Identifying Incorrect Classifications with Balanced Uncertainty

Bolian Li Zige Zheng Changqing Zhang Tianjin University The Chinese University
of Hong Kong, Shenzhen Tianjin University

###### Abstract

Uncertainty estimation is critical for cost-sensitive deep-learning
applications (i.e. disease diagnosis). It is very challenging partly due to
the inaccessibility of uncertainty groundtruth in most datasets. Previous
works proposed to estimate the uncertainty from softmax calibration, Monte
Carlo sampling, subjective logic and so on. However, these existing methods
tend to be over-confident about their predictions with unreasonably low
overall uncertainty, which originates from the imbalance between positive
(correct classifications) and negative (incorrect classifications) samples.
For this issue, we firstly propose the distributional imbalance to model the
imbalance in uncertainty estimation as two kinds of distribution biases, and
secondly propose Balanced True Class Probability ($BTCP$) framework, which
learns an uncertainty estimator with a novel Distributional Focal Loss ($DFL$)
objective. Finally, we evaluate the $BTCP$ in terms of failure prediction and
out-of-distribution (OOD) detection on multiple datasets. The experimental
results show that $BTCP$ outperforms other uncertainty estimation methods
especially in identifying incorrect classifications.

##  1 Introduction

Deep learning has fundamentally changed the way we co-operate with computing
devices. Deep-learning-based applications have emerged into a variety of
fields, including computer vision (Forsyth and Ponce, 2012), natural language
processing (Deng and Liu, 2018) and data mining (Han et al., 2011). However,
in some risk-sensitive applications, deploying traditional deep-learning
models may bring disastrous outcomes since their predictions are not
trustworthy (Floridi, 2019), such as disease diagnosis (Ao et al., 2020),
automatic driving (Yasunobu and Sasaki, 2003) and robotics (Davies, 2000).
Deep-learning models need to be interpretable (Molnar, 2020) and trustworthy
(Abdar et al., 2020). To this end, uncertainty estimation proposes to enable
deep-learning models to be self-aware about their predictions. They provide
estimated uncertainty for users to choose to trust or not to trust the
predictions. The recent uncertainty estimation methods propose to obtain the
uncertainty scores from softmax calibration (Hendrycks and Gimpel, 2016),
Monte Carlo sampling (Gal and Ghahramani, 2016) and subjective logic (Şensoy
et al., 2018).

However, there is one limitation for current uncertainty estimation methods.
These models tend to be overly confident about all predictions (provide
unreasonably low overall uncertainty), even for the incorrect predictions
(Mukhoti et al., 2020). For this issue, we conduct analytical experiments on
the origin of this problem, and find that the imbalance problem (He and
Garcia, 2009) is the primary cause. We also noticed that the traditional
imbalance problem as mentioned in (He and Garcia, 2009) is based on
classification task and is modeled with discrete mathematical forms, while in
uncertainty estimation, the observed imbalance problem is between correct
classifications (with continual low uncertainty groundtruth) and incorrect
classifications (with continual high uncertainty groundtruth), which is based
on regression task. For this issue, we propose the distributional imbalance to
accurately model the imbalance problem in uncertainty estimation, which
presents two kinds of distribution biases in the uncertainty distribution.
Then, we propose the Balanced True Class Probability ($BTCP$) framework to
learn an uncertainty estimator on the Distributional Focal Loss ($DFL$)
objective adapted from Focal Loss (FL, Lin et al. (2017)), which calibrates
the uncertainty distribution by end-to-end training. Finally, we evaluate the
$BTCP$ in terms of failure prediction and out-of-distribution (OOD) detection
on multiple datasets. The experimental results show that $BTCP$ achieves the
state-of-the-art performance, and outperforms other uncertainty estimation
methods in identifying incorrect classifications.

The main contributions of this paper are summarized as follows:

  * •

We propose the distribution imbalance to model the imbalance problem in
uncertainty estimation, presenting two kinds of distribution biases in the
uncertainty distribution.

  * •

We propose the Balanced True Class Probability ($BTCP$) framework to learn an
uncertainty estimator in end-to-end training on a novel Distributional Focal
Loss ($DFL$) objective.

  * •

We evaluate our model in terms of failure prediction and out-of-distribution
(OOD) detection on various datasets. Our model achieves the state-of-the-art
performance.

The rest of the paper is organized as: related works (Sec 2), preliminary (Sec
3), the introduction of distributional imbalance (Sec 4), the $BTCP$ framework
(Sec 5), experiments (Sec 6), and finally the conclusion and future work (Sec
7).

Figure 1: Overview of $BTCP$. The classifier and uncertainty estimator share
the encoder. In training uncertainty estimator, the output uncertainty is
supervised by the $DFL$ objective with groundtruth from the classifier (TCP
scores).

##  2 Related Works

In this section, we review the related works, including uncertainty estimation
(Sec 2.1) and imbalanced learning (Sec 2.2).

###  2.1 Uncertainty Estimation

Uncertainty estimation is a critical topic with long history. Blatz et al.
(2004) summarized traditional methods which are not based on deep learning.
Abdar et al. (2020) summarized the techniques, applications and challenges of
uncertainty estimation. Recently, MCP (Hendrycks and Gimpel, 2016) has become
a widely used baseline by viewing the maximal value of softmax distribution as
confidence. Probabilistic methods with Bayesian modeling obtain the
uncertainty by post-process, such as Bayesian neural network (Blundell et al.,
2015), Monte-Carlo dropout (Gal and Ghahramani, 2016), prior networks (Malinin
and Gales, 2018) and variational inference (Posch et al., 2019). Subjective
logic (Şensoy et al., 2018) employs evidence theory to obtain the uncertainty
from Dirichlet distribution. Another approach of uncertainty estimation is
based on deep regression, such as TCP (Corbière et al., 2019). Moon et al.
(2020) developed a new form of loss function to regularize class probabilities
for uncertainty estimation.

###  2.2 Imbalanced Learning

The imbalance problem is summarized in Han et al. (2005). threshold-moving
proposed to adjust the decision threshold adaptively. Liu et al. (2008)
introduced an under-sampling method with ensemble model. Han et al. (2005)
developed an over-sampling way via data interpolation. Both of them
concentrated on rebuilding balanced training data. Lin et al. (2017) raised
the significance of the scarce classes by applying low weights to the classes
with massive samples. Shrivastava et al. (2016) raised the significance of the
scarce classes by applying high weights to the classes with few samples.

##  3 Preliminary

In this section, we introduce the concepts and definitions of uncertainty in
section 3.1 and illustrate the task of failure prediction in section 3.2.

###  3.1 Uncertainty in Deep Learning

Uncertainty in deep learning is the extent to which predictions cannot be
trusted. The source of uncertainty is 2-fold: data uncertainty and knowledge
uncertainty (Abdar et al., 2020). There are multiple definitions of
uncertainty (Malinin and Gales, 2018; Corbière et al., 2019; Koh and See,
1994). In this paper, we define uncertainty as real number that is
complementary with the confidence score, which holds:

$u_{A}+c_{A}=1,$ (1)

where $u_{A}$ and $c_{A}$ are the uncertainty and confidence of a prediction
$A$ respectively. Moreover, we obtain the uncertainty groundtruth by TCP
(Corbière et al., 2019), which uses the class probability of the labeled class
as confidence score:

$u_{A}=TCP(\hat{y}_{A},y_{A})=1-{\hat{y}_{A}}^{T}y_{A},$ (2)

where $\hat{y_{A}}$ and $y_{A}$ are the predicted and labeled class
probability vectors respectively.

###  3.2 Failure Prediction

Failure prediction aims at evaluating the ability for the neural network to
recognize when its predictions are wrong (Corbière et al., 2019). The
classification results of a neural network are the groundtruth of failure
prediction. The uncertainty is used to predict whether the classification
result is correct. When uncertainty is lower than a given threshold: $u<\tau$,
it is predicted as correct classification (positive), and when $u\geq\tau$, it
is predicted as incorrect classification (negative). The evaluation of failure
prediction is based on 4 pairs of conditions (i.e. true positive, false
positive, false negative and true negative), which are shown in the confusion
matrix in table 1.

Table 1: Confusion matrix of failure prediction. uncertainty | correct classification | incorrect classification
---|---|---
$u<\tau$ | true positive (TP) | false positive (FP)
$u\geq\tau$ | false negative (FN) | true negative (TN)

##  4 Distributional Imbalance

In this section, we introduce the concept of distributional imbalance in
section 4.1 and provide visual illustration for it in section 4.2.

###  4.1 Concept of Distributional Imbalance

The traditional classificatory imbalance (He and Garcia, 2009) is based on
classification task, in which the different numbers of samples in different
classes are the primary cause. We propose the distributional imbalance to
accurately model the imbalance problem in uncertainty estimation, considering
the continuity of uncertainty scores. Specifically, the imbalanced uncertainty
distribution of a neural network would have such 2 kinds of distribution
biases:

  * •

The means of uncertainty distribution would be low. This is because the
uncertainty groundtruth is dominated by massive correct classifications and
would be over-fitted on the low uncertainty groundtruth.

  * •

The standard deviation of uncertainty distribution would be low. This is due
to the unreasonably low overall uncertainty scores, and would lead to the
difficulty in distinguishing between easy and hard samples by uncertainty.

The above distribution biases are visualized in figure 2, where the red
distribution suffers from these 2 kinds of biases. Besides, our proposed
objective ($DFL$) aims at adjusting the uncertainty distribution from the red
one to the blue one. The demonstration of distributional imbalance on real-
world datasets is shown in section 4.2.

Figure 2: Distributional imbalance.

The harm of distributional imbalance is that, estimated uncertainty tends to
be low even for incorrect classifications. It would cause the neural network
to give wrong predictions confidently, which is unacceptable in some risk-
sensitive applications like disease diagnosis (Ao et al., 2020).

###  4.2 Visualization of Distributional Imbalance

First, we visualize the TCP uncertainty scores (Eq.2) on Fashion-MNIST with
the uncertainty histogram (50 bins, shown in Fig.3). The uncertainty scores
are distributed just as what we talked in section 4.1. The real-world
uncertainty distribution suffers from the distributional imbalance with low
means and standard deviation.

Figure 3: Uncertainty histogram on Fashion-MNIST.

Second, we visualize the false positive rate (FPR) and false negative rate
(FNR) of 4 methods in figure 4. The FPR is used to measure how likely
incorrect classifications are viewed as correct ones, which is computed by

$\displaystyle FPR=1-R_{incorrect}=\frac{FP}{FP+TN},$ (3)

and FNR is used to measure how likely correct classifications are viewed as
incorrect ones, which is computed by

$\displaystyle FNR=1-R_{correct}=\frac{FN}{TP+FN}.$ (4)

Here, $R_{correct}$ and $R_{incorrect}$ are the recall ratios of correct
classification and incorrect classification respectively. We train each model
for totally 20 epochs and make a checkpoint at each epoch. The evaluated FPRs
and FNRs are based on uncertainty obtained from each checkpoint. Generally,
the TCP suffers from high FPR, which is the consequence of distributional
imbalance.

(a) $BTCP$

(b) TCP

(c) OHEM Loss

(d) thresholding

Figure 4: Visualization of FPR and FNR on Fashion-MNIST.

Moreover, the other 3 methods ($BTCP$, thresholding and OHEM Loss (Shrivastava
et al., 2016)) all consider the imbalance problem in uncertainty estimation.
Among them, our proposed $BTCP$ outperforms OHEM Loss woth lower FPR, and has
better overall performance compared to thresholding with averagely lower FNR
and FPR.

##  5 Method

In this section, we propose a novel Distributional Focal Loss (DFL) in section
5.1 and introduce the training scheme of uncertainty estimator in section 5.2.
Our discussion about objective and training process are based on the framework
in figure 1.

###  5.1 Distributional Focal Loss

Previous work. To address the imbalance problem in classification tasks, Lin
et al. (2017) proposed Focal Loss (FL) to adaptively reweight the CrossEntropy
objective based on different class volumes. It is defined as

$FL(\hat{p}_{d})=-(1-\hat{p}_{d})^{\gamma}log(\hat{p}_{d}),$ (5)

where $\hat{p}_{d}$ is the predicted probability of the $d$-th class, and
$\gamma$ is a fixed hyperparameter controlling the sensitivity of reweighting.
This objective eliminates the imbalance originating from different class
volumes.

Motivation. We are inspired by FL and are motivated to adapt the FL to
uncertainty estimation. Given that uncertainty estimation is inherently a kind
of regression tasks, and that it suffers from the distributional imbalance
(see Sec 4), we propose a novel Distributional Focal Loss ($DFL$) objective
and argue that:

  * •

First, we should employ the classic Mean Squared Error (MSE, Wang and Bovik
(2009)) to compute the unweighted losses of all uncertainty scores, since
uncertainty estimation is a kind of regression tasks.

  * •

Second, we should adjust the unweighted losses by a density factor similar to
Eq.5. We use the density factor to measure the local frequency of given
uncertainty scores in the uncertainty distribution.

Formulation. Following the above motivations, we define the $DFL$ objective
as:

$DFL(\hat{u},u)=\left[1-\frac{|N(u)|}{|U|}\right]^{\gamma}||\hat{u}-u||^{2},$
(6)

where $\hat{u}$ is the estimated uncertainty score and $u$ is the groundtruth
(TCP score). $N(u)$ is a small continual subset centering at $u$:
$u-\epsilon\sim u+\epsilon$, consisting of samples with similar uncertainty.
The relative volume of this subset (or density factor): $|N(u)|/|U|$, is a
rational approximation of local frequency in the uncertainty distribution. The
$DFL$ objective can lead the neural network to focus less on the massive low
uncertainty groundtruth, which originates from too many correct
classifications.

###  5.2 Learning Uncertainty Estimator

We train the $BTCP$ framework to learn a balanced uncertainty estimator
following the training scheme of Corbière et al. (2019). First, we train the
classifier on CrossEntropy objective with class labels. Second, we train the
uncertainty estimator on $DFL$ objective with uncertainty groundtruth (TCP
scores). Finally, we fine-tune the entire model by carrying out the above 2
steps simultaneously. The training process of $BTCP$ is summarized in
algorithm 1.

#_classifier_

for  _epoch = 1,2,……_ do

Predict the output softmax by classifier;

Compute the CrossEntropy loss;

Update the parameters of classifier and encoder;

end for

#_uncertainty estimator_

for  _epoch = 1,2,……_ do

Predict the output softmax by classifier;

Predict the uncertainty by uncertainty estimator;

Comupte TCP scores as uncertainty groundtruth;

Compute the $DFL$ loss;

Update the parameters of uncertainty estimator;

end for

#_fine-tuning_

for  _epoch = 1,2,……_ do

Predict the output softmax by classifier;

Predict the uncertainty by uncertainty estimator;

Comupte TCP scores as uncertainty groundtruth;

Compute CrossEntropy and $DFL$ loss, then combine them as a joint loss;

Update the parameters of all modules.

end for

Algorithm 1 Training process of $BTCP$.

##  6 Experiments

In this section, we conduct experiments to evaluate the $BTCP$ framework in
failure prediction and out-of-distribution (OOD) detection on multiple
datasets. We list the specifications of experiments in section 6.1, and
provide detailed quantitative and visual results in section 6.2 and section
6.3 respectively. Our code is available at
https://github.com/lblaoke/balanced-TCP

###  6.1 Experimental Setup

Datasets. For comprehensive and fair comparison with other uncertainty
estiamtion methods, we use 5 real-world image datasets:

  * •

CIFAR-10 and CIFAR-100111http://www.cs.toronto.edu/~kriz/cifar.html consist of
60000 32x32 colour images. There are 50000 training images and 10000 test
images (Krizhevsky, 2009).

  * •

SVHN222http://ufldl.stanford.edu/housenumbers/ is a real-world image dataset
for developing machine learning and object recognition algorithms, obtained
from house numbers in Google Street View images (Netzer et al., 2011).

  * •

MNIST333http://yann.lecun.com/exdb/mnist/ is a database of handwritten digits,
and has a training set of 60000 examples, and a test set of 10000 examples
(LeCun, 1998).

  * •

Fashion-MNIST444https://github.com/zalandoresearch/fashion-mnist is a dataset
consisting of a training set of 60000 examples and a test set of 10000
examples. Each example is a 28x28 grayscale image, associated with a label
from 10 classes (Xiao et al., 2017).

Metrics. For failure prediction, we evaluate models on balanced accuracy
(BACC, Brodersen et al. (2010)), the area under curve (AUC, McClish (1989))
and false positive rate (FPR, defined in Eq.3). Specifically, BACC is computed
by the average of recall obtained on each class:

$\displaystyle BACC$ $\displaystyle=(R_{correct}+R_{incorrect})/2$ (7)
$\displaystyle=\frac{1}{2}\left[\frac{TP}{TP+FN}+\frac{TN}{FP+TN}\right],$

where $R_{correct}$ and $R_{incorrect}$ are the recall ratios of correct
classification and incorrect classification respectively.

For OOD detection, we evaluate models on accuracy (ACC) and average
uncertainty (AU). The larger AU a model has, the better it is since larger AU
shows that this model is less likely to give low uncertainty to OOD data.

Compared methods. We compare the $BTCP$ model with 2 sets of baseline methods:
First, uncertainty estimation methods including MCP (Hendrycks and Gimpel,
2016), MCDropout (Gal and Ghahramani, 2016)), TrustScore (Jiang et al., 2018)
and TCP (Corbière et al., 2019); second, imbalanced learning methods such as
OHEM Loss (Shrivastava et al., 2016).

Implementation. The basic framework of $BTCP$ is based on Corbière et al.
(2019), which consists of a classifier, an uncertainty estimator and a shared
encder (shown in Fig.1). To be specific, for all datasets, we build the
encoder module on a series of convolutional layers. For classifier and
uncertainty estimator, we build them both on fully connected layers. The
details of network architecture are listed in the supplementary material. In
evaluations, we evaluate each method on each metric for 5 times and report the
average performances for fair comparison. We set the decision threshold $\tau$
as $0.5$ by default. Moreover, we recommend to set the hyperparameter $\gamma$
following

$\gamma=\frac{1}{12\cdot var(U)},$ (8)

where $var(U)$ is the variance of uncertainty scores. It is rational since we
need to make the objective in equation 6 sensitive when the variance of
uncertainty distribution is low, which indicates strong distributional
imbalance. The detailed deduction of equation 8 is listed in the supplementary
material.

###  6.2 Quantitative Evaluations

Failure prediction. The quantitative results on failure prediction are shown
in table 2. Our method achieves the state-of-the-art performance on each
dataset for almost all metrics. Specifically, for datasets with complex visual
details (i.e. SVHN), our method outperforms the others significantly, while
for datasets with simple pixel-level details (i.e. Fashion-MNIST), our method
outperforms the others slightly. This is due to the lack of complex visual
details, which causes the classifier to give less incorrect classifications
and therefore has negative effects on methods that do not consider imbalance
problem in uncertainty estimation.

Table 2: Overall performance on failure prediction. Dataset | Metric | MCP | MCDropout | TrustScore | OHEM Loss | TCP | BTCP
---|---|---|---|---|---|---|---
CIFAR-10 | BACC | 64.32 | 78.62 | 78.21 | 78.00 | 71.93 | 78.80
AUC | 84.23 | 89.08 | 88.47 | 85.71 | 85.60 | 89.09
FPR↓ | 10.62 | 9.02 | 5.70 | 2.13 | 7.72 | 2.01
CIFAR-100 | BACC | 72.86 | 75.78 | 74.70 | 77.14 | 77.51 | 77.80
AUC | 85.67 | 86.09 | 84.17 | 87.17 | 86.28 | 88.07
FPR↓ | 7.20 | 4.68 | 4.74 | 3.53 | 3.19 | 3.11
SVHN | BACC | 61.26 | 84.55 | 85.35 | 80.27 | 83.86 | 87.97
AUC | 93.20 | 92.85 | 92.16 | 90.60 | 93.44 | 93.66
FPR↓ | 8.66 | 5.60 | 3.74 | 1.16 | 2.41 | 0.67
MNIST | BACC | 85.54 | 89.33 | 90.52 | 87.16 | 90.75 | 91.52
AUC | 97.13 | 97.15 | 97.52 | 97.29 | 97.83 | 97.33
FPR↓ | 3.80 | 2.26 | 3.00 | 0.52 | 0.34 | 0.26
Fashion-MNIST | BACC | 60.63 | 76.65 | 78.16 | 80.78 | 65.16 | 81.09
AUC | 82.93 | 87.04 | 87.53 | 87.53 | 87.99 | 88.36
FPR↓ | 7.87 | 4.87 | 4.64 | 2.78 | 3.93 | 0.20

OOD detection. The quantitative results on OOD detection are shown in table 3.
We use 4 pairs of in-distribution and OOD datasets. Our method ($BTCP$)
achieves the state-of-the-art performance on each dataset for almost all
metrics. Such results show that the uncertainty of $BTCP$ still performs well
on traditional OOD detection tasks while having a balanced distribution.

Table 3: Overall performance on out-of-distribution (OOD) detection. Dataset/OOD | CIFAR-10/SVHN | SVHN/CIFAR-10 | MNIST/Fashion-MNIST | Fashion-MNIST/MNIST
---|---|---|---|---
Metric | ACC(%) | AU↑ | ACC(%) | AU↑ | ACC(%) | AU↑ | ACC(%) | AU↑
MCP | 43.97 | 0.3652 | 43.56 | 0.3690 | 53.46 | 0.2745 | 38.42 | 0.2146
MCDropout | 85.86 | 0.5056 | 74.28 | 0.4927 | 92.48 | 0.5147 | 46.09 | 0.3488
TrustScore | 87.82 | 0.5919 | 79.38 | 0.5166 | 85.64 | 0.4168 | 42.78 | 0.3261
OHEM Loss | 99.80 | 0.7056 | 85.93 | 0.6074 | 93.66 | 0.5826 | 51.97 | 0.4437
TCP | 99.64 | 0.7190 | 75.18 | 0.5476 | 88.88 | 0.5434 | 47.09 | 0.4146
BTCP | 99.83 | 0.7239 | 86.04 | 0.5823 | 99.56 | 0.8216 | 56.24 | 0.4789

###  6.3 Visual Evalutions

Uncertainty distribution. To show the effectiveness of $BTCP$ in addressing
distributional imbalance, we visualize the means and standard deviation of
uncertainty distribution from 4 methods. We train 4 models on Fashion-MNIST
for totally 20 epochs and make a checkpoint at each epoch. Then we obtain
uncertainty scores based on these checkpoints. The uncertainty distribution of
$BTCP$ has higher means and standard deviation compared with the other 3
methods, which demonstrates that $BTCP$ is the best method in addressing
distributional imbalance.

(a) Means

(b) Standard deviation

Figure 5: Visualization of the means and standard deviation of uncertainty
distribution on Fashion-MNIST.

OOD confidence distribution. We also visualize the confidence distribution of
OOD data (red) and in-distribution data (blue) from our method ($BTCP$) and
TCP (Corbière et al., 2019). The in-distribution and OOD datasets are MNIST
and fashion-MNIST respectively. Our method outperforms TCP with lower overall
confidence (higer overall uncertainty) in the OOD data and distinguishes more
clearly between the in-distribution and OOD confidence distributions.
Moreover, the in-distribution confidence of our method is also better since it
is distributed more scattered and therefore more distinguishable for each
sample.

(a) $BTCP$

(b) TCP

Figure 6: Visualization of OOD confidence distribution on the MNIST/Fashion-
MNIST pair.

##  7 Conclusion and Future Work

In this paper, we propose the Balanced True Class Probability ($BTCP$)
framework. We argue that the distributional imbalance between correct and
incorrect classifications consists of two kinds of distribution biases, and is
an inevitable problem for uncertainty estimation methods. We model uncertainty
estimation as a regression task, propose to learn an uncertainty estimator to
obtain the uncertainty scores, and propose to balance the objective for
correct and incorrect classifications by a novel Distributional Focal Loss
($DFL$). The experimental results verify the effectiveness of $BTCP$. Our
future work will concentrate on introducing probabilistic modeling into the
uncertainty estimator and attempting to estimate aleatoric and epistemic
uncertainties in a separate way.

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# Current-induced torques and interfacial spin-orbit coupling

Paul M. Haney$,^{1}$ Hyun-Woo Lee,2 Kyung-Jin Lee,1,3,4,5 Aurélien Manchon,6
and M. D. Stiles1 1Center for Nanoscale Science and Technology, National
Institute of Standards and Technology, Gaithersburg, Maryland 20899-6202, USA
2PCTP and Department of Physics, Pohang University of Science and Technology,
Kyungbuk 790-784, Korea
3Korea University, Department of Material Science & Engineering, Seoul 136701,
South Korea
4Korea Institute of Science and Technology, Seoul 136-791, Korea
5Univeristy of Maryland, Maryland Nanocenter, College Pk, MD 20742 USA
6Core Labs, King Abdullah University of Science and Technology (KAUST), Thuwal
23955-6900, Saudi Arabia

###### Abstract

In bilayer systems consisting of an ultrathin ferromagnetic layer adjacent to
a metal with strong spin-orbit coupling, an applied in-plane current induces
torques on the magnetization. The torques that arise from spin-orbit coupling
are of particular interest. Here, we calculate the current-induced torque in a
Pt-Co bilayer to help determine the underlying mechanism using first
principles methods. We focus exclusively on the analogue to the Rashba torque,
and do not consider the spin Hall effect. The details of the torque depend
strongly on the layer thicknesses and the interface structure, providing an
explanation for the wide variation in results found by different groups. The
torque depends on the magnetization direction in a way similar to that found
for a simple Rashba model. Artificially turning off the exchange spin
splitting and separately the spin-orbit coupling potential in the Pt shows
that the primary source of the “field-like” torque is a proximate spin-orbit
effect on the Co layer induced by the strong spin-orbit coupling in the Pt.

###### pacs:

85.35.-p, 72.25.-b,

##  I Introduction

Spintronics has had significant impact on information technology, the most
common example being the read heads in hard disk drives. The field is poised
to have even greater impact, led by the development of devices such as spin
transfer torque-based magnetic random access memory. A crucial component of
this next generation of spintronic applications is the control of magnetic
orientation with an electrical currentstiles ; katine  (or electric fieldchien
; ohno0 ). The approach is furthest developed in magnetic tunnel junctions,
which utilize spin transfer torque to reversibly switch the magnetization in
one of the layers. An alternative approach has been demonstrated in recent
experiments on bilayer systems consisting of ultrathin ferromagnetic layers
adjacent to heavy metals such as Pt.miron1 ; miron2 ; miron3 ; liu1 ; ando ;
kim0 ; parkin ; beach ; fan  In these systems, spin-orbit coupling is
responsible for current-induced torques. There are indications that the
efficiency of these current-induced torques (measured as, for example, the
torque per current density) may be larger than the conventional spin transfer
torque.ralph  For this reason among others, these bilayer and related systems
offer a possible route for spintronics to fulfill its full promise in
technological applications.

The spin-orbit coupling in these systems leads to multiple effects. For one,
there is a spin Hall effect in the Pt layer, so that spin current flows from
the Pt into the magnetic layer, with spin orientation perpendicular to the
charge current direction and the interface normal. This spin current flux
induces magnetization dynamics via the conventional spin transfer torque. A
distinct effect originates from the simultaneous presence of magnetization,
spin-orbit coupling, and broken inversion symmetry at the interface. These
ingredients result in an electronic structure in which current carrying states
acquire a spin accumulation transverse to the magnetization, resulting in a
torque.manchon  Both the spin Hall effect and interfacial spin-orbit torque
are proportional to the charge current. An additional consequence of the
interfacial spin-orbit coupling is the current-independent Dzyaloshinskii-
Moriya (DM) interaction,Dzyaloshinskii ; Moriya  which has been argued to be
importantthiaville ; parkin ; beach  for current-induced domain wall motion.

The vector components of the spin-orbit torques can be decomposed into two
vector fields as a function of ${\bf{\hat{M}}}$, the magnetization direction:
$\hat{\bf M}\times\left(\hat{\bf j}\times\hat{\bf z}\right)$, which we refer
to as a field-like torque and $\hat{\bf M}\times\left[\hat{\bf
M}\times\left(\hat{\bf j}\times\hat{\bf z}\right)\right]$, which we refer to
as a damping-like torque (${\bf\hat{j}}$ is the charge current direction and
${\hat{\bf z}}$ is the interface normal).footnote1  These names derive from
the similarity of the first form to the torque due to a field along $\hat{\bf
j}\times\hat{\bf z}$, and the second to the damping that would result from
precession around that field.

Depending on the assumed scattering processes, both the mechanism that
combines the spin Hall effect plus spin-transfer torque and the interfacial
mechanism can give torques in both directions.macdonald ; wang ; duine2012 ;
Kim2012  However in the clean limit of the relaxation time approximation,
Freimuth et al.Freimuth  have shown that the physics behind two torque
components separate rather cleanly: The damping torque originates
predominantly from the spin Hall effect, and is a consequence of the
perturbation of electronic states by the applied electric field, while the
field-like torque originates mostly from the spin-orbit coupling at the
interface, in conjunction with the perturbation of the electron distribution
function by the applied field. In this paper, we focus on the underlying
physics behind the interfacial spin-orbit torque, neglecting the spin Hall
effect. In the terminology of Ref. Freimuth, , we calculate the odd torque
only [$\bm{\Gamma}(\hat{\bf M})=-\bm{\Gamma}(-\hat{\bf M})$], and show that
this component yields predominantly “field-like” torques for realistic systems
(within the relaxation time approximation).

Which of these mechanisms (field-like or damping-like) is responsible for
these current-induced torques is controversial. Measurements of the reversal
of magnetic layers are interpreted in terms of a damping-like torque due to
the spin Hall effect.ralph  Some experiments on current-induced domain wall
motion in these systems are interpreted in terms of a damping-like torque in
combination with the DM interaction.parkin ; beach  Other experiments are
interpreted in terms of a combination of field-like and damping-like torques
arising from the interface.miron1  Experimental measurements of the torque
vector show varied results for the magnitude of the field-like and damping
torques, depending sensitively on the sample details.kim0
CalculationsFreimuth  show both torques to be present and sensitive to
structural details. The spin-orbit coupling affects the magnetization dynamics
in multiple ways, and clearly distinguishing the different contributions
requires more careful experimental and theoretical efforts. In this work, we
calculate the current-induced torques present at the interface between Co and
Pt using first principles methods. As discussed in Sec. II, we do not include
the contributions from the spin Hall effect. The calculation is therefore
analogous to that of the 2DEG Rashba model, but with the full electronic
structure of the Co-Pt interface taken into account. With this approach, we
explore the sensitivity of the current-induced torques to the system
structure, the angular dependence of the current-induced torques, and attempt
to identify the most important physical ingredients of the system which lead
to the current-induced torques.

##  II Method

A quantitative description of any system using density functional theory
requires specific knowledge of the atomic structure. In the absence of
experimental characterization at this level of detail, we study a variety of
structures to reach qualitative conclusions. Our aim with this approach is to
extract semi-quantitative estimates of the current-induced torques, and to
identify the trends and most important physical mechanisms underlying the
current-induced torques.

There is significant mismatch between the lattice constants of bulk Co and Pt
($a_{\rm Co}=0.354~{}{\rm nm},~{}a_{\rm Pt}=0.392~{}{\rm nm}$). Studies of Co
growth on Pt observe a structure that is generally inhomogeneous and quite
sensitive to the Co coverage.ferrer ; lundgren  For simplicity, we assume a
uniform Co layer with the in-plane Pt lattice constant. The distance between
interface Co and Pt layers is taken from Ref. wu, , and the distance between
Co planes was chosen to match the bulk Co density. Our qualitative conclusions
do not depend sensitively on these choices, as we discuss in Sec. III. We
present results for 1, 2 and 3 monolayers (ML) of Co on 8 layers of Pt,
stacked along the [111] direction. We generally take the Co layer stacking to
be fcc. We also study systems with an intermixed interface (see Fig. 1b).
Computational limitations preclude the use of a larger super cells; however
these smaller unit cells reduce the overall symmetry of the system, removing
some artifacts present only for ideal systems.

Figure 1: (a) System geometry for an ideal interface, (b) Example of an
alloyed interface. Black (gray) dots represent Pt (Co), and the larger dots
are in a layer above the smaller dots.

We use the local spin density approximation,perdew  with full non-collinear
spin, and the spin-orbit coupling is included using the on-site approximation
as described in Ref. sanvito, . We include $1.1~{}{\rm nm}$ of vacuum along
the $\hat{\bf z}$ direction and use a minimal localized atomic orbital basis
set (a single $s,p,d$ basis set for each atom) and norm-conserving
pseudopotentials. Adequately converging the ground state energy requires a 2d
k-point mesh with mesh spacing $dk=0.18~{}{\rm nm}^{-1}$.

Using the ground state potential, we find the eigenstates at the Fermi energy
using an approach described in Refs. taylor1, ; taylor2, . We evaluate the
charge current and spin torquenunez  from a Boltzmann distribution of the
states (within the relaxation time approximation):

$\displaystyle I$ $\displaystyle=$
$\displaystyle\frac{e^{2}\tau}{\hbar\left(2\pi\right)^{2}}\int d{\bf
k}_{\parallel}\left({v}_{\bf k_{\parallel}}^{R}-{v}_{\bf
k_{\parallel}}^{L}\right)\cdot{E_{x}}$ (1) $\displaystyle{\bf\Gamma}$
$\displaystyle=$ $\displaystyle\frac{e\tau}{\hbar\left(2\pi\right)^{2}}\int
d{\bf k}_{\parallel}\left(\left(\bf s\times\bf\Delta\right)^{R}_{\bf
k_{\parallel}}-\left(\bf s\times\bf\Delta\right)^{L}_{\bf
k_{\parallel}}\right){E_{x}}$ (2)

where $\tau$ is the lifetime, ${\bf k}_{\parallel}$ is the 2-dimensional Bloch
wave vector normal to the current direction (so that the resulting integral is
1-dimensional). ${\bf s}$ is the spin operator, and $\Delta$ is the spin-
dependent exchange-correlation potential. The superscript $R$ ($L$) refers to
states with positive (negative) group velocity in the $\hat{\bf x}$-direction.
The resulting torque per current is independent of scattering time $\tau$.
Converging the 1-dimensional transport integrals requires a finer mesh spacing
($dk=0.044~{}{\rm nm}^{-1}$) than needed for the ground state energy. While
the torque varies strongly with the magnetization direction, the effective
field only varies weakly. We present our results in terms of a scaled
effective field

$H_{R}/J=\frac{\bf\Gamma\cdot[{\bf M}\times(\hat{\bf j}\times\hat{\bf
z})]}{|{\bf M}\times(\hat{\bf j}\times\hat{\bf z})|^{2}}\frac{d}{M_{\rm s}I}$
(3)

where $M_{\rm s}$ is the computed magnetization density and $d$ is the slab
thickness. The result has units of field per bulk current density, a quantity
commonly used to report experimental results.

Eq. 2 captures some - but not all - contributions to the current-induced
torque. In the language of Ref. Freimuth, , we only include the odd torques,
and hence neglect any intrinsic spin Hall effect in the Pt layers or at the
interface. We also neglect effects from interband scattering on the
torquesjungwirth  as well as higher order corrections due to momentum
scattering that were considered in Refs. macdonald, ; wang,  for a Rashba
model. Ref. Freimuth,  shows that for Co/Pt, the even torque is dominated by
the spin Hall effect and the interfacial contributions are negligible. We also
neglect skew scattering that could also give a spin Hall effect. Thus, we only
capture the current-induced torque related to the Rashba model and neglect all
contributions from the spin Hall effect in the bulk of the Pt. Recent models
of this system employing a Boltzmann model show that the Rashba and spin Hall
effect torques are largely independent of each other.stiles:2013  We use the
approach described here to focus on understanding the contribution to the
current-induced torque from the spin-orbit coupling near the interface.

Our calculations are complementary to those of Ref. Freimuth,  Those
calculations compute the torque for the magnetization parallel and
antiparallel to the interface normal and consider the even and odd components.
These are equivalent to the damping-like and field-like torques. Their
calculations include the spin Hall effect in the Pt and so describe the
damping-like torque that we neglect. Where the calculations can be compared,
we find similar results. The reduced computational demands of our approach
enable us to explore a wider range of systems, including different geometries
and different magnetization orientations.

##  III Results

We first consider the important energy scales in the system: the spin-orbit
coupling and the exchange spin splitting. Fig. 2 shows the calculated bulk
band structure of bulk Pt with and without spin-orbit coupling. The spin-orbit
energy splittings are large at points of high-symmetry: the spin-orbit
splitting at ${\bf k}=0$ of the ${\Gamma}_{25}^{\prime}$ band is on the order
of $1~{}{\rm eV}$. For bulk hcp Co, we find an exchange splitting energy
$\Delta$ of about $1~{}{\rm eV}$. We show below that the exchange splitting in
the Co induces exchange splitting (and a small moment) in the Pt and that the
spin-orbit coupling in the Pt induces a transverse moment in the Co. Of these
proximity effects, the transverse moment on the Co induced by the spin-orbit
coupling in the Pt plays the dominant role in determining the current-induced
torques.

Figure 2: (Color online) Band structure of bulk Pt. Solid lines: with spin-
orbit coupling, dotted lines: without spin-orbit coupling.

The layer resolved magnetic moments are shown in Fig. 3a. The induced moment
on the interface Pt layer varies with the Co coverage, with values of
$0.30~{}\mu_{\rm B},0.22~{}\mu_{\rm B},~{}{\rm and}~{}0.25~{}\mu_{\rm B}$ for
1 ML, 2 ML, and 3 ML of Co, respectively. These values are similar to those
found in previous calculationswu ; moulas ; lehnert  and to experimental
measurements (Refs. ferrer, ; geissler, ; valvidares,  measure a Pt interface
moment of $\approx 0.2~{}\mu_{\rm B}$, while Ref. suzuki,  measures a moment
of $\approx 0.6~{}\mu_{\rm B}$). Fig. 3b shows the moments in each layer for
the disordered interface geometry showed in Fig. 1b. The moments on the Pt
atoms in the alloyed interface range from 0.25 $\mu_{\rm B}$ to 0.4 $\mu_{\rm
B}$. The importance of this proximity magnetization in the metal is an open
question. In a recent experiment, a Au layer was placed between the Pt and the
Co.parkin  Au has a lower magnetic susceptibility and a much smaller induced
moment. It is found that increasing the Au layer thickness reduces the offset
in the current-domain wall velocity curves, which is explained by a reduction
in the DM interaction. It’s concluded that proximate magnetization plays a
central role in one of the important spin-orbit coupling effects at the
interface (the DM interaction).parkin  On the other hand, our calculations
suggest that the transverse spin in the Co induced by the Pt plays a more
important role for the field-like torque, as we discuss at the end of this
section.

Figure 3: Magnetic moment versus layer for (a) 8 layers of Pt (P) - 2 layers
of Co (C), and (b) 8 layers of Pt - 2 alloyed (A) layers - 1 layer Co, (see
Fig. 2b for the alloy coordinates). The four atomic moments in each layer are
plotted individually (since the disordered interface has inequivalent atoms in
each layer), where filled (open) symbols represent Pt (Co).

Fig. 4a shows the layer-resolved effective field per current for 1, 2, and 3
ML of Co. The largest field (or largest torque) is on the Co atoms, although
there is also torque present on the magnetized Pt layer. The effective field
per current for various geometries are listed in Table 1. Experimental values
range from a highmiron1  of $10^{-12}~{}{\rm T\cdot m^{2}/A}$ to slightly
abovekim0 ; garello  $10^{-14}~{}{\rm T\cdot m^{2}/A}$, although the precise
value is highly dependent on system details like layer thicknesses: Ref.
ohno2,  finds an order of magnitude difference in the current-induced
effective field when the magnetic layer thickness changes from 1 nm to 1.2 nm.
We also find the magnitude depends sensitively on coverage. Fig. 4b shows the
effective field for an alloyed interface. The total magnitude is decreased in
all alloyed interfaces we have investigated, relative to the ideal interface.
The torques are again predominantly localized on the Co atoms, and they are
nearly oppositely oriented for Co atoms in the alloy layer adjacent to the
pure Pt. This cancelation is largely responsible for the decrease in the total
current-induced torques. Note that our result for the 8-3 ML Pt-Co system is
similar to that of Ref. Freimuth, , despite some slight differences in the
structures used in the two calculations.

Figure 4: (color online) Layer resolved effective field per current density
for (a) 1, 2, and 3 ML of Co with an ideal interface, and (b) 2 alloyed
interfaces.

The torques are largely unchanged if we use hcp stacking for the Co layers
(see Table 1). They do change in calculations for larger Co-Pt and Co-Co
interplane distance, but the trends with respect to Co coverage and alloying
are similar. We highlight the trend that the torques are decreased for
imperfect interfaces, and that these decreased values are in semi-quantitative
agreement with experiment.

Structure | SO on Pt |  $\Delta$ on Pt | $H_{R}/J\left[10^{-14}~{}{\rm T\cdot m^{2}/A}\right]$
---|---|---|---
8-0-1 | $\bullet$ | $\bullet$ | -22.4
8-0-2 | $\bullet$ | $\bullet$ | -17.9
8-0-2 | $\bullet$ |  | -19.5
8-0-2 |  | $\bullet$ | -3.2
8-0-2(hcp) | $\bullet$ | $\bullet$ | -18.9
8-1-1 | $\bullet$ | $\bullet$ | -11.1
8-2-1 (a) | $\bullet$ | $\bullet$ | -3.3
8-2-1 (a) | $\bullet$ |  | -3.9
8-2-1 (a) |  | $\bullet$ | 2.9
8-2-1 (b) | $\bullet$ | $\bullet$ | -3.7
8-0-3 | $\bullet$ | $\bullet$ | -7.7
8-2-2 | $\bullet$ | $\bullet$ | -1.9
Table 1: Field-like torque for different film geometries. The first column
gives the number of pure Pt layers, the number of alloy layers, and the number
of pure Co layers. In the second and third columns, no “$\bullet$” appears if
the spin-orbit coupling (second) or exchange potential (third) has been set to
zero. The two different structures considered for the 8-2-1 geometry are
designated (a) and (b).

As noted above, our calculations include only the odd contribution to the
torque, $\bm{\Gamma}(\hat{\bf M})=-\bm{\Gamma}(-\hat{\bf M})$. This
contribution is dominated by the field-like torque, although we also find a
nonzero damping-like torque for certain orientations of the magnetization.
Fig. 5a shows the angular dependence of the current-induced fields responsible
for field-like and damping-like torques. The left panels are for an ideal 8-2
ML Co-Pt system, and the right panels are for a disordered interface as shown
in Fig. 1b. For the Rashba model in the limit where the exchange potential is
much larger than the spin-orbit coupling energy, the equivalent fields are
independent of the magnetic orientation.manchon  However, when the exchange
potential and spin-orbit coupling are similar in magnitude, the angular
dependence of the fields found from the Rashba model is similar to that shown
here.Lee2  Based on Fig. 5, we conclude that the simple Rashba model
description of bilayers accounts reasonably well for many properties of the
torque (at least for an ideal interface).

Figure 5: (color online) The angular dependence of the effective field (solid
blue) and “damping field” (dashed red) magnitude per current density. The left
panel is for 2 ML of Co with an ideal interface, while the right panel is for
2 layers of alloy+1 layer Co. The angles are conventional spherical
coordinates for the coordinate system of Fig. 2.

We next comment on the damping-like torque calculated in our system. We first
note that if a system is isotropic in the $x-y$ plane, the odd torque is
entirely field-like. This is because the eigenstate have spin components
$s_{x},~{}s_{z}$ components which are even functions of $k_{x}$, and $s_{y}$
which is an odd function of $k_{x}$.footnote2  Forming a current-carrying
distribution in the $\hat{x}$-direction leads to a spin accumulation purely in
the $\hat{y}$-direction, yielding a field like torque. The inequivalence of
$+\hat{x}$ and $-\hat{x}$ directions of our lattice implies that the spins
have no such symmetries when the magnetization has a $\hat{y}$ component, so
that a spin accumulation of any direction is allowed by symmetry.footnote3
This in turn leads to both field-like and damping-like torques. For real
systems with disordered interfaces, we expect the in-plane direction to be
relatively isotropic, so that the odd torque is primarily field-like (at least
within the relaxation time approximation).

Figure 6: (color online) (a) Plot of states at the Fermi level versus wave
vector. (b) Zoom in of states near the zone center. The dark blue (light red)
color indicates states with positive (negative) group velocity in the
$x$-direction. The dot size of each state is proportional to the magnitude of
the state’s $z$ component of spin on the interface Co layer. (c) The $x$
component of spin on the interface Co layer. The magnitude of each state’s
spin is proportional to the dot size, and the colors (shading) indicate the
sign of $s_{x}$. (d) The same plot for the $y$ component of each state’s spin.
The torque on the interface Co layer contributed by each state is proportional
to the in-plane spin component for that state.

To further illustrate the similarities and differences of the system with
simple Rashba model, we plot the states at the Fermi energy in Fig. 6. Despite
the enormous complexity of the electronic structure, there are similarities to
the Rashba model. This is shown in Figs. 6c-d, which depict the spin structure
of states near ${\bf k}$=0. The symmetry of the Fermi surface and ${\bf
k}$-dependence of the spin follows from the system symmetry.footnote3  The
current-induced torque arises when summing over a current carrying
distribution of these states (recall the current to be in the $\hat{x}$
direction). The sizeable transverse moments of the states, shown in panels (c)
and (d) of Fig. 6, are the result of the interaction of the Co orbitals with
the spin-orbit potential localized on the Pt.

Summing over states leads to significant cancellation of the torques. We find
that the average absolute value of torque from each state is about 10 to 100
times greater than the integrated total, depending on the specific system. As
described in Ref. park1, , this cancellation can be understood qualitatively
in a tight-binding model, where different bands have different signs of
orbital chirality $\pm\left({\bf L}\times{\bf k}\right)\cdot\hat{\bf
z}$.kim:2011  The addition of spin-orbit coupling ${\bf L}\cdot{\bf S}$ then
(roughly speaking) leads to alignment of the spin in the $\pm\left({\bf
S}\times{\bf k}\right)\cdot\hat{\bf z}$ direction, resulting in different
signs of an effective Rashba parameter for different states.park1

We next attempt to identify the primary source of the torque present at the
interface. A current-induced torque from a Rashba-like model requires the
simultaneous presence of exchange splitting and spin-orbit coupling. It’s not
a priori obvious if the induced exchange present in the Pt is more or less
important than the induced spin-orbit coupling in the Co. We first make a
general remark about this distinction.

Figure 7: (color online) Layer resolved torque for 8 layer Pt - 2 layer Co
system. “Default” refers to the system with full exchange and spin-orbit
coupling. The other torques are calculated by removing the specified potential
from the Pt atoms, as described in the text.

The magnetic proximity effect is one of the most dramatic examples of the
effect of hybridization between neighboring materials’ orbitals. However, as
discussed in Ref. park1, , there is a similar energy splitting on the Co
orbitals due to the interaction with the spin-orbit coupling in the Pt. This
effect is not as obvious as the magnetic proximity effect for two reasons: the
Co levels with which Pt hybridize are pure spin states, which leads to
appreciable spin splitting for all Pt states, and a macroscopic magnetization
in the Pt interface layer. On the other hand, the eigenstates of the atomic
spin-orbit potential ${\bf L}\cdot{\bf S}$ are those of the total angular
momentum operator ${\bf J}={\bf L}+{\bf S}$. ${\bf J}$ is not a good quantum
number in the symmetry broken crystal field, so the Pt states with which Co
hybridize generally do not directly reflect the spin orbit potential. This
obscures the effect of the Pt spin-orbit on the levels in the Co. Despite
being less “obvious” in this way, the proximate effect of the spin-orbit
coupling is quite important, as we show next.

To determine the role of magnetic proximity effect, we remove the exchange
splitting on the Pt atoms from the ground state Hamiltonian, and the resulting
current-induced torques are calculated as described in Sec. II. To determine
the role of the spin-orbit coupling proximity effect, we remove the spin-orbit
potential from the Pt atoms, perform a new ground state calculations, and
calculated the current-induced torques. (We find that removing the Pt spin-
orbit potential from the initial ground state and calculating the current-
induced torques without performing a new ground state calculations yields very
similar results.)

Fig. 7 shows the results for the ideal interface. The current-induced
effective field is nearly unchanged when the Pt magnetization is removed, and
greatly diminished when the Pt spin-orbit coupling is removed. We conclude
that the spin-orbit in the Pt is the main agent behind the total torque. We
additionally repeat this calculation for an alloyed system, with the layer and
atom-resolved torques shown in Fig. 8. A similar scenario holds in this case.
(Note however that the magnitude of the total torques are not so different in
the alloyed case (see Table 1) when we remove the spin-orbit from the Pt; this
is due to the significant cancelations that occur when adding the
contributions from all the states in the default system.) It’s instructive to
evaluate the average absolute value of the torque from all states for these
different scenarios. We find the torques are nearly unchanged for no
magnetization on Pt, while they’re about 5 times smaller when the spin-orbit
is removed from the Pt. To rationalize this result, we note that the exchange
splitting in the Pt is 10 times smaller than in the Co. On the other hand, the
transverse spin density is 3x smaller in the Co than in the Pt. These
properties of the states indicate that the Pt spin-orbit potential is the
primary source of the overall torque.

Figure 8:  Similar plot as Fig. 7, but with an alloyed interface. The
individual torques on the 4 atoms of each layer are shown (the torques from
the bottom 5 Pt layers are omitted from the plot, as they vanish). Open
(filled) circles represent Pt (Co).

##  IV Conclusion

In summary, we performed first principles calculations of the field-like
current-induced torque on a series of Co-Pt bilayers, and from this we
conclude that: 1. the torque is very sensitive to system details, such as Co
coverage. This is consistent with experimental data, and the general magnitude
of the calculated effective fields is similar to the experimental values. 2.
The angular dependence of the torque is very similar to that predicted by the
simple 2-d Rashba model for an ideal interface, but more complicated for
alloyed interfaces. 3. The primary source of the torque is derived from the
spin-orbit coupling localized on the Pt interface atoms, which affects the
nonequilibrium spin density in the Co interface layer, and drives the current-
induced torques. The extent to which the trends revealed by the first
principles calculations can be made more revealing in simpler models is a
question for future work. In addition, this approach may be used to screen
different materials combinations in order to anticipate which materials may
show the strongest effect.

## References

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