Theoretical Lower Limit of Coercive Field in Ferroelectric Hafnia

The high coercive field (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi mathvariant="script">E</a:mi><a:mi>c</a:mi></a:msub></a:math>) of hafnia-based ferroelectrics presents a major obstacle to their applications. The ferroelectric switching mechanisms in hafnia that dictate <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:msub><d:mi mathvariant="script">E</d:mi><d:mi>c</d:mi></d:msub></d:math>, especially those related to domain nucleation in the nucleation-limited-switching (NLS) model and domain-wall motion in the Kolmogorov-Avrami-Ishibashi (KAI) model, have remained elusive. We develop a deep-learning-assisted multiscale approach, incorporating atomistic insights into the critical nucleus, to predict both NLS- and KAI-type coercive fields. The theoretical NLS-type <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi mathvariant="script">E</g:mi><g:mi>c</g:mi></g:msub></g:math> values agree with previous experimental results as well as our own measurements and also exhibit the correct thickness scaling for films between 3 and 20 nm. Combined theoretical and experimental investigations reveal that the giant <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:msub><j:mi mathvariant="script">E</j:mi><j:mi>c</j:mi></j:msub></j:math> in hafnia-based ferroelectrics arises from the ultrathin geometry, which confines switching to the NLS mechanism. We predict that the theoretical lower limit for KAI-type <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi mathvariant="script">E</m:mi><m:mi>c</m:mi></m:msub></m:math> is <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mrow><p:mn>0.1</p:mn><p:mtext> </p:mtext><p:mtext> </p:mtext><p:mi>MV</p:mi><p:mo>/</p:mo><p:mi>cm</p:mi></p:mrow></p:math> arising from mobile domain walls. The activation of KAI-type switching to achieve lower <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:msub><r:mi mathvariant="script">E</r:mi><r:mi>c</r:mi></r:msub></r:math> is supported by our experimental demonstration of a low coercive field of <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mrow><u:mn>1</u:mn><u:mtext> </u:mtext><u:mtext> </u:mtext><u:mi>MV</u:mi><u:mo>/</u:mo><u:mi>cm</u:mi></u:mrow></u:math> in 60 nm ferroelectric <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mrow><w:mo stretchy="false">(</w:mo><w:mrow><w:msub><w:mrow><w:mi>HfO</w:mi></w:mrow><w:mrow><w:mn>2</w:mn></w:mrow></w:msub></w:mrow><w:msub><w:mrow><w:mo stretchy="false">)</w:mo></w:mrow><w:mrow><w:mi>n</w:mi></w:mrow></w:msub><w:mo>/</w:mo><w:mo stretchy="false">(</w:mo><w:mrow><w:msub><w:mrow><w:mi>ZrO</w:mi></w:mrow><w:mrow><w:mn>2</w:mn></w:mrow></w:msub></w:mrow><w:msub><w:mrow><w:mo stretchy="false">)</w:mo></w:mrow><w:mrow><w:mi>n</w:mi></w:mrow></w:msub></w:mrow></w:math> (<cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi>n</cb:mi><cb:mo>=</cb:mo><cb:mn>3</cb:mn></cb:math> unit cells) superlattices. These findings establish a comprehensive framework for understanding ferroelectric switching in hafnia and highlight the potential of geometry and domain-wall engineering to achieve low-<eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:msub><eb:mi mathvariant="script">E</eb:mi><eb:mi>c</eb:mi></eb:msub></eb:math> devices.