11institutetext:  Informatics Institute, University of Amsterdam, Amsterdam,
Netherlands. 22institutetext: Amsterdam UMC, University of Amsterdam, dept
Medical informatics, Amsterdam Public Health research institute, Meibergdreef
9, Amsterdam, Netherlands 22email<EMAIL_ADDRESS>{i.vagliano,a.abu-
<EMAIL_ADDRESS>
# Soft-prompt tuning to predict lung cancer using primary care free-text Dutch
medical notes

Auke Elfrink 1122 0009-0006-1805-5597 Iacopo Vagliano 22 0000-0002-3066-9464
Ameen Abu-Hanna 22 0000-0003-4324-7954
Iacer Calixto 22 0000-0001-6244-7906

###### Abstract

We investigate different natural language processing (NLP) approaches based on
contextualised word representations for the problem of early prediction of
lung cancer using free-text patient medical notes of Dutch primary care
physicians. Because lung cancer has a low prevalence in primary care, we also
address the problem of classification under highly imbalanced classes.
Specifically, we use large Transformer-based pretrained language models (PLMs)
and investigate: 1) how soft prompt-tuning—an NLP technique used to adapt PLMs
using small amounts of training data—compares to standard model fine-tuning;
2) whether simpler static word embedding models (WEMs) can be more robust
compared to PLMs in highly imbalanced settings; and 3) how models fare when
trained on notes from a small number of patients. We find that 1) soft-prompt
tuning is an efficient alternative to standard model fine-tuning; 2) PLMs show
better discrimination but worse calibration compared to simpler static word
embedding models as the classification problem becomes more imbalanced; and 3)
results when training models on small number of patients are mixed and show no
clear differences between PLMs and WEMs. All our code is available open source
in https://bitbucket.org/aumc-kik/prompt˙tuning˙cancer˙prediction/.111A short
version of this paper has been published at the 21st International Conference
on Artificial Intelligence in Medicine (AIME 2023).

###### Keywords:

Prediction models Natural Language Processing Cancer.

##  1 Introduction

In the Netherlands, general practitioners (GPs) act as a gatekeeper between
patients and the secondary healthcare system: GPs decide who should be
referred to specialist care. They also oversee the same patients for many
years. A patient’s visit to their GP is registered in narrative form as free-
text notes, and collectively these provide a unique lens into years of a
patient’s health and medical history. In previous work, Luik et al. [16] have
used free-text patient notes in the early prediction of lung cancer with
context-independent word embedding methods (WEMs) and a simple logistic
regression objective, with promising preliminary results. Ideally, one would
like to use state-of-the-art contextualised pretrained language models (PLMs)
instead of simpler context-independent WEMs, since PLMs typically show
improved performance across a range of different Natural Language Processing
tasks [23, 22] including tasks in the (bio)medical domain [4, 19]. However,
there are at least two important reasons that make using PLMs difficult for
predicting lung cancer from patient free-text notes: i) Even though PLMs are
language models pretrained on large amounts of raw text using self-supervised
learning [3, 21], such models are data-hungry and typically need large amounts
of supervised training data to achieve good predictive performance on any
specific task. Conversely, lung cancer has a low prevalence (about $0.4\%$ in
our dataset), which makes adapting PLMs to this task difficult. ii) A single
patient may have many years worth of health records, meaning possibly
thousands of words in their medical notes. A PLM’s complexity is quadratic
with respect to input length, making them non-trivial to apply in this use-
case.

In this work, we propose methods to deal with the aforementioned issues by
addressing the following research questions. RQ1. How does soft-prompt tuning
compare to standard model fine-tuning in terms of discrimination and
calibration? In this work, we use soft prompt-tuning [10, 13], an NLP
technique used to adapt PLMs using small amounts of training data, and compare
it to standard model fine-tuning. Overall, we find that soft-prompt tuning
compares favourably to standard model fine-tuning, but similar to fune-tuning
when models are trained on small numbers of patients. RQ2. How does the number
of patients used for model training affect model performance in terms of
discrimination and calibration? We empirically evaluate models trained on very
small numbers of patients—a setting also known as few-shot learning [24]—and
models trained on datasets with different degrees of class imbalance. We
report discrimination and calibration metrics, since both should be assessed
for clinical prediction models [18]. RQ3. How do PLMs compare to simpler
static WEMs in terms of discrimination and calibration? Overall,
contextualised PLMs outperform static WEMs in terms of discrimination, but
WEMs show better calibration in both balanced and imbalanced classification
settings. However, results we obtain for few-shot experiments show mixed
results, indicating that WEMs may be an alternative to PLMs in this setting.

##  2 Methodology

We now provide details on the patient selection and data cleaning before
elaborating on our experimental settings.

###  2.1 Data

We have unique access to data from patients of General Practitioners (GPs)
that are associated with the two hospitals of the Amsterdam University Medical
Centers, the Free University Medical Center (VUMC) and the Academic Medical
Center of the University of Amsterdam (AMC). Our patient cohort has an entry
date of 01/01/2002 and an exit date of 31/12/2020, and includes patient
demographics and free-text GP notes for every patient visit to their GP.
Henceforth, we only consider free-text notes that fall within the entry and
exit dates. We include patients who conform with the following criteria: i)
the patient must be at least 40 years-old—i.e., we define the patient age as
the date of their most recent free-text note minus their year of birth—, ii)
the patient must have at least one valid free-text note.222All notes were
anonymized with a modified version of the DEDUCE algorithm [17]. Furthermore,
we define how old a patient’s free-text note is by the number of days between
the date of the note and that patient’s last available note.

A free-text note is considered valid depending on whether the note belongs to
a patient diagnosed with lung cancer or not. For a free-text note of a patient
diagnosed with lung cancer to be considered valid, the note must be dated at
least 150 days ($\sim 5$ months) and at most 730 days ($\sim 2$ years) before
the date of the diagnosis. In case the patient is not diagnosed with lung
cancer, the note must be dated at most 730 days ($\sim 2$ years) after the
date of the patient’s last available note. Finally, we also exclude any notes
that are undated, or have a date outside of the collection period.333For
example, a reminder for a 5-year checkup. We use these cut-off periods to
ensure the predictive value of the model, and since GPs typically start to
suspect lung cancer from about 4 months before diagnosis, and symptoms are
expected to be present from 4 months to 2 years before diagnosis [7]. Finally,
we denote by $\mathcal{D}$ the set of patients after we apply all the above
mentioned procedures.

We establish lung cancer diagnoses for the patients in our cohort by linking
to a central database maintained by Integraal Kankercentrum Nederland
(Integrated Cancer Center, IKNL). Henceforth, we refer to patients with a lung
cancer diagnosis as ‘positive’, and all other patients as ‘negative’.

####  2.1.1 Data splits

In Table 1 we show key characteristics for the different datasets used in our
experiments: $\mathcal{D}^{bal}$, $\mathcal{D}^{1:10}$, $\mathcal{D}^{1:100}$,
$\mathcal{D}^{1:250}$, $\mathcal{F}^{2}$, $\mathcal{F}^{4}$,
$\mathcal{F}^{8}$, $\mathcal{F}^{16}$, $\mathcal{F}^{32}$, $\mathcal{F}^{64}$,
and $\mathcal{F}^{128}$. Train, validation, and test splits are always
stratified, i.e., they inherit the same ratio of patients with/without cancer
as the original dataset. Below we describe different data splits we use in
different experiments in more detail.

##### Balanced dataset

We first create the subset $\mathcal{D}^{bal}$ $\subset$ $\mathcal{D}$ with
the same number of positive and negative patients. In $\mathcal{D}^{bal}$ we
include all $1,733$ positive patients in $\mathcal{D}$ and an equal number of
negative patients, i.e., we randomly subsample patients without lung cancer
from $\mathcal{D}$ to have a balanced dataset.

##### Imbalanced test sets

To study the effect of class imbalance in our models, we build test sets where
the class imbalance between the positive and negative class (which in
$\mathcal{D}^{bal}$ has a $1$:$1$ ratio) becomes closer and closer to the true
ratio in the dataset population (which has approximately a $1$:$250$ ratio).
More concretely, we propose different test sets $\mathcal{D}^{pos:neg}$ with
different positive-to-negative ratios, and build the test sets
$\mathcal{D}^{1:10}$, $\mathcal{D}^{1:100}$, and $\mathcal{D}^{1:250}$.

##### Few-shot datasets

Finally, we also build few-shot datasets $\mathcal{F}^{k}$ $\subset$
$\mathcal{D}^{bal}$ that include a small number of patients, and where $k$ is
the number of patients in each class (positive and negative). We use
$k\in\\{2,4,8,16,32,64,128\\}$, and therefore have few-shot datasets
$\mathcal{F}^{2}$, $\mathcal{F}^{4}$, $\mathcal{F}^{8}$, $\mathcal{F}^{16}$,
$\mathcal{F}^{32}$, $\mathcal{F}^{64}$, and $\mathcal{F}^{128}$.

Dataset name # patient notes mean (min, max) total patients (#positive,
#negative) train valid test_1 test_2 $\mathcal{D}^{bal}$ 32.2 (1, 284) 1384
(692, 692) 518 (259, 259) 170 (85, 85) 1394 (697, 697) $\mathcal{D}^{1:10}$
28.1 (1, 293) — — — 7667 (697, 6970) $\mathcal{D}^{1:100}$ 29.2 (1, 572 — — —
70396 (697, 69699) $\mathcal{D}^{1:250}$ 29.5 (1, 1403) — — — 174946 (697,
174249) $\mathcal{F}^{2}$ 30.7 (1, 284) 2 (1, 1) 2 (1, 1) — —
$\mathcal{F}^{4}$ 30.7 (1, 284) 4 (2, 2) 4 (2, 2) — — $\mathcal{F}^{8}$ 30.9
(1, 284) 8 (4, 4) 8 (4, 4) — — $\mathcal{F}^{16}$ 30.9 (1, 284) 16 (8, 8) 16
(8, 8) — — $\mathcal{F}^{32}$ 30.9 (1, 284) 32 (16, 16) 32 (16, 16) — —
$\mathcal{F}^{64}$ 31.1 (1, 284) 64 (32, 32) 64 (32, 32) — —
$\mathcal{F}^{128}$ 31.5 (1, 284) 128 (64, 64) 128 (64, 64) — —

Table 1: Dataset statistics: we show characteristics for our balanced dataset
split $\mathcal{D}^{bal}$, our imbalanced test sets $\mathcal{D}^{pos:neg}$,
and our few-shot datasets $\mathcal{F}^{k}$. For each dataset, we show the
number of patients in the train, valid, test_1 and test_2 splits, as well as
the mean, minimum and maximum number of free-text notes associated to each
patient. Overall, there are a total of 1,733 positive patients in our cohort
(692+259+85+697).

###  2.2 Aggregating per-note into per-patient predictions

In our cohort introduced in Section 2.1, patients have on average
$28.1$–$32.2$ medical notes in their medical history (see Table 1), where each
note has on average $26.2$–$34.2$ tokens. Both PLMs we use in this work,
MedRoBERTa.nl and RobBERT, process texts with a maximum length of $512$
tokens. Nonetheless, we wish to build models that predict risk of lung cancer
for a patient and thus use all medical notes in the patient’s medical history
as predictors. In practice, we train our models to predict lung cancer
independently from each note available for a patient, and apply a simple but
effective post-hoc aggregation step to compute per-patient predictions from
per-note predictions.

First, we construct each training instance $\\{X^{P}_{n},Y^{P}\\}_{n=1}^{N}$
by propagating the label $Y^{P}$ of patient $P$ to each of the $N$ notes
available for that patient, i.e., $X^{P}_{n}$ is the $n$-th note belonging to
patient $P$, and $Y^{P}$ is a binary variable indicating whether patient $P$
has lung cancer. After the model is trained, we aggregate per-note into per-
patient predictions following Equation 1 below.

$P(Y^{P}=1|\\{X^{P}_{n}\\}_{n=1}^{N})=P_{min},$ (1)

where $P_{min}$ is the lowest per-note probability predicted across all notes
$\\{X^{P}_{n}\\}_{n=1}^{N}$.

###  2.3 Soft-prompt tuning

Soft-prompt tuning is a method of model adaptation where a small number of new
parameters are added to a model throughout that model’s architecture, as
opposed to standard model fine-tuning where a classification layer is
typically appended at the “end” of the model [14, 11, 13]. Concretely, we
append continuous, trainable embeddings to the input sequence which are
optimised via backpropagation, while the original model parameters remain
frozen and are not updated. We refer the reader to [13] for details.

###  2.4 Models

####  2.4.1 Pretrained language models (PLMs)

We use two Dutch-language PLMs based on the RoBERTa architecture [15]: RobBERT
[3], which was created by retraining the RoBERTa tokenizer and model from
scratch on a Dutch general language corpus, and MedRoBERTa.nl [21], which was
created with a similar procedure, but using Dutch hospital notes instead of
general language data.

####  2.4.2 Static word embedding models (WEMs)

FastText [2] is a static WEM where word representations encode subword
information. After FastText has been trained, however, the meaning of a word
will not change when changing its context. However, FastText shows strong
results across text classification tasks, especially in tasks where noisy
input texts are common, e.g. texts containing typos, mispellings, and acronyms
[8]. For that reason, we use FastText as one of the baselines in our
experiments.

We follow two steps: 1) We pretrain FastText from scratch using all free-text
notes available for all patients from all splits in $\mathcal{D}^{bal}$. 2) We
use the training notes in $\mathcal{D}^{bal}$ to fine-tune FastText to predict
risk of lung cancer for that dataset.

##  3 Results and Discussion

Model AUROC ($\uparrow$) AUPRC ($\uparrow$) Brier score ($\downarrow$) N P N P
N P FastText (FT) — $90.8$ — $\underline{90.4}$ — $\bm{44.7}$ RobBERT (FT)
$81.8$ $90.2$ $65.7$ $84.8$ $18.2$ $\underline{54.3}$ MedRoBERTa.nl (FT)
$82.4$ $90.2$ $68.6$ $82.7$ $17.6$ $56.8$ RobBERT (ST) $\bm{84.5}$ $\bm{94.3}$
$\underline{72.3}$ $\bm{91.4}$ $\underline{16.7}$ $55.9$ MedRoBERTa.nl (ST)
$\underline{84.3}$ $\underline{92.7}$ $\bm{75.5}$ $86.5$ $\bm{15.9}$ $58.0$

Table 2: Results on $\mathcal{D}^{bal}$ test_1 for models fine-tuned (FT) to
predict lung cancer vs. using soft-prompt tuning (ST). We highlight best
results in bold and second-best by underscoring. N denotes per-note and P per-
patient results. We aggregate per-note into per-patient predictions following
Equation 1.

###  3.1 Full data availability and balanced classes

In Table 2, we report results on our experiments where we use the
$\mathcal{D}^{bal}$ dataset. The $\mathcal{D}^{bal}$ dataset includes all the
positive patients in our cohort and an equal number of negative patients
sampled randomly, i.e., we build a version of the data set with balanced
positive/negative classes (see Table 1 for more details). Here, we aim to
investigate how the methods used in this work fare in this idealised scenario
where class imbalance is not an issue. Moreover, we compare models when
directly fine-tuned or when soft-prompt tuned on the task of predicting
whether a patient has lung cancer. We use the train, valid, and test_1 splits
for model training, model selection, and testing, respectively.

####  3.1.1 From per-note to per-patient predictions

From Table 2, we first highlight that per-patient predictions—which aggregates
per-note predictions for a patient according to Equation 1—consistently
improve on per-note predictions, which suggests that the offline aggregation
method we use can suppress possibly noisy per-note predictions well. We also
experimented with aggregating per-note into per-patient predictions as 1) the
mean, 2) the maximum, and 3) Clinical BERT [6]’s aggregation over all note
probabilities, however using Equation 1 performs best (we do not report those
other experiments due to space constraints).

####  3.1.2 Discussion

First of all, we note that FastText performs comparably to or better than our
fine-tuned PLMs according to all metrics; however, when compared to our soft
prompt-tuned PLMs, FastText is slightly worse (AUROC), comparable (AUPRC) or
much better in terms of calibration (Brier score). Moreover, soft-prompt
tuning tends to consistently outperform standard model fine-tuning, according
to AUROC (by $\sim 2$–$4\%$) and AUPRC (by $\sim 4$–$6\%$), but with no clear
trend according to the Brier score.

Finally, per-note Brier scores are considerably better than per-patient Brier
scores. This suggests that the aggregation method we use (see Equation 1)
consistently improves AUROC and AUPRC at the expense of model calibration.

###  3.2 Balanced vs. imbalanced classes

In Table 3, we show results for our FastText baseline and the best performing
RobBERT model trained on $\mathcal{D}^{bal}$ (train) and selected according to
AUROC scores on $\mathcal{D}^{bal}$ (valid). We provide results on the test_2
splits of $\mathcal{D}^{bal}$, $\mathcal{D}^{1:10}$, $\mathcal{D}^{1:100}$,
and $\mathcal{D}^{1:250}$.

As expected, we note that both AUROC and AUPRC tend to decrease as we increase
the number of negative patients, with more pronounced effects on AUPRC (i.e.,
when test sets have roughly 1:250 patient-to-positive ratios, AUPRC scores are
below $1\%$). However, we also note an opposite trend: Brier scores improve as
test sets become more imbalanced. This makes sense when taking our per-patient
aggregation method into account: since we use the lowest note probability, the
prediction for true negatives will likely be much closer to 0 than the
prediction for true positives will be to 1.

AUROC ($\uparrow$) AUPRC ($\uparrow$) Brier score ($\downarrow$) Test set
$\mathcal{D}^{?}$ bal 1:10 1:100 1:250 bal 1:10 1:100 1:250 bal 1:10 1:100
1:250 Random 50.0 50.0 50.0 50.0 50.0 10.0 1.0 0.4 — — — — Per-note
predictions RB (FT) 76.8 78.3 61.3 54.4 63.9 15.9 1.2 $\bm{0.4}$ 19.9 18.7
28.3 32.4 RB (ST) $\bm{80.1}$ $\bm{81.4}$ $\bm{64.5}$ $\bm{57.8}$ $\bm{70.6}$
$\bm{22.0}$ $\bm{1.5}$ $\bm{0.4}$ $\bm{18.3}$ $\bm{16.8}$ $\bm{27.4}$
$\bm{32.3}$ Per-patient predictions FaT (FT) 78.7 81.0 60.1 56.1 77.2 32.3 1.4
0.5 $\bm{44.8}$ $\bm{8.2}$ $\bm{1.2}$ $\bm{0.7}$ RB (FT) 88.2 90.1 70.3 65.4
81.9 40.0 1.8 0.6 54.0 67.7 24.3 18.0 RB (ST) $\bm{89.7}$ $\bm{91.1}$
$\bm{71.2}$ $\bm{67.1}$ $\bm{86.2}$ $\bm{47.0}$ $\bm{1.9}$ $\bm{0.7}$ 56.3
71.0 25.1 17.8

Table 3: Results on imbalanced vs. balanced test datasets with the best
performing fine-tuning (FT) and soft prompt-tuning (ST) model (RobBERT)
trained on $\mathcal{D}^{bal}$ train split. N denotes per-note and P per-
patient results. RB: RobBERT. FaT: Fast-text. We aggregate per-note into per-
patient results with Equation 1.

###  3.3 Few-shot learning

Few-shot learning results show that when models are trained on 32 or less
patients, PLMs tend to clearly outperform FastText, but when models are
trained on 128 patients we already note that FastText outperforms all PLMs
(according to both AUROC and AUPRC). This happens regardless of the PLM being
fine-tuned or soft-prompt tuned, which suggests that soft-prompt tuning is
still not clearly more resilient to ‘noisy’ inputs than fine-tuned models, at
least according to our experiments.

Moreover, we note that fine-tuned and soft-prompt tuned models seem to perform
comparably in our few-shot experiments. Oscillations in both AUROC and AUPRC
from smaller to higher $k$’s suggest that there is some degree of randomness
in model performance, which is an undesirable feature which we would like to
address in future work.

(a) Number of training patients vs. AUROC scores on $\mathcal{D}^{bal}$
test_1.

(b) Number of training patients vs. AUPRC scores on $\mathcal{D}^{bal}$
test_1.

Figure 1: Results for models trained on $\mathcal{F}^{k}$ (train),
$k\in\\{2,4,8,16,32,64,128\\}$. ST: soft-prompt tuning. FT: fine-tuning.
RobBERT ’full’ is the best RobBERT model trained and tested on
$\mathcal{D}^{bal}$ train and test_1 splits, respectively.

##  4 Related Work

Recent years have seen the release of many English-language PLMs (pre)trained
on clinical data [5, 20, 12, 9]. Dutch-language PLMs are much more rare, and
even more so in the medical domain: we are only aware of Verkijk and Vossen’s
PLM [21], which is pretrained on hospital patient notes. Whereas a variety of
models predict cancer incidence or survival based on hospital notes [9, 1], we
only know of Torec et al. [16]’s model that use free-text GP notes as
predictors. In recent years, a large variety of prompt tuning methods have
become available. Prompt tuning approaches use a fraction of the data and
compute compared to models that fine-tune all parameters in a PLM, which is
one of the main reasons we investigate prompt tuning in our work [11, 14].

##  5 Conclusions and Future Work

In this work, we investigate the application of soft-prompt tuning to perform
the early prediction of lung cancer using free-text patient medical notes of
Dutch primary care physicians. We find that using contextualised pretrained
language models (RobBERT and MedRoBERTa.nl) outperforms strong static word
embedding models (FastText) according to AUROC and AUPRC, however FastText
shows much better calibration than both PLMs. Results we obtained in few-shot
experiments are not so clear cut, and here the difference between PLMs and
WEMs is less clear. Soft prompt tuning consistently outperforms standard model
fine-tuning with PLMs, which we find promising and believe warrants further
research. When testing our models on datasets with increased class imbalance,
performance deteriorates as expected; nonetheless, the best performing PLM
still achieves a reasonable $67.1$ AUROC when tested on a $1$:$250$
(positive:negative) ratio—virtually the same ratio as that of the population
in our dataset—, though the corresponding $0.7$ AUPRC shows modest
improvements upon the random baseline ($0.4$ AUPRC).

As future work, we plan to investigate soft prompt tuning techniques with PLMs
and how to best combine these models with static word embedding models to try
to obtain both high discrimination and calibration. We will also investigate
how to stabilise predictions in few-shot experiments as we increase the number
of training examples, possibly training models with different patient
populations (but the same number of patients) as a form to estimate
uncertainty.

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# Helicity current generation by distorted Rashba coupling

Takumi Funato  Kavli Institute for Theoretical Sciences, University of Chinese
Academy of Sciences, Beijing, 100190, China.  Center for Spintronics Research
Network, Keio University, Yokohama 223-8522, Japan  Mamoru Matsuo  Kavli
Institute for Theoretical Sciences, University of Chinese Academy of Sciences,
Beijing, 100190, China.  CAS Center for Excellence in Topological Quantum
Computation, University of Chinese Academy of Sciences, Beijing 100190, China
RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195,
Japan

###### Abstract

We theoretically study spin transport in two-or three-dimensional Rashba
systems dynamically distorted by surface acoustic waves. The spin currents in
the linear response to lattice distortion dynamics are calculated on the basis
of a microscopic theory combined with local coordinate transformations. As a
result, we find a mechanism of direct spin-current generation from lattice
distortion not associated with a charge current or spin accumulation.
Moreover, the in-plane helicity currents are generated by shear surface
acoustic waves via the present mechanism. The generated helicity currents are
not parallel to the vorticity of the lattice, and cannot be created with the
conventional methods. Thus, our findings offer an alternative functionality of
the conventional Rashba systems in the field of spintronics.

###### pacs:

Valid PACS appear here

In the field of spintronics, spin current creation and manipulation is a
central research topic. The interconversion between spin and charge via spin-
orbit interaction (SOI) plays a key role specifically the direct and inverse
spin Hall effects (SHE)d'yakonov1971 ; hirsch1999 ; zhang2000 ; murakami2003 ;
sinova2004 ; engel2005 ; saitoh2006 ; sinova2015 , Edelstein
effectedelstein1990 ; li2014 ; fan2014 ; karube2016 ; manchon2015 , and spin
galvanic effectganichev2002  among others.

Recently, spin-current generation due to the conversion of macroscopic angular
momentum associated with mechanical motion (e.g, rigid rotation, vibration,
and vortex) into spin angular momentum has been considered. In particular,
spin current generation via spin-vorticity coupling (SVC) has received broad
attention because it does not require SOI. SVC is a coupling of electron spin
and effective magnetic fields arising in the rotating (non-inertial) frame
which is locally fixed on the moving materialsmatsuo2011 ; matsuo2013 .
Takahashi et al. measured electric voltage in the shear flow of liquid metal,
and proposed that the spin current was generated from the vorticity motion of
the liquid metal via SVCsvc_expt1 ; svc_expt2 ; svc_expt3 ; svc_theo1 .
Kobayashi et al. observed spin-torque spin-wave resonance due to an AC spin
current driven by the lattice vorticity motion associated with surface
acoustic waves (SAWs) via SVCkobayashi2017 ; tateno2020 .

The effect of SOI on mechanical spin-current generation has also been studied.
Kawada et al. applied SAWs to a heavy metal/ferromagnetic metal bilayer under
an in-plane magnetic field and investigated the angular dependence of DC
voltage on the in-plane magnetic fieldkawada2021 . They found characteristic
magnetic dependence, which was not seen when using weak SOI metal instead of
heavy metal , indicating mechanical spin-current generation via intrinsic SOI.
Moreover, a previous work theoretically studied a spin current driven by
lattice distortion dynamics via extrinsic SOI, and presented two mechanisms:
one via the SHE from the charge current and the other via the diffusion of the
spin accumulationfunato2018 . It has also been suggested that spin current
generation via extrinsic SOI is comparable with that via SVC in strong SOI
systems. Nonetheless, the effects of intrinsic SOI remain an open problem.
Direct spin-current generation by mechanical means, e.g, via modulation of
intrinsic SOI, is expected.

Figure 1:  Helicity currents $j^{x}_{\text{s},x}$, $j^{y}_{\text{s},y}$ are
generated by the SSAW in Rashba systems. In three-dimensional Rashba systems,
the longitudinal spin current $j^{x}_{\text{s},z}$ is also induced.

The purpose of this research is to explore the mechanisms of mechanical spin-
current generation involved in intrinsic SOI. As a first step, we focused on
Rashba SOIrashba1 ; rashba2  originating from spatial inversion symmetry
breaking. We considered a Rashba electron system, in which lattice distortion
dynamics are induced. Treating the effect of lattice distortion with local
coordinate transformation, we calculated the non-equilibrium spin current in
the linear response to lattice displacement. The results suggested that the
spin current was generated directly from the lattice displacement instead of
via the charge current or spin accumulation. A helicity current flows in the
in-plane direction by shear SAW (SSAW), which could not be created via
conventional methods, such as SVC and SHE. The spin current generated by the
present mechanism is large enough to be observable and expected to be
experimentally observed using spin wave resonance.

We consider two- and three-dimensional electrons subjected to impurity
potential in the presence of the lattice periodic potential and its SOI:

$\displaystyle
H=\frac{p^{2}}{2m}+V_{\text{p}}(\bm{r}^{\prime})+V_{\text{i}}(\bm{r}^{\prime})+\lambda_{\text{so}}\bm{\sigma}\cdot[\nabla^{\prime}V_{\text{p}}(\bm{r}^{\prime})\times\bm{p}],$
(1)

where $V_{\text{p}}(\bm{r}^{\prime})$ is the periodic potential,
$V_{\text{i}}(\bm{r}^{\prime})$ is the impurity potential,
$\lambda_{\text{so}}$ is the SOI strength, and
$\bm{\sigma}=(\sigma^{x},\sigma^{y},\sigma^{z})$ are the Pauli matrices. The
point like impurity potentials,
$V_{\text{i}}(\bm{r}^{\prime})=u_{\text{i}}\sum_{j}\delta(\bm{r}^{\prime}-\bm{R}_{j})$,
are assumed where $u_{\text{i}}$ is the strength of the impurity potential and
$\bm{R}_{j}$ is the position of the $j$-th impurity. When lattice distortion
dynamics were applied, e.g, because of sound waves and external forces, the
Hamiltonian is deformed as

$\displaystyle
H_{\text{T}}=\frac{p^{2}}{2m}+V_{\text{p}}(\bm{r}^{\prime}-\delta\bm{R})+V_{\text{i}}(\bm{r}^{\prime}-\delta\bm{R})$
$\displaystyle+\lambda_{\text{so}}\bm{\sigma}\cdot[\nabla^{\prime}V_{\text{p}}(\bm{r}^{\prime}-\delta\bm{R})\times\bm{p}],$
(2)

where $\delta\bm{R}(\bm{r}^{\prime},t)$ is the displacement vector of the
lattice.

To treat the lattice distortion, we perform a local coordinate transformation,
$\bm{r}=\bm{r}^{\prime}-\delta\bm{R}(\bm{r}^{\prime},t)$, from the laboratory
frame (with $\bm{r}^{\prime}$) to a ”material” frame (with $\bm{r}$) locally
fixed to the deformed lattice. We assume the free-electron systems with
$z-$direction spatial symmetry breaking. Up to the first order in
$\delta\bm{R}$, the total Hamiltonian is given by

$\displaystyle H_{\text{T}}=H+H^{\prime}(t).$ (3)

The first and second terms are a time-independent Hamiltonian and an external
perturbation Hamiltonian, respectively, due to the lattice distortion
dynamics. Time-independent term $H=H_{0}+H_{\text{i}}$ consists of an
unperturbed (kinetic energy and Rashba SOI) term
$H_{0}=\sum_{\bm{k}}c^{\dagger}_{\bm{k}}[\frac{k^{2}}{2m}+\alpha_{\text{R}}\hat{z}\cdot(\bm{k}\times\bm{\sigma})]c_{\bm{k}}$
and an impurity potential term
$H_{\text{i}}=\sum_{\bm{k},\bm{k}^{\prime}}V_{\bm{k}^{\prime}-\bm{k}}c^{\dagger}_{\bm{k}^{\prime}}c_{\bm{k}}$,
where $c_{\bm{k}}^{\dagger}(c_{\bm{k}})$ is the creation (annihilation)
operator of the electrons, $\alpha_{\text{R}}=\lambda_{\text{so}}\int
d^{3}r\partial_{z}V_{\text{p}}$ is the strength of the Rashba SOI, and
$V_{\bm{k}^{\prime}-\bm{k}}$ is the Fourier component of the impurity
potential.

The external perturbed Hamiltonian due to the lattice distortion dynamics
consists of two parts,
$H^{\prime}(t)=H^{\prime}_{\text{K}}(t)+H^{\prime}_{\text{R}}(t)$ (the
detailed derivation can be found in the Supplementary Material):

$\displaystyle
H^{\prime}_{\text{K}}(t)=-\sum_{\bm{k}}c^{\dagger}_{\bm{k}+}W^{\text{K}}_{j}(\bm{k})c_{\bm{k}-}u^{j}_{\bm{q},\omega},$
(4) $\displaystyle
H^{\prime}_{\text{R}}(t)=-\sum_{\bm{k}}c^{\dagger}_{\bm{k}_{+}}W_{j}^{\text{R}}(\bm{k})c_{\bm{k}_{-}}u^{j}_{\bm{q},\omega},$
(5)

where $\bm{k}_{\pm}=\bm{k}\pm\frac{\bm{q}}{2}$ and
$\bm{u}_{\bm{q},\omega}=-i\omega\delta\bm{R}_{\bm{q},\omega}$ are the Fourier
components of the lattice velocity field. Here, $W^{\text{K}}_{j}$ and
$W^{\text{R}}_{j}$ are given by

$\displaystyle
W^{\text{K}}_{j}(\bm{k})=\left(1-\frac{\bm{v}\cdot\bm{q}}{\omega}\right)k_{j},$
(6) $\displaystyle
W^{\text{R}}_{j}(\bm{k})=\frac{\alpha_{\text{R}}}{\omega}\left[\bm{k}\cdot(\bm{\sigma}\times\bm{q})\delta_{jz}-\hat{z}\cdot(\bm{\sigma}\times\bm{q})k_{j}\right].$
(7)

Throughout this study, we assume that the wavenumber $q$ and the frequency
$\omega$ are much smaller than inverse of mean-free path $l$ and relaxation
time $\tau$ of electrons, $q\ll l^{-1}$ and $\omega\ll\tau^{-1}$,
respectively.

Assuming uniform random distributed impurities, the impurity averaged
equilibrium retarded/advanced Green functions are given by

$\displaystyle
G^{\text{R/A}}_{\bm{k}}(\epsilon)=\frac{1}{2}\sum_{s=\pm}\frac{1+s\hat{\Gamma}_{\bm{k}}}{\mu+\epsilon-
E_{(s)}\pm i\gamma},$ (8)

where $E_{(s)}=\frac{k^{2}}{2m}+s\alpha_{\text{R}}k_{\text{t}}$ with
$k_{\text{t}}=(k_{x}^{2}+k_{y}^{2})^{\frac{1}{2}}$ is the eigenenergy, $\mu$
is the chemical potential, and $\hat{\Gamma}_{\bm{k}}=\epsilon_{\beta
l}\sigma^{\beta}\frac{k_{l}}{k_{\text{t}}}$ with $\epsilon_{\beta
l}=\epsilon_{z\beta l}$ being the Levi-Civita symbol. Here,
$\gamma=\frac{\pi}{2}n_{\text{i}}u_{\text{i}}^{2}\nu(\mu)$ is the damping rate
calculated in the Born approximation, where $n_{\text{i}}$ is the impurity
concentration and $\nu(\mu)=\nu_{0}a_{1}$ is the Fermi-level density of
states, with $\nu_{0}=\frac{mk_{\text{F}}}{2\pi^{2}}$(for 3D) and
$\frac{m}{2\pi}$ (for 2D) being the density of states in free-electron systems
without Rashba SOI, $k_{\text{F}}=\sqrt{2m\mu}$ being Fermi wavenumber, and
$a_{1}$ being a dimensionless parameter defined after Eq. (14). Here, we use
the ensemble average for impurity positions $\langle
V_{\bm{k}}V_{\bm{k}^{\prime}}\rangle_{\text{av}}=n_{\text{i}}u_{\text{i}}^{2}\delta_{\bm{k},\bm{k}^{\prime}}$.

The Spin current operator is given by

$\displaystyle\hat{j}^{\alpha}_{\text{s},i}(\bm{q})=\sum_{\bm{k}}c^{\dagger}_{\bm{k}_{-}}j^{\alpha}_{\text{s},i}c_{\bm{k}_{+}},$
(9)

where $\alpha$($=x,y,z$) specifies the spin direction, $i$($=x,y,z$) specifies
the flow direction, and
$j^{\alpha}_{\text{s},i}=\sigma^{\alpha}v_{i}+\alpha_{\text{R}}\epsilon_{z\alpha
i}$ represents the spin-current vertex.

The expectation value of the spin-current density in non-equilibrium states is
given by

$\displaystyle\langle\hat{j}^{\alpha}_{\text{s},i}(\bm{q})\rangle_{\text{ne},\omega}=\int^{\infty}_{-\infty}\frac{d\epsilon}{2\pi
i}\sum_{\bm{k}}\text{tr}\left[j^{\alpha}_{\text{s},i}G^{<}_{\bm{k}_{+},\bm{k}_{-}}(\epsilon_{+},\epsilon_{-})\right],$
(10)

where $\epsilon_{\pm}=\epsilon\pm\frac{\omega}{2}$. The trace is taken for the
spin space. Moreover,
$G^{<}_{\bm{k}_{+},\bm{k}_{-}}(\epsilon_{+},\epsilon_{-})$ is the lesser
component of the non-equilibrium path-ordered Green function defined by

$\displaystyle G_{\bm{k},\bm{k}^{\prime}}(t,t^{\prime})=-i\langle
T_{C}c_{\bm{k}}(t)c^{\dagger}_{\bm{k}^{\prime}}(t^{\prime})\rangle_{H_{\text{T}}},$
(11)

where $T_{C}$ is a path-ordering operator and
$\langle\cdots\rangle_{H_{\text{T}}}$ represents the expectation value in the
non-equilibrium state. We expand the lesser Green function for external
perturbation $H_{\text{R}}^{\prime}$, and calculate the linear response to
lattice velocity field $\bm{u}_{\bm{q},\omega}$ up to the first order in
$\bm{q}$.

Considering only the most dominant term (the so-called ”Fermi surface term”),
the non-equilibrium spin current is given by

$\displaystyle\langle\hat{j}^{\alpha}_{\text{s},i}\rangle^{\text{K/R}}_{\text{ne}}=\frac{i\omega}{2\pi}\sum_{\bm{k}}\text{tr}\left[j^{\alpha}_{\text{s},i}G^{\text{R}}_{\bm{k}}(0)W^{\text{K/R}}_{j}G^{\text{A}}_{\bm{k}}(0)\right]u^{j}_{\bm{q},\omega},$
(12)

where $\langle\cdots\rangle^{\text{K/R}}$ represents the linear response to
$H^{\prime}_{\text{K}}(H^{\prime}_{\text{R}})$. The results are given byvcs

$\displaystyle\langle\hat{j}^{\alpha}_{\text{s},i}\rangle_{\text{ne}}^{\text{K}}=j_{\text{s}0}\frac{\tilde{\alpha}_{\text{R}}}{4}\bigl{(}4\tilde{c}_{2}\eta^{z}_{\alpha
ijl}+\tilde{a}_{4}\eta^{\text{t}}_{\alpha
ijl}\bigr{)}iq_{l}u^{j}_{\bm{q},\omega},$ (13)
$\displaystyle\langle\hat{j}^{\alpha}_{\text{s},i}\rangle_{\text{ne}}^{\text{R}}=-j_{\text{s}0}\frac{\tilde{\alpha}_{\text{R}}}{4}a_{3}\bigl{(}2\epsilon_{\alpha
l}\delta^{\text{t}}_{ij}+3\epsilon_{\alpha
i}\delta^{z}_{jl}+\Delta^{xy}_{\alpha
l}\lambda^{xy}_{ij}\bigr{)}iq_{l}u^{j}_{\bm{q},\omega},$ (14)

where $j_{\text{s}0}=v_{\text{F}}\mu\nu_{0}\tau$ is a material constant and
$\tilde{\alpha}_{\text{R}}=\alpha_{\text{R}}/v_{\text{F}}$ is the
dimensionless Rashba parameter. In the aforementioned expression, we defined
$\delta^{\text{t}}_{ij}=\delta_{ij}^{x}+\delta_{ij}^{y}$,
$\delta^{i}_{jl}=\delta_{ij}+\delta_{il}$, $\eta^{z/\text{t}}_{\alpha
ijl}=\epsilon_{\alpha l}\delta^{z/\text{t}}_{ij}+\epsilon_{\alpha
j}\delta^{z/\text{t}}_{il}+\epsilon_{\alpha i}\delta^{z/\text{t}}_{jl}$,
$\Delta^{xy}_{\alpha l}=\delta_{\alpha x}\delta_{ly}-\delta_{\alpha
y}\delta_{lx}$, and
$\lambda^{xy}_{ij}=\delta_{ix}\delta_{jy}+\delta_{iy}\delta_{jx}$.
$a_{n}=\nu_{0}^{-1}\sum_{s,\bm{k}}(s\tilde{k}_{\text{t}})^{n-1}\delta(\mu-
E_{(s)})$ and
$c_{n}=\nu_{0}^{-1}\sum_{s,\bm{k}}\tilde{k}_{z}^{2}(s\tilde{k}_{\text{t}})^{n-1}\delta(\mu-
E_{(s)})$ were dimensionless parameters (see the Supplementary Materials),
with $\tilde{k}=k/k_{\text{F}}$ as the dimensionless wavenumber. For the
calculation, we also defined $\tilde{a}_{n}=-\tilde{\alpha}_{\text{R}}a_{n}$
and $\tilde{c}_{n}=-\tilde{\alpha}_{\text{R}}c_{n}$ for even $n$. We confirmed
that the charge current and spin accumulation did not arise within this
calculation; i.e, the spin current was generated directly by the lattice
distortion dynamics. Note that the AC spin current was generated from the
present mechanism.

Figure 2:  Rashba parameter $\tilde{\alpha}_{\text{R}}$ dependence of the spin current generated by the SSAW, which is normalized by $j_{\text{s}0}qu_{0}$ or $j_{\text{s}0}\kappa u_{0}$. The red (blue) line represents the helicity current polarized in the $x$ direction ($y$ direction), and the green line represents the perpendicular spin current polarized in the $x$ direction. The solid (dashed) line represents the spin current generated in 3D (2D) systems.  Table 1:  Symmetry of the spin current generated by the applied SSAW via each mechanism. Clearry, the spin current generated via the present mechanism is not generated by conventional mechanisms. The last line describes SHE case due to the Rashba SOI for reference.  Ref. | Input | Interaction | Helicity current | Spin current
---|---|---|---|---
$j^{x}_{x}$, $j^{y}_{y}$ | $j^{x}_{z}$ | $j^{y}_{z}$ | $j^{z}_{x}$
Present | SSAW | Rashba SOI | ✓ | ✓ | – | –
funato2018  | Extrinsic SOI | – | ✓ | – | ✓
matsuo2013  | Rayleigh SAW | SVC | – | – | ✓ | –
sinova2004  | Charge current | Rashba SOI | – | – | – | ✓

We consider a spin current induced via the present mechanism when an SSAW is
applied. The velocity field is given by
$\bm{u}(\bm{r},t)=\hat{y}u_{0}e^{-\kappa z}e^{i(qx-\omega t)}$, where
$\hat{y}$ is the unit vector in the $y$ direction, $\kappa$ is the damping
constant of the SAW in the $z$ direction, and $u_{0}$ is the amplitude. Then,
the spin current generated by the SSAW is given by

$\displaystyle\langle\hat{j}^{\alpha}_{\text{s},i}\rangle^{\text{sw}}_{\text{ne}}=j_{\text{s}0}\frac{\tilde{\alpha}_{\text{R}}}{8}\left[-(2\tilde{a}_{4}+3a_{3})\Delta^{xy}_{\alpha
i}+a_{3}\lambda_{\alpha i}^{xy}\right]\nabla_{x}u^{y}$
$\displaystyle+j_{\text{s}0}\tilde{\alpha}_{\text{R}}\tilde{c}_{2}\delta_{\alpha
x}\delta_{iz}\nabla_{z}u^{y}.$ (15)

The first line is an in-plane helicity current whose spin and flow directions
are parallel, and the second line is a perpendicular spin current whose spin
and flow directions are orthogonal, as illustrated in Fig. 1. Inparticular,
the in-plane helicity current cannot be produced by conventional methods. The
SVC mechanism produces a spin current whose spin direction is parallel to the
vorticity of the lattice, and the SHE produces a spin current whose spin and
flow directions are orthogonal, as listed in Table 1. The reason for such an
unconventional spin current is that the present mechanisms produces the spin
current directly from the lattice distortion dynamics via the Rashba SOI
without a charge current or spin accumulation. However, the spin current
flowing to the $z$ direction is along the SAW decay, which is generated only
in the case of the 3D systems. This current can also be induced via the
extrinsic SHE, but cannot be understood by the Rashba SHE because the Rashba
SHE produces only a $z$ polarized spin currentsinova2004 .

Let us estimate the generated spin current in specific materials: For a
Cu/Bi2O3 bilayerkarube2016 , the Rashba parameter is
$\tilde{\alpha}_{\text{R}}=0.1$. With material parameters of bulk Cu (Fermi
energy $\mu\sim 7.03$[eV], Fermi velocity $v_{\text{F}}\sim 1.57\times
10^{6}$[m/s], density of states $N_{0}(\mu)=1.81\times 10^{28}$[m-3eV-1], and
relaxation time $\tau\sim 10^{-15}$[s]), we estimate $j_{\text{s}0}qu_{0}\sim
3.0\times 10^{27}$[m-2s-1] for lattice displacement $\delta R\sim 1$Å,
frequency $f\sim 3.8$[GHz], and phase velocity $v_{\text{a}}\sim 3.8$[km/s].
The spin currents have magnitudes that is large enough to be observable.
However, they are expected to be smaller than $j_{\text{s}0}qu_{0}$ obtained
here because of the spin current flowing throughout the bulk. Regarding BiTeI
as a 3D Rashba system with $\tilde{\alpha}_{\text{R}}\sim 1$demko2012 ;
shibata2018 , the normalized parameter for the helicity current is estimated
as $j_{\text{s}0}qu_{0}\sim 6\times 10^{16}$-$10^{19}$[m-2s-1] with
$\mu=0.2$[eV], $v_{\text{F}}\sim 10^{5}$[ms-1], $N_{0}(\mu)\sim 3\times
10^{26}$[m-3eV-1], and $\tau\sim 10^{-15}$-$10^{-18}$[s]. The normalized
parameter for the orthogonal spin current can be estimated as
$j_{\text{s}0}\kappa u_{0}\sim j_{\text{s}0}qu_{0}$ because damping constant
$\kappa$ is approximately equal to the wavenumber $\kappa\sim q$. As a
detection of AC spin currents, spin wave resonances have been observed by
injecting spin currents into ferromagnetic metalskobayashi2017 ; tateno2020 .
Moreover, AC spin currents have been detected by the rectification effect due
to magnetostrictionkawada2021 . The spin current produced by the present
mechanism may be observed by using these methods.

In summary, we studied the spin current generation due to dynamical lattice
distortion in 2D and 3D Rashba electron systems using nonequilibrium quantum
field theory. We found a process in which spin currents are generated directly
by the lattice distortion dynamics. Because of the interplay between the
Rashba SOI and the dynamical lattice distortion, unlike conventional spin
transport, the in-plane helicity currents are generated without an
accompanying charge current or spin accumulation. We also found that the in-
plane helicity currents do not flow parallel to the vorticity of the lattice
distortions. The obtained spin currents were experimentally detectable in
strong Rashba systems and are expected to be able to be observed by spin-wave
resonance. Overall, our theory revealed alternative spintronic functionalities
of Rashba systems.

We would like to thank M. Hayashi, T. Kawada, Y. Nozaki, K. Yamanoi, and T.
Horaguchi for enlightening discussions. We also thank H. Kohno, A. Yamakage,
Y. Imai, J. J. Nakane. Y. Yamazaki, Y. Ogawa, Y. Ozu, and Y. Hayakawa for
daily discussions. This work was partially supported by JST CREST Grant Number
JPMJCR19J4, Japan. TF is supported by Grant-in Aid for JSPS Fellows Grant
Number 19J15369, and by a Program for Leading Graduate Schools ”Integrative
Graduate Education and Research in Green Natural Sciences”. MM is supported by
Grant-in-Aid for Scientific Research B (20H01863).

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