k=1}^{K}\nabla\mathcal{L}_{aux_{k}}\right),\text{
for }t<T_{switch}$ (4)

Post $T_{switch}$, the update focuses solely on the primary task:

$\theta^{(t+1)}=\theta^{(t)}-\eta\nabla\mathcal{L}_{primary},\text{ for }t\geq
T_{switch}$ (5)

This strategy leverages the initial shared learning from multiple tasks and
then refines the model’s capabilities to focus more intently on the primary
task by switch.

input : Initial parameters $\theta^{(0)}$, switch epoch $T_{switch}$, sequence
input $X$

output : Optimized parameters $\theta^{*}$

initialize : $\theta\leftarrow\theta^{(0)}$

for  _$t\leftarrow 1$ to $T$_ do

$Emb\leftarrow$ Encoder(_$X,\theta$_);

if  _$t <T_{switch}$_ then

$\nabla\mathcal{L}_{primary}\leftarrow$ BackwardPass(_$Emb,\theta$ ,
primary_);

for  _$k\leftarrow 1$ to $K$_ do

$\nabla\mathcal{L}_{aux_{k}}\leftarrow$ BackwardPass(_$Emb,\theta$ ,
$aux_{k}$_);

end for

$\Delta\theta\leftarrow\nabla\mathcal{L}_{primary}+\sum_{k=1}^{K}\nabla\mathcal{L}_{aux_{k}}$;

else

$\nabla\mathcal{L}_{primary}\leftarrow$ BackwardPass(_$Emb,\theta$ ,
primary_);

$\Delta\theta\leftarrow\nabla\mathcal{L}_{primary}$;

end if

$\theta\leftarrow\theta-\eta\cdot\Delta\theta$;

if  _CheckConvergence(_$\nabla\mathcal{L}_{primary}$_)_ then

return _$\theta$_ ;

end if

end for

Algorithm 2 Switch Strategy for MT2ST

##  IV Result Analysis

To evaluate the performance of MT2ST, various experimental setups were
implemented (experimental settings are available in the appendix).

In the first experimental group, the approach was to directly fine-tune the
pre-pretrained embeddings on a dataset dedicated to a single task.
Observations from this experiment indicated a steady decrease in Loss, but
there were fluctuations in accuracy, indicating overfitting in the model.

Figure 5: Accuracy and Loss change of Original Mode

In the second experimental series, a multi-task learning approach was applied.
This approach led to a consistent improvement in both Loss and accuracy
metrics. The model achieved a notable accuracy rate of 68% on the test set.

Figure 6: Loss change (Multi-Task Learning)

Figure 7: Accuracy change (Multi-Task Learning)

In the third set of experiments, the MT2ST-diminish strategy was employed.
Unlike previous tests, the Loss did not decrease uniformly, which was
attributed to the reduction of auxiliary task Loss during the training
process. However, the model’s accuracy consistently improved, ultimately
achieving a higher test set accuracy of 71% compared to the standard multi-
task learning approach. This result highlights the effectiveness of the MT2ST-
diminish strategy in facilitating a smooth transition from multi-task to task-
specific learning, thereby enhancing the model’s overall performance.

Figure 8: Loss change in Mode (MT2ST: Dimish)

Figure 9: Accuracy change (MT2ST: Dimish)

The fourth experimental group implemented fine-tuning on the dataset using
predefined embeddings through the MT2ST-switch strategy. This approach
consisted of two phases: initial multi-task learning followed by task-specific
learning, with an abrupt transition between the two. As indicated by the red
line in the graph, a step change in Loss occurred after the switch, yet there
was a steady improvement in both phases. This method achieved a high score of
77% on the test set, demonstrating its effectiveness.”

Figure 10: Loss change (MT2ST: Switch)

Figure 11: Accuracy change (MT2ST: Swtich)

##  V Discussion

Our empirical experiments reveal that the MT2ST model notably outperforms
conventional embedding training methods in terms of training efficiency and
overall quality. The newly developed embeddings exhibit superior performance
and robustness across a range of downstream NLP tasks, including sentiment
analysis, part-of-speech tagging, and named entity recognition. These results
underscore the effectiveness of MT2ST in reducing training times and lessening
computational resource requirements. This efficiency makes MT2ST a valuable
solution for fast training and high-quality word representations.

## References

  * [AX23] Felipe Almeida and Geraldo Xexéo.  Word embeddings: A survey, 2023.
  * [BOHG19] Lu Bai, Yew-Soon Ong, Tiantian He, and Abhishek Gupta.  A multi-task gradient descent method for multi-label learning, 2019.
  * [CPKC09] Xi Chen, Weike Pan, James T. Kwok, and Jaime G. Carbonell.  Accelerated gradient method for multi-task sparse learning problem.  In 2009 Ninth IEEE International Conference on Data Mining, pages 746–751, 2009.
  * [EWVD19] Seth Ebner, Felicity Wang, and Benjamin Van Durme.  Bag-of-words transfer: Non-contextual techniques for multi-task learning.  In Colin Cherry, Greg Durrett, George Foster, Reza Haffari, Shahram Khadivi, Nanyun Peng, Xiang Ren, and Swabha Swayamdipta, editors, Proceedings of the 2nd Workshop on Deep Learning Approaches for Low-Resource NLP (DeepLo 2019), pages 40–46, Hong Kong, China, November 2019. Association for Computational Linguistics.
  * [LJD19] Shikun Liu, Edward Johns, and Andrew J. Davison.  End-to-end multi-task learning with attention, 2019.
  * [LLJ+21] Bo Liu, Xingchao Liu, Xiaojie Jin, Peter Stone, and Qiang Liu.  Conflict-averse gradient descent for multi-task learning, 2021.
  * [MST+20] Sara Meftah, Nasredine Semmar, Mohamed-Ayoub Tahiri, Youssef Tamaazousti, Hassane Essafi, and Fatiha Sadat.  Multi-task supervised pretraining for neural domain adaptation.  In Lun-Wei Ku and Cheng-Te Li, editors, Proceedings of the Eighth International Workshop on Natural Language Processing for Social Media, pages 61–71, Online, July 2020. Association for Computational Linguistics.
  * [SK18] Ozan Sener and Vladlen Koltun.  Multi-task learning as multi-objective optimization.  CoRR, abs/1810.04650, 2018.
  * [SSM+22] Yixuan Su, Lei Shu, Elman Mansimov, Arshit Gupta, Deng Cai, Yi-An Lai, and Yi Zhang.  Multi-task pre-training for plug-and-play task-oriented dialogue system.  In Smaranda Muresan, Preslav Nakov, and Aline Villavicencio, editors, Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 4661–4676, Dublin, Ireland, May 2022. Association for Computational Linguistics.
  * [TLJ+23] Marcos Treviso, Ji-Ung Lee, Tianchu Ji, Betty van Aken, Qingqing Cao, Manuel R. Ciosici, Michael Hassid, Kenneth Heafield, Sara Hooker, Colin Raffel, Pedro H. Martins, André F. T. Martins, Jessica Zosa Forde, Peter Milder, Edwin Simpson, Noam Slonim, Jesse Dodge, Emma Strubell, Niranjan Balasubramanian, Leon Derczynski, Iryna Gurevych, and Roy Schwartz.  Efficient Methods for Natural Language Processing: A Survey.  Transactions of the Association for Computational Linguistics, 11:826–860, 07 2023.
  * [ZYY+23] Zhihan Zhang, Wenhao Yu, Mengxia Yu, Zhichun Guo, and Meng Jiang.  A survey of multi-task learning in natural language processing: Regarding task relatedness and training methods.  In Andreas Vlachos and Isabelle Augenstein, editors, Proceedings of the 17th Conference of the European Chapter of the Association for Computational Linguistics, pages 943–956, Dubrovnik, Croatia, May 2023. Association for Computational Linguistics.

##  VI Appendix

TABLE I: The Result for Embedding Training Acceleration Performance (Multi-Task Learning) Methodology | Epoches | Accuracy | macro | average
---|---|---|---|---
Precision | Recall | F1 | Precision | Recall | F1
$CBOW$ | original | 25 | 49% | 49% | 49% | 48% | 50% | 49% | 48%
update0 | 25 | 69% | 69% | 67% | 65% | 69% | 69% | 66%
update1 | 25 | 67% | 69% | 68% | 65% | 61% | 67% | 64%
update2 | 25 | 71% | 71% | 71% | 71% | 71% | 71% | 71%
update3 | 25 | 71% | 68% | 66% | 68% | 73% | 74% | 73%
$SkipGram$ | original | 25 | 48% | 51% | 50% | 50% | 51% | 48% | 32%
update0 | 25 | 68% | 68% | 67% | 67% | 68% | 68% | 68%
update1 | 25 | 74% | 74% | 73% | 73% | 74% | 74% | 73%
update2 | 25 | 69% | 74% | 68% | 69% | 64% | 69% | 67%
update3 | 25 | 76% | 76% | 76% | 76% | 76% | 76% | 76%
$CBOWFastText$ | original | 25 | 45% | 45% | 46% | 42% | 45% | 45% | 41%
update0 | 25 | 70% | 74% | 71% | 69% | 65% | 70% | 79%
update1 | 25 | 62% | 66% | 63% | 61% | 66% | 62% | 61%
update2 | 25 | 69% | 61% | 71% | 68% | 71% | 69% | 69%
update3 | 25 | 70% | 68% | 74% | 71% | 68% | 68% | 68%
$SkipGramFastText$ | original | 25 | 51% | 51% | 51% | 51% | 51% | 51% | 51%
update0 | 25 | 61% | 64% | 63% | 62% | 65% | 64% | 62%
update1 | 25 | 64% | 63% | 62% | 63% | 63% | 65% | 64%
update2 | 25 | 62% | 62% | 62% | 62% | 62% | 62% | 62%
update3 | 25 | 72% | 73% | 71% | 72% | 71% | 72% | 71%
$GloveTwitter$ | original | 25 | 48% | 52% | 50% | 35% | 53% | 48% | 33%
update0 | 25 | 74% | 74% | 74% | 74% | 74% | 74% | 74%
update1 | 25 | 68% | 68% | 68% | 68% | 68% | 68% | 68%
update2 | 25 | 70% | 72% | 65% | 68% | 73% | 69% | 71%
update3 | 25 | 73% | 72% | 73% | 73% | 73% | 73% | 73%
TABLE II: The Result for Embedding Training Acceleration Performance (Multi-Task to Single-Task Dimish) Methodology | Epoches | Accuracy | macro | average
---|---|---|---|---
Precision | Recall | F1 | Precision | Recall | F1
$CBOW$ | original | 25 | 49% | 49% | 49% | 48% | 50% | 49% | 48%
update0 | 25 | 63% | 62% | 63% | 63% | 63% | 63% | 63%
update1 | 25 | 72% | 70% | 69% | 68% | 71% | 70% | 68%
update2 | 25 | 73% | 71% | 71% | 71% | 74% | 72% | 71%
update3 | 25 | 75% | 74% | 72% | 74% | 77% | 74% | 74%
$SkipGram$ | original | 25 | 48% | 51% | 50% | 36% | 51% | 48% | 32%
update | 25 | 68% | 77% | 67% | 64% | 76% | 68% | 65%
update0 | 25 | 71% | 79% | 70% | 67% | 78% | 69% | 68%
update1 | 25 | 74% | 82% | 73% | 73% | 79% | 74% | 73%
update2 | 25 | 75% | 83% | 76% | 74% | 81% | 75% | 75%
update3 | 25 | 77% | 85% | 79% | 76% | 82% | 76% | 76%
$CBOWFastText$ | original | 25 | 45% | 45% | 46% | 42% | 45% | 45% | 41%
update0 | 25 | 70% | 74% | 71% | 69% | 65% | 70% | 79%
update1 | 25 | 71% | 77% | 73% | 70% | 68% | 72% | 82%
update2 | 25 | 74% | 79% | 74% | 72% | 71% | 74% | 84%
update3 | 25 | 77% | 81% | 75% | 74% | 73% | 75% | 85%
$SkipGramFastText$ | original | 25 | 51% | 51% | 51% | 51% | 51% | 51% | 51%
update0 | 25 | 61% | 64% | 63% | 62% | 65% | 64% | 62%
update1 | 25 | 64% | 65% | 66% | 65% | 67% | 65% | 64%
update2 | 25 | 66% | 67% | 69% | 67% | 70% | 66% | 66%
update3 | 25 | 72% | 73% | 71% | 72% | 71% | 72% | 71%
$GloveTwitter$ | original | 25 | 47% | 45% | 49% | 34% | 45% | 47% | 32%
update0 | 25 | 71% | 74% | 72% | 70% | 74% | 71% | 70%
update1 | 25 | 74% | 74% | 74% | 74% | 74% | 74% | 74%
update2 | 25 | 75% | 75% | 75% | 75% | 75% | 75% | 75%
update3 | 25 | 76% | 76% | 76% | 76% | 76% | 76% | 76%
TABLE III: The Result for Embedding Training Acceleration Performance (Multi-Task to Single Task - Switch) Methodology | Epoches | Accuracy | macro | average
---|---|---|---|---
Precision | Recall | F1 | Precision | Recall | F1
$CBOW$ | original | 25 | 49% | 49% | 49% | 48% | 50% | 49% | 48%
update0 | 25 | 68% | 68% | 68% | 68% | 68% | 68% | 68%
update1 | 25 | 70% | 69% | 69% | 69% | 70% | 70% | 70%
update2 | 25 | 71% | 71% | 71% | 70% | 71% | 72% | 71%
update3 | 25 | 74% | 74% | 74% | 74% | 74% | 74% | 74%
$SkipGram$ | original | 25 | 48% | 51% | 50% | 36% | 51% | 48% | 32%
update | 25 | 68% | 68% | 68% | 68% | 68% | 68% | 68%
update0 | 25 | 70% | 69% | 69% | 70% | 70% | 70% | 69%
update1 | 25 | 71% | 71% | 71% | 72% | 71% | 72% | 71%
update2 | 25 | 74% | 74% | 74% | 74% | 74% | 74% | 74%
$CBOWFastText$ | original | 25 | 45% | 45% | 46% | 42% | 45% | 45% | 41%
update0 | 25 | 68% | 68% | 68% | 68% | 68% | 68% | 68%
update1 | 25 | 69% | 69% | 70% | 70% | 70% | 70% | 70%
update2 | 25 | 71% | 71% | 71% | 72% | 71% | 72% | 72%
update3 | 25 | 74% | 74% | 74% | 74% | 74% | 74% | 74%
$SkipGramFastText$ | original | 25 | 51% | 51% | 51% | 51% | 51% | 51% | 51%
update0 | 25 | 68% | 68% | 68% | 68% | 68% | 68% | 68%
update1 | 25 | 70% | 70% | 70% | 69% | 69% | 69% | 69%
update2 | 25 | 72% | 71% | 71% | 70% | 70% | 70% | 71%
update3 | 25 | 74% | 74% | 74% | 74% | 74% | 74% | 74%
$GloveTwitter$ | original | 25 | 51% | 46% | 49% | 40% | 46% | 51% | 41%
update0 | 25 | 68% | 69% | 68% | 68% | 69% | 68% | 68%
update1 | 25 | 70% | 70% | 69% | 69% | 70% | 70% | 69%
update2 | 25 | 74% | 75% | 73% | 73% | 74% | 74% | 73%
update3 | 25 | 77% | 77% | 77% | 77% | 77% | 77% | 77%

###  VI-A Update Functions Settings in MT2ST Experiments

This section elaborates on the update functions in MT2ST experiments. These
functions serve as auxiliary tasks, each targeting specific aspects of NLP
models.

UPDATE 0: Semantic/Syntactic Meaning
Description: This task evaluates the differences in semantic and syntactic
tasks in word embedding representations.

$UPDATE0=\sum\left(BE_{4},(BE_{2}-BE_{1}+BE_{3})^{T}\right)$ (6)

UPDATE 1: Misalignment Error
Description: This task is designed to tackle the issue of misalignment errors,
a common challenge in translation tasks.

$UPDATE1=-\text{not\\_aligned}+\gamma\times\text{mis\\_aligned}$ (7)

UPDATE 2: Word Analogy
Description: The task is developed for comprehensive evaluation of word
analogy task by a linear combination of three key metrics: Semantic Accuracy,
Syntactic Accuracy, and Total Accuracy.

$\begin{split}UPDATE3=Linear\\{Semantic\\_Acc,\\\
Syntactic\\_Acc,Total\\_Acc\\}\end{split}$ (8)

UPDATE 3: Spearsman’s Correlation
Description: This task focuses on Spearman Correlation as an auxiliary task to
measure the relationship between ranked variables.

Input : Word Embedding Matrix from encoder: $\mathcal{WE}$;

Word correlation information: $\mathcal{CORR}$

Output : Spearman’s Correlation Loss: $SMLoss$, considering $v$ for all
$v\in\mathcal{WE}$ with correlation data $\mathcal{CORR}$

Initialize $SMLoss=0$

for  _$v_{1}$ in $\mathcal{WE}$_ do

for  _$v_{2}$ in $\mathcal{WE}[\mathcal{CORR}[v_{1}]]$_ do

$R_{1}$ = RankElem($v_{1}$)

$R_{2}$ = RankElem($v_{2}$)

$SMLoss$ += $\frac{\text{Cov}(R_{1},R_{2})}{\sigma_{R_{1}}*\sigma_{R_{2}}}$

end for

end for

return _$SMLoss$_

Algorithm 3 Spearman’s Correlation Loss Calculation

###  VI-B Update Function Design

$w_{i}^{(t+1)}=w_{i}^{t}-\delta*\nabla Loss^{(t)}{w_{i}^{(t)}}$ (9)

$Loss^{(t+1)}=\begin{cases}Loss_{task}^{(t)}+WALoss^{(t)}+SMLoss^{(t)},&\text{if
}0<t<T_{cutoff}\\\ Loss_{task},&\text{if }T_{cutoff}<t<T_{train}\end{cases}$
(10)

$Loss^{(t+1)}=Loss_{task}^{(t)}+\alpha^{t}*WALoss^{(t)}+\beta^{t}*SMLoss^{(t)}$
(11)


# Gate-voltage switching of non-reciprocal transport in oxide-based Rashba
interfaces

Julien Bréhin  Luis M. Vicente Arche  Sara Varotto  Srijani Mallik  Unité
Mixte de Physique, CNRS, Thales, Université Paris-Saclay, Palaiseau, France
Jean-Philippe Attané  Laurent Vila  University Grenoble Alpes, CEA, CNRS,
Grenoble INP, SPINTEC, 38000 Grenoble, France  Agnès Barthélémy  Unité Mixte
de Physique, CNRS, Thales, Université Paris-Saclay, Palaiseau, France  Nicolas
Bergeal  Laboratoire de Physique et d’Etude des Matériaux, UMR 8213 - CNRS,
ESPCI-Paris, PSL, Sorbonne Université, Paris, FRANCE  Manuel Bibes  Author to
whom correspondence should be addressed<EMAIL_ADDRESS>Unité
Mixte de Physique, CNRS, Thales, Université Paris-Saclay, Palaiseau, France

(August 30, 2024)

###### Abstract

The linear magnetoelectric effect (ME) is the phenomenon by which an electric
field produces a magnetization. Its observation requires both time-reversal
and space-inversion symmetries to be broken, as in multiferroics. While the ME
effect has only been studied in insulating materials, it can actually exist in
non-centrosymmetric conductors such as two-dimensional electron gases (2DEGs)
with Rashba spin-orbit coupling. It is then coined the Edelstein effect (EE),
by which a bias voltage – generating a charge current – produces a transverse
spin density, i.e. a magnetization. Interestingly, 2D systems are sensitive to
voltage gating, which provides an extra handle to control the EE. Here, we
show that the sign of the EE in a SrTiO3 2DEG can be controlled by a gate
voltage. We propose various logic devices harnessing the dual control of the
spin density by current and gate voltages and discuss the potential of our
findings for gate-tunable non-reciprocal electronics.

††preprint: PRApplied-NER1048/Bréhin

##  I Introduction

As classical semiconductor electronics reaches its physical limits and the
power consumption of computing architectures increases dramatically [1], new
paradigms of computation are emerging, notably harnessing the electron’s spin
in addition to its charge [2]. Realizing beyond-CMOS devices operating on spin
requires the implementation of schemes to electrically generate, control and
detect spin-encoded information. Key concepts in this endeavour are the
magnetoelectric effect (ME, to generate and control a magnetization by an
electric field) and spin-charge interconversion. The latter can be achieved by
the direct and inverse spin Hall (SHE)[3] and Edelstein effects (EE)[4]. The
SHE directly generates a spin current from a charge current while the inverse
SHE (ISHE) does the reciprocal operation. The EE generates a spin density
(that is, a magnetization) from a charge current; the spin density can then
diffuse in adjacent layers as a spin current. Reciprocally, the inverse EE
(IEE) generates a charge current from an injected spin density.

Examples of spin-based logic devices include Intel’s MESO transistor [5, 6]
(in which the ME is used to write magnetic information and the ISHE or IEE is
used to read it out) or the FESO concept [7] (that harnesses the ferroelectric
control of the IEE [7] or the ISHE [8, 9]).

While the ME [10] and EE [4] look a priori quite distinct, the EE may appear
as a form of ME arising in conducting systems (the ME is usually restricted to
insulating polar magnets such as Cr2O3 or multiferroics), as was actually
pointed out by Edelstein himself [4]. The EE has been reported in surface
states of topological insulators [11] and at Rashba interfaces including
Ge(111) [12] and oxide interfaces [13, 14, 15, 16]. However, the sign of the
EE, determining the direction of the non-equilibrium magnetization generated
by the applied charge current, is intrinsic to the band structure of the
material and is thus usually fixed.

Here, by injecting an electrical current in an AlOx/SrTiO3 two-dimensional
electron gas (2DEG) patterned into a Hall bar, we demonstrate the generation
of a magnetization via the EE, that manifests as an additional longitudinal
resistance upon applying a transverse magnetic field. This resistance term
depends linearly on the current, indicating that the sign of the EE-generated
magnetization changes when switching the current direction. We show that it is
possible to also change its sign by applying a gate voltage, which tunes the
Fermi energy across the competing Rashba-split electronic band pairs of the
SrTiO3 (STO) 2DEG[17, 18]. This provides a unique dual control of the EE-
generated magnetization, which we propose to exploit in various types of
spintronic logic gate devices free from ferromagnets.

##  II Theoretical background

Figure 1: (a) Fermi contours resulting from the direct Edelstein effect (DEE)
in a simple Rashba system with spin-split parabolic bands. Upon injection of a
charge current $\mathbf{j^{c}}$ along $-x$, the Fermi contours are shifted in
the same direction by $\Delta k_{x}$, creating a net density of spins with
transverse direction, and resulting in a net magnetization. (b) Temperature
dependence of the sheet resistance of a device based on a-LAO/Al/STO 2DEG
(LAO: LaAlO3). (c) Schematic of the sample and measurement configuration used
throughout the paper. The inset shows a microscope image of the actual device
used hereafter.

The Rashba effect [19] results from the coexistence of a broken inversion
symmetry – which is typically present at a heterostructure interface and
manifests as an electric field perpendicular to that interface – and a
sizeable spin-orbit interaction (SOI). The direct consequence is the lifting
of spin-degeneracy in momentum-space, introduced by a new term
$H_{R}=\frac{\alpha_{R}}{\hbar}(\sigma\crossproduct
p)$$\dotproduct\hat{\mathbf{{z}}}$ in the Hamiltonian, where $\hbar$ is the
reduced Planck constant, ${\bf\it\sigma}$ is the vector of Pauli matrices,
$\mathbf{p}$ is the electron momentum and $\hat{\mathbf{z}}$ is a unit vector
normal to the plane of the heterointerface (and thus, parallel to the
symmetry-breaking electric field) and $\alpha_{R}$ is a constant that
quantifies the strength of the Rashba SOI and the resulting spin-splitting. It
appears from this term that the spin and momentum degrees of freedom will be
locked perpendicular to each other and, in a system where the Rashba splitting
occurs near the Fermi level, the Fermi surface will consist of two concentric
contours with opposite chirality of the spin winding [20].

In the semiclassical picture of transport, applying an electric field across
the 2DEG and thus injecting a charge current $\mathbf{j^{c}}$ along one
direction, say $\mathbf{-x}$, results in a shift of both inner and outer
contours by the same amount $\Delta k_{x}$. This generates an excess spin
densities $\delta s_{+}$ and $\delta s_{-}$ for each contour as seen in Fig.
1a. Since these two contours do not coincide, these two densities are not
equivalent, they do not compensate each other and a finite spin density (a
magnetization) transverse (i.e. along the $y$-axis) and proportional to the
charge current appears in the system. This magneto-electric effect converting
a charge current into a spin current was first described in Ref. [4] and is
referred to as the direct EE or the inverse spin galvanic effect.

Figure 2: Angle dependence of the longitudinal resistance obtained by harmonic
transport with an in-plane magnetic field, at a temperature
$T=$2\text{\,}\mathrm{K}$$ and gate voltage $V_{g}=$200\text{\,}\mathrm{V}$$.
(a) First harmonic signal normalized by the (zero-field) sheet resistance
$R_{0}=R_{s}(B=0)$, for magnetic fields at $0.5\text{\,}\mathrm{T}$ (in blue)
and increasing from $2\text{\,}\mathrm{T}$ up to $9\text{\,}\mathrm{T}$ (in
red) with steps of $1\text{\,}\mathrm{T}$. Continuous lines are fits to the
experimental data (filled circles). (b) Fitted amplitudes of the first
harmonic resistance signal showing a quadratic dependence with the in-plane
magnetic field amplitude $\mu_{0}H_{\parallel}$. (c) and (e) angular
dependences of the second harmonic signal $R_{xx}^{2\omega}$ at different
fields (c) and current densities (e), showing unidirectional $2\pi$-periodic
magnetoresistance. (d) and (f) display the corresponding fits of their
amplitude as a function of field and current, respectively. Some points on (f)
at low current densities correspond to curves not shown on (e) for clarity.

It is important to note that, upon reversal of the sign of the injected charge
current, the net magnetization generated will change its orientation
accordingly. Thus, an external magnetic field $\mathbf{B}$ applied in-plane
will interact constructively or destructively with the generated magnetization
depending on the sign of the current and the applied field orientation,
leading to a decrease or to an increase of the sample resistance. This
unidirectional magnetoresistance term then has two extrema as a function of
the angle $\Phi$ between the current and the field, namely when they are
transverse to each other, and it is proportional to both the injected current
density and the applied magnetic field. As such, it is also called bilinear
magnetoresistance (BMR) and its angular dependence has been found by Vaz _et
al._ [14] to follow:

$\textup{BMR}=A_{BMR}\frac{j^{c}_{x}}{j^{c}}\sin{\Phi},$ (1)

where

$A_{BMR}=\frac{3}{2}\pi\frac{g\mu_{B}}{\absolutevalue{e}\hbar}\frac{\alpha_{R}\tau^{2}}{\epsilon_{F}}j^{c}B,$
(2)

in the case where one uses normalized magnetoresistance
$\textup{MR}=[R_{s}(B)-R_{s}(B=0)]/R_{s}(B=0)$ and with $\Phi$ the angle
between the injected charge current $\mathbf{j^{c}}$ and the applied in-plane
magnetic field $\mathbf{B}=\mu_{0}\mathbf{H_{\parallel}}$. For a given set of
experimental conditions, the amplitude of this effect can thus be used to
estimate the Rashba coefficient $\alpha_{R}$. However, it is also highly
dependent on other physical parameters of the system such as the scattering
time $\tau$, the $g$-factor and Fermi energy $\epsilon_{F}$. Importantly, in
the case of oxide 2DEGs, it is known that these three parameters can vary when
applying a gate voltage, in addition to $\alpha_{R}$ [21, 22, 17].

Concurrently to the BMR, the magnetoresistance is also modulated by an
anisotropic $\pi$-periodic term that is independent on the current but that
scales quadratically with the applied magnetic field, i.e. a quadratic
magnetoresistance (QMR) term [14, 23]:

$\textup{QMR}=A_{QMR}\cos{2\Phi},$ (3)

where

$A_{QMR}=\frac{3}{4}\left(\frac{g\mu_{B}}{\hbar}\right)^{2}\tau^{2}B^{2}.$ (4)

Since both the BMR and QMR are increasing with $\tau^{2}$, taking the ratio
between their amplitudes removes one system-dependent variable and simplifies
the expression into a function depending mainly on universal constants and
fixed experimental parameters:

$\frac{A_{BMR}}{A_{QMR}}=\frac{2\pi\hbar}{\absolutevalue{e}g\mu_{B}}\frac{\alpha_{R}}{\epsilon_{F}}\frac{j^{c}}{B}.$
(5)

This model provides an efficient method to extract a Rashba coefficient in
different systems, and has been applied already to LAO/STO and KTaO3 2DEGs,
yielding values consistent with the literature[21, 24, 25]. In LAO/STO, Vaz
_et al._ have shown a gate-induced modulation by a factor $\sim 2.5$ of the
Rashba coefficient extracted from the BMR[14]. In the next section, we show
results for AlO_x/STO 2DEGs with a carrier density typically higher than in
LAO/STO [26] and evidence a change of sign of the BMR and thus of the
effective $\alpha_{R}$ by a gate voltage.

##  III Experimental details

###  III.1 Sample preparation

To characterize the charge-to-spin current conversion, we have performed
transport measurements in a 2DEG formed by depositing
$2\text{\,}\mathrm{n}\mathrm{m}$ of Al on a STO substrate [26]. We fabricated
Hall bar devices via a single-step lift-off process by exposing the design
with the SmartPrint projection lithography setup after spin-coating a SPR 700
1.0 photoresist on the bare substrate. Al was deposited at room temperature by
dc magnetron sputtering on as received STO (001) substrates from SurfaceNet.
During Al deposition, the Ar partial pressure and the dc power were kept fixed
at $5.2\times 10^{-4}$ mbar and 11 W, respectively so that the deposition rate
was with a rate of $\sim$ 1 Å/s.

The Al in contact with the STO substrate oxidizes into AlO_x, which reduces
the STO and dopes its surface with electrons, forming the 2DEG[26]. To help
increasing the visibility of the devices, a layer of
$20\text{\,}\mathrm{n}\mathrm{m}$ of amorphous LaAlO3 (a-LAO) was then
deposited by PLD at room temperature with a KrF Excimer laser at an energy of
$42\text{\,}\mathrm{mJ}$, before proceeding to the lift-off with acetone.

###  III.2 Magnetotransport measurements

The samples were bonded with Al wire after the patterning process and inserted
in a Quantum Design PPMS to measure their transport properties. They show the
usual metallic behaviour typical of AlO_x/STO 2DEGs, with a residual
resistivity ratio of $\sim\mathrm{1}\mathrm{0}$ between
$300\text{\,}\mathrm{K}2\text{\,}\mathrm{K}$, as seen in Fig. 1 (b).

At the lowest temperature of $2\text{\,}\mathrm{K}$, a back-gate forming step
was applied: the gate voltage was first increased to its maximum value
$V_{G,max}$ and then cycled repeatedly within the gate range of the experiment
$[-V_{G,max};+V_{G,max}]$ to suppress hysteresis and ensure the full
reversibility of the measurements. Figure 2 and 3 were obtained from two
distinct cooldowns, with $V_{G,max}=$+200\text{\,}\mathrm{V}$$ and
$V_{G,max}=$+150\text{\,}\mathrm{V}$$, respectively.

The magnetotransport measurements were then performed by rotating the sample
under an in-plane constant magnetic field while injecting an a.c. current
along the main channel of the device and measuring the longitudinal voltage
using Stanford Research 830 lock-in amplifiers combined with a Keithley 6221
current source and a Keithley 2400 to apply a gate voltage. With this
technique, any resistance oscillation that is linearly dependent on the
current amplitude (or more broadly any voltage signal with a quadratic current
dependence) will appear in the second harmonic (i.e. at twice the frequency of
the injected a.c. current, $2\omega$). In contrast, symmetric terms
independent of the current should be found in the first harmonic, $\omega$.

##  IV Results

Figure 2a shows the angular dependence of the sheet resistance of the device
obtained from the first harmonic and normalized by the sheet resistance
measured at $B=$$0\text{\,}\mathrm{T}$, and for a gate voltage of
$V_{g}=$$200\text{\,}\mathrm{V}$. Following the model from Vaz et al. [14],
the data can be fitted with a cosine of $2\Phi$ as in Eq. 3. The amplitudes
$A_{QMR}$ obtained from such fits follow a quadratic dependence with the
applied magnetic field as seen in Fig. 2b.

Using Eq. 4 and assuming the $g$-factor to be equal to 2 as was reported in
LAO/STO at high gate voltages [21], we obtain a momentum relaxation time
$\tau_{m}\approx$ $1.6\text{\times}{10}^{-14}\text{\,}\mathrm{s}$ which goes
up to $6.4\text{\times}{10}^{-14}\text{\,}\mathrm{s}$ for $g=0.5$ (as used in
ref. [14]). This is in the range of values obtained from a classical transport
model with magnetotransport data obtained in the same sample (see Fig S1 in
Supplemental Material [27]), with $\tau_{m}\approx$
$5.5\text{\times}{10}^{-14}\text{\,}\mathrm{s}1.2\text{\times}{10}^{-13}\text{\,}\mathrm{s}$
at the highest gate voltage (for a $d_{xy}$-only mass $m^{*}=0.6m_{e}$ [17] or
an estimated $m^{*}=1.3m_{e}$ used in ref. [14], respectively).

In addition, as expected from the Edelstein effect, we obtain a
$2\pi$-periodic signal from the second harmonic signal which we fit to extract
the amplitude and plot as a function of the magnetic field in Fig. 2d and as a
function of the current density in Fig. 2f. We see that this signal is mostly
linear in both current density and magnetic field, hence it corresponds to the
BMR term from charge-to-spin conversion. Note that here, we chose to use the
same sign convention as in Ref. [14] where a negative sine function is
associated to a positive amplitude and as a consequence, the positive sine
function obtained at a gate voltage of $200\text{\,}\mathrm{V}$ yields a
negative BMR amplitude.

Since the Rashba spin-orbit properties of STO 2DEGs are known to vary
significantly when applying a gate voltage[21, 22, 17, 14], we have also
performed similar measurements at different back-gate voltages, as shown in
Fig. 3.

Figure 3: Gate dependence of the bilinear magnetoresistance obtained from
second harmonic transport measurements with
$I_{ac}=$150\text{\,}\mathrm{\SIUnitSymbolMicro A}$$, at a temperature
$T=$2\text{\,}\mathrm{K}$$. (a) Angle-dependence of the second harmonic signal
at fixed gate voltages from $-150\text{\,}\mathrm{V}+125\text{\,}\mathrm{V}$,
applying $9\text{\,}\mathrm{T}$ in the plane of the sample. Circles represent
the measured data, after removing some background contributions (see Methods),
while the fits are indicated by black dashed lines. Only few curves are
displayed at gate voltages separated by $25\text{\,}\mathrm{V}$ for the sake
of clarity. (b) BMR amplitude extracted from the fits to the angular
dependences as a function of the gate voltage. (c) Sheet resistance from the
second harmonic channel as the in-plane magnetic field is swept between +9;-9,
at $V_{g}=$-100\text{\,}\mathrm{V}$$ (in orange) and $+75\text{\,}\mathrm{V}$
(in green). The lines are a guide to the eye.

Fig. 3a shows the second-harmonic signal measured at different gate voltages
varying from +125-150. It is clear that the amplitude of the BMR signal
changes significantly and even switches sign, passing from positive to
negative amplitude as the gate voltage is increased. The full dependence of
$A_{BMR}$ is summarized in Fig. 3b where we can identify the cross-over to be
around $V_{g}\approx\qtyrange{0}{20}{}$. We also measured this second-harmonic
signal at a fixed angle $\Phi=$90\text{\,}\mathrm{\SIUnitSymbolDegree}$$, i.e.
keeping the current and field perpendicular to each other, while sweeping the
magnetic field. Fig. 3c confirms the linear dependence of the BMR with the
field with opposite slope between $V_{g}=$-100\text{\,}\mathrm{V}$$ and
$+75\text{\,}\mathrm{V}$.

Figure 4: Examples of three arms logic gate devices exploiting the direct and
inverse Edelstein effects. (a) A gate voltage and the charge current sign of
the first branch are used as inputs, and the sign of the current in the third
branch as output, realizing the XNOR logic function. (b) Two independent gate
voltages as inputs are applied at the crossing points, with the sign of the
current in the third branch as output (XNOR logic gate). (c) Two charge
currents are summed together and used as inputs to gate the first crossing,
with the sign of the current in the third arm as the output (NOR logic gate).

Checking upon Eq. 2, we see that the BMR amplitude may depend on different
systems parameters which can be affected under the application of a back-gate
voltage $V_{g}$. First, variations of the $g$-factor have been reported from
0.5 to 2.5 in LAO/STO 2DEGs as $V_{g}$ is increased towards positive values
[21]. Yet, this variation cannot explain the sign change of the BMR. In the
simple parabolic band model used in ref. [14], $A_{BMR}$ is also inversely
proportional to the Fermi energy and should thus decrease in amplitude as the
carrier density is increased but should remain positive. Finally, even
considering the quadratic dependence of $A_{BMR}$ with the momentum relaxation
time $\tau$, and an increase of the overall mobility as the second type of
carrier (from $d_{xz}$/$d_{yz}$ subbands) arises, one could explain the fast
increase of the BMR amplitude in absolute value but not the sign change.

In fact, it has been shown previously that the complex band structure of STO
2DEGs with topologically trivial and non-trivial avoided crossings and
opposite contributions to the Edelstein tensor between the different subbands
with $d_{xy}$ and $d_{xz}$/$d_{yz}$ character could give rise to varying
amplitude and sign of the effective Rashba coefficient $\alpha_{R}$ and
resulting charge-spin interconversion [17, 14, 18, 28].

Thus, the sign change observed in the BMR signals upon varying the applied
voltage most likely reflects a change of $\alpha_{R}$ as the gate voltage
tunes the chemical potential of the system, populating higher states in the
band-structure with $d_{xz,yz}$ character and opposite Rashba spin texture to
that of $d_{xy}$ lower energy states.

More quantitatively, in the explored gate range and using g=0.5, within the
model of Ref. [14] the value of $\alpha_{R}$ varies from $\sim
50\text{\,}\mathrm{meV}\text{${}\cdot{}$}\mathrm{\text{Å}}$ at large negative
voltage to $\sim-200\text{\,}\mathrm{meV}\text{${}\cdot{}$}\mathrm{\text{Å}}$
at large positive voltage. This former value is in the range of what was found
in previous reports [21]. However,
$\sim-200\text{\,}\mathrm{meV}\text{${}\cdot{}$}\mathrm{\text{Å}}$ clearly
exceeds what is expected for STO 2DEGs. Using $g$ factor values of 2 reported
at high carrier densities[21] would make this value even larger, signaling the
limits of the model with single parabolic Rashba bands in this case. Overall,
these results thus point to the need for developing more elaborate models to
compute the BMR from the band structure, beyond the simple one-band scenario
from Ref. [14], i.e., models able to capture better the physics of this system
when 3 or 4 band pairs are occupied, with a complex electronic structure
involving trivial and non-trivial avoided crossings [17, 28].

##  V Discussion

In any case, the observed BMR sign change implies that the magnetization
generated via the direct Edelstein effect, when injecting a given charge
current along a device main channel, changes sign as well. Since the BMR
depends linearly on the current, our results thus demonstrate a dual control
of the BMR (and thus of the current-generated magnetization) by both the
current and the gate voltage. This offers interesting possibilities to design
new spintronic devices. In a H-shaped device such as those sketched in Fig. 4,
consisting of two longitudinal arms (left, input and right, output) connected
by a transverse arm, applying a charge current in the left arm generates a
transverse magnetization able to diffuse as a spin current in the transverse
arm over a certain length $\lambda_{s}$ in the absence of any net charge
displacement. It is then possible to use the IEE at the intersecton of the
transverse and right arms to convert this spin current back to a charge
current whose sign will depend on $\alpha_{R}$ and on the direction of the
spins generated by the EE.

In Fig. 4, we present devices utilizing the direct and inverse Edelstein
effects, combined with a gate electric field tuning of the 2DEG band filling,
allowing a modulation of the spin-charge interconversion sign and efficiency
accordingly. Here, we have demonstrated that the BMR sign can be modulated by
a back gate, and for the devices, we propose to use a top gate as it was also
shown to efficiently modulate transport properties in STO 2DEGs (see e.g. [29,
30]) and would be advantageous to integrate and scale down such devices while
tuning the required voltages given some adequate material engineering. In Fig.
4a, a top gate is thus placed at the intersection between the input and
transverse arms to modulate the direction of the spins generated by the direct
EE. Depending on their direction, the charge current generated by the IEE at
the intersection of the transverse and output arms will be positive or
negative. Since changing the sign of the charge current in the input arm also
reverses the direction of the generated spins and thus of the sign of the
output current, we can exploit this dual control to perform logic operations,
with the sign of the input charge current $J_{C}^{in}$ and the sign of the
gate voltage $V_{G}$ as inputs and the sign of the output current as the
ouput. Compiling the different input and output logic states in a truth table
shows that this type of device realizes an exclusive-inverted-OR (XNOR) logic
function and could be chained with other similar devices.

In Fig. 4(b), a similar logic operation is achieved but now by having two
independent top gates, one controlling the sign of charge-spin conversion at
the intersection of the input and transverse arms (VG1) and another one
controlling the sign of charge-spin conversion at the intersection of the
transverse and output arms (VG2). Finally, Fig. 4(c) sketches another device
with just one top gate, in which the two inputs are currents $I_{1}$ and
$I_{2}$ accumulating or depleting charges at the top gate. In absence of any
current, the logic gate is in its high state as per the truth table in Fig
4(a). As soon as one of these two currents becomes greater than zero, an
effective electric field develops across the gate capacitor, so that the
output current of the device $J_{out}$ becomes negative and the logic state
becomes zero. The same results is obtained if both currents $I_{1}$ and
$I_{2}$ are above zero, completing the NOR universal logic function.

##  VI Conclusion

Our work expands the topic of nonreciprocal transport phenomena[31] that is
pervading the realm of quantum materials. Nonreciprocal transport has been
reported in non-magnetic topological insulators[11, 32], magnetic topological
insulators [33, 34], non-centrosymmetric superconductors[35], ferroelectric
Rashba semiconductors [36] and Rashba interfaces[13, 12, 14, 16]. While the
mechanism may be different depending on the materials systems, the possibility
to tune the nonreciprocal transport response by a gate voltage in amplitude
_and_ sign has been reported in a few systems[32, 34] which could ultimately
bring the proposed devices to higher operating temperatures. The theoretical
efficiency of such devices is discussed in the Supplemental Material
[[See][foradiscussionontheRashbavs.Fermienergiesandonthedeviceoutput/inputefficiency.]supplemental_material]
(see also references [40, 41, 42, 37, 38, 39] therein). This body of results
offers exciting possibilities to design spin–orbitronics applications
operating at low power, from logic devices such as those proposed here to spin
circulators, mixers, isolators, etc, in analogy with nonreciprocal devices
operating with photons or magnons[31].

###### Acknowledgements.

This project received funding from the ERC Advanced grant “FRESCO”
$\\#$833973, from the European Commission under the H2020 FETPROACT Grant
TOCHA (824140), and the French Research Agency (ANR) as part of the projects
CONTRABASS (ANR-20-CE24-0023) and QUANTOP (ANR-19-CE47-0006-02).

## Author contributions

JB prepared the samples with help from LMVA. JB collected and analysed the
data with SV, LVMA, SM, NB and MB. MB conceptualized and led the study with NB
and help from AB, LV and JPA. JB and MB wrote the paper with inputs from all
authors.

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