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{"title":"In-plane sliding ferroelectricity and piezoelectricity in bilayer and trilayer𝑔−C3N4","authors":"Yuhua Wei, Zuhui Hu, Xiaonan Ma, Haotian Wang, Feng Gao, Xiaoxuan Ma, Yin Wang, Wei Ren","doi":"10.1103/physrevb.110.174103","DOIUrl":null,"url":null,"abstract":"The graphenelike form of carbon nitride is known as\n<mjx-container ctxtmenu_counter=\"43\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"(10 0 1 (9 (4 2 3) 8 (7 5 6)))\"><mjx-mrow data-semantic-children=\"0,9\" data-semantic-content=\"1\" data-semantic- data-semantic-owns=\"0 1 9\" data-semantic-role=\"subtraction\" data-semantic-speech=\"g minus normal upper C 3 normal upper N 4\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑔</mjx-c></mjx-mi><mjx-mtext data-semantic-annotation=\"general:text\" data-semantic- data-semantic-operator=\"infixop,−\" data-semantic-parent=\"10\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" style='font-family: MJX-STX-ZERO, \"Helvetica Neue\", Helvetica, Roboto, Arial, sans-serif;'><mjx-utext style=\"font-size: 90.6%; padding: 0.828em 0px 0.221em; width: 7px;\" variant=\"-explicitFont\">−</mjx-utext></mjx-mtext><mjx-mrow data-semantic-added=\"true\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"4,7\" data-semantic-content=\"8\" data-semantic- data-semantic-owns=\"4 8 7\" data-semantic-parent=\"10\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-msub data-semantic-children=\"2,3\" data-semantic- data-semantic-owns=\"2 3\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>C</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c>3</mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"9\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\"5,6\" data-semantic- data-semantic-owns=\"5 6\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>N</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c>4</mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow></mjx-mrow></mjx-math></mjx-container>, a single sheet of which may be composed of an <mjx-container ctxtmenu_counter=\"44\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"s\" data-semantic-type=\"identifier\"><mjx-c>𝑠</mjx-c></mjx-mi></mjx-math></mjx-container>-triazine (<mjx-container ctxtmenu_counter=\"45\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper S\" data-semantic-type=\"identifier\"><mjx-c>𝑆</mjx-c></mjx-mi></mjx-mrow></mjx-math></mjx-container>-type) or tri-<mjx-container ctxtmenu_counter=\"46\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"s\" data-semantic-type=\"identifier\"><mjx-c>𝑠</mjx-c></mjx-mi></mjx-math></mjx-container>-triazine (<mjx-container ctxtmenu_counter=\"47\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper T\" data-semantic-type=\"identifier\"><mjx-c>𝑇</mjx-c></mjx-mi></mjx-mrow></mjx-math></mjx-container>-type) periodic lattice with a different size of hole. By employing the first-principles calculation, we find that the difference in the out of plane polarization caused by the symmetry breaking of the <mjx-container ctxtmenu_counter=\"48\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper S\" data-semantic-type=\"identifier\"><mjx-c>𝑆</mjx-c></mjx-mi></mjx-mrow></mjx-math></mjx-container>-type or <mjx-container ctxtmenu_counter=\"49\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper T\" data-semantic-type=\"identifier\"><mjx-c>𝑇</mjx-c></mjx-mi></mjx-mrow></mjx-math></mjx-container>-type structure of bilayer <mjx-container ctxtmenu_counter=\"50\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"(10 0 1 (9 (4 2 3) 8 (7 5 6)))\"><mjx-mrow data-semantic-children=\"0,9\" data-semantic-content=\"1\" data-semantic- data-semantic-owns=\"0 1 9\" data-semantic-role=\"subtraction\" data-semantic-speech=\"g minus normal upper C 3 normal upper N 4\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑔</mjx-c></mjx-mi><mjx-mtext data-semantic-annotation=\"general:text\" data-semantic- data-semantic-operator=\"infixop,−\" data-semantic-parent=\"10\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" style='font-family: MJX-STX-ZERO, \"Helvetica Neue\", Helvetica, Roboto, Arial, sans-serif;'><mjx-utext style=\"font-size: 90.6%; padding: 0.828em 0px 0.221em; width: 7px;\" variant=\"-explicitFont\">−</mjx-utext></mjx-mtext><mjx-mrow data-semantic-added=\"true\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"4,7\" data-semantic-content=\"8\" data-semantic- data-semantic-owns=\"4 8 7\" data-semantic-parent=\"10\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-msub data-semantic-children=\"2,3\" data-semantic- data-semantic-owns=\"2 3\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>C</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c>3</mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"9\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\"5,6\" data-semantic- data-semantic-owns=\"5 6\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>N</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c>4</mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow></mjx-mrow></mjx-math></mjx-container> through interlayer sliding may be related to the size of the in-plane hole. The nonpolar bilayer <mjx-container ctxtmenu_counter=\"51\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"(10 0 1 (9 (4 2 3) 8 (7 5 6)))\"><mjx-mrow data-semantic-children=\"0,9\" data-semantic-content=\"1\" data-semantic- data-semantic-owns=\"0 1 9\" data-semantic-role=\"subtraction\" data-semantic-speech=\"g minus normal upper C 3 normal upper N 4\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑔</mjx-c></mjx-mi><mjx-mtext data-semantic-annotation=\"general:text\" data-semantic- data-semantic-operator=\"infixop,−\" data-semantic-parent=\"10\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" style='font-family: MJX-STX-ZERO, \"Helvetica Neue\", Helvetica, Roboto, Arial, sans-serif;'><mjx-utext style=\"font-size: 90.6%; padding: 0.828em 0px 0.221em; width: 7px;\" variant=\"-explicitFont\">−</mjx-utext></mjx-mtext><mjx-mrow data-semantic-added=\"true\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"4,7\" data-semantic-content=\"8\" data-semantic- data-semantic-owns=\"4 8 7\" data-semantic-parent=\"10\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-msub data-semantic-children=\"2,3\" data-semantic- data-semantic-owns=\"2 3\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>C</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c>3</mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"9\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\"5,6\" data-semantic- data-semantic-owns=\"5 6\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>N</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c>4</mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow></mjx-mrow></mjx-math></mjx-container> (<mjx-container ctxtmenu_counter=\"52\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper S\" data-semantic-type=\"identifier\"><mjx-c>𝑆</mjx-c></mjx-mi></mjx-mrow></mjx-math></mjx-container>- or <mjx-container ctxtmenu_counter=\"53\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper T\" data-semantic-type=\"identifier\"><mjx-c>𝑇</mjx-c></mjx-mi></mjx-mrow></mjx-math></mjx-container>-type) state needs to slide about two bond lengths before it is electrically driven to the polar state, resulting in ferroelectric structures with opposite polarization directions. It has been confirmed experimentally and theoretically [Zelisko <i>et al.</i>, <span>Nat. Commun.</span> <b>5</b>, 4284 (2014)] that the anomalous piezoelectric properties of <mjx-container ctxtmenu_counter=\"54\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper T\" data-semantic-type=\"identifier\"><mjx-c>𝑇</mjx-c></mjx-mi></mjx-mrow></mjx-math></mjx-container>-type <mjx-container ctxtmenu_counter=\"55\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"(10 0 1 (9 (4 2 3) 8 (7 5 6)))\"><mjx-mrow data-semantic-children=\"0,9\" data-semantic-content=\"1\" data-semantic- data-semantic-owns=\"0 1 9\" data-semantic-role=\"subtraction\" data-semantic-speech=\"g minus normal upper C 3 normal upper N 4\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑔</mjx-c></mjx-mi><mjx-mtext data-semantic-annotation=\"general:text\" data-semantic- data-semantic-operator=\"infixop,−\" data-semantic-parent=\"10\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" style='font-family: MJX-STX-ZERO, \"Helvetica Neue\", Helvetica, Roboto, Arial, sans-serif;'><mjx-utext style=\"font-size: 90.6%; padding: 0.828em 0px 0.221em; width: 7px;\" variant=\"-explicitFont\">−</mjx-utext></mjx-mtext><mjx-mrow data-semantic-added=\"true\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"4,7\" data-semantic-content=\"8\" data-semantic- data-semantic-owns=\"4 8 7\" data-semantic-parent=\"10\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-msub data-semantic-children=\"2,3\" data-semantic- data-semantic-owns=\"2 3\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>C</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c>3</mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"9\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\"5,6\" data-semantic- data-semantic-owns=\"5 6\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>N</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c>4</mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow></mjx-mrow></mjx-math></mjx-container> monolayer come from their noncentrosymmetric triangular holes. Interestingly, we find here that the <mjx-container ctxtmenu_counter=\"56\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper S\" data-semantic-type=\"identifier\"><mjx-c>𝑆</mjx-c></mjx-mi></mjx-mrow></mjx-math></mjx-container>-type <mjx-container ctxtmenu_counter=\"57\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"(10 0 1 (9 (4 2 3) 8 (7 5 6)))\"><mjx-mrow data-semantic-children=\"0,9\" data-semantic-content=\"1\" data-semantic- data-semantic-owns=\"0 1 9\" data-semantic-role=\"subtraction\" data-semantic-speech=\"g minus normal upper C 3 normal upper N 4\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑔</mjx-c></mjx-mi><mjx-mtext data-semantic-annotation=\"general:text\" data-semantic- data-semantic-operator=\"infixop,−\" data-semantic-parent=\"10\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" style='font-family: MJX-STX-ZERO, \"Helvetica Neue\", Helvetica, Roboto, Arial, sans-serif;'><mjx-utext style=\"font-size: 90.6%; padding: 0.828em 0px 0.221em; width: 7px;\" variant=\"-explicitFont\">−</mjx-utext></mjx-mtext><mjx-mrow data-semantic-added=\"true\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"4,7\" data-semantic-content=\"8\" data-semantic- data-semantic-owns=\"4 8 7\" data-semantic-parent=\"10\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-msub data-semantic-children=\"2,3\" data-semantic- data-semantic-owns=\"2 3\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>C</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c>3</mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"9\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\"5,6\" data-semantic- data-semantic-owns=\"5 6\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>N</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c>4</mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow></mjx-mrow></mjx-math></mjx-container> exhibits the opposite piezoelectric property to the <mjx-container ctxtmenu_counter=\"58\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper T\" data-semantic-type=\"identifier\"><mjx-c>𝑇</mjx-c></mjx-mi></mjx-mrow></mjx-math></mjx-container>-type <mjx-container ctxtmenu_counter=\"59\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"(10 0 1 (9 (4 2 3) 8 (7 5 6)))\"><mjx-mrow data-semantic-children=\"0,9\" data-semantic-content=\"1\" data-semantic- data-semantic-owns=\"0 1 9\" data-semantic-role=\"subtraction\" data-semantic-speech=\"g minus normal upper C 3 normal upper N 4\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑔</mjx-c></mjx-mi><mjx-mtext data-semantic-annotation=\"general:text\" data-semantic- data-semantic-operator=\"infixop,−\" data-semantic-parent=\"10\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" style='font-family: MJX-STX-ZERO, \"Helvetica Neue\", Helvetica, Roboto, Arial, sans-serif;'><mjx-utext style=\"font-size: 90.6%; padding: 0.828em 0px 0.221em; width: 7px;\" variant=\"-explicitFont\">−</mjx-utext></mjx-mtext><mjx-mrow data-semantic-added=\"true\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"4,7\" data-semantic-content=\"8\" data-semantic- data-semantic-owns=\"4 8 7\" data-semantic-parent=\"10\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-msub data-semantic-children=\"2,3\" data-semantic- data-semantic-owns=\"2 3\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>C</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c>3</mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"9\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\"5,6\" data-semantic- data-semantic-owns=\"5 6\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>N</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c>4</mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow></mjx-mrow></mjx-math></mjx-container>, that is, a negative longitudinal piezoelectricity (NLP). Our computational analysis shows that the superposition of two reversed triangular holes in the <mjx-container ctxtmenu_counter=\"60\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper S\" data-semantic-type=\"identifier\"><mjx-c>𝑆</mjx-c></mjx-mi></mjx-mrow></mjx-math></mjx-container>-type <mjx-container ctxtmenu_counter=\"61\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"(10 0 1 (9 (4 2 3) 8 (7 5 6)))\"><mjx-mrow data-semantic-children=\"0,9\" data-semantic-content=\"1\" data-semantic- data-semantic-owns=\"0 1 9\" data-semantic-role=\"subtraction\" data-semantic-speech=\"g minus normal upper C 3 normal upper N 4\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑔</mjx-c></mjx-mi><mjx-mtext data-semantic-annotation=\"general:text\" data-semantic- data-semantic-operator=\"infixop,−\" data-semantic-parent=\"10\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" style='font-family: MJX-STX-ZERO, \"Helvetica Neue\", Helvetica, Roboto, Arial, sans-serif;'><mjx-utext style=\"font-size: 90.6%; padding: 0.828em 0px 0.221em; width: 7px;\" variant=\"-explicitFont\">−</mjx-utext></mjx-mtext><mjx-mrow data-semantic-added=\"true\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"4,7\" data-semantic-content=\"8\" data-semantic- data-semantic-owns=\"4 8 7\" data-semantic-parent=\"10\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-msub data-semantic-children=\"2,3\" data-semantic- data-semantic-owns=\"2 3\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>C</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c>3</mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"9\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\"5,6\" data-semantic- data-semantic-owns=\"5 6\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>N</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c>4</mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow></mjx-mrow></mjx-math></mjx-container> plane will result in the emergence of NLP. Hence, the origin of different piezoelectric properties between <mjx-container ctxtmenu_counter=\"62\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper S\" data-semantic-type=\"identifier\"><mjx-c>𝑆</mjx-c></mjx-mi></mjx-mrow></mjx-math></mjx-container>-type and <mjx-container ctxtmenu_counter=\"63\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper T\" data-semantic-type=\"identifier\"><mjx-c>𝑇</mjx-c></mjx-mi></mjx-mrow></mjx-math></mjx-container>-type <mjx-container ctxtmenu_counter=\"64\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"(10 0 1 (9 (4 2 3) 8 (7 5 6)))\"><mjx-mrow data-semantic-children=\"0,9\" data-semantic-content=\"1\" data-semantic- data-semantic-owns=\"0 1 9\" data-semantic-role=\"subtraction\" data-semantic-speech=\"g minus normal upper C 3 normal upper N 4\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑔</mjx-c></mjx-mi><mjx-mtext data-semantic-annotation=\"general:text\" data-semantic- data-semantic-operator=\"infixop,−\" data-semantic-parent=\"10\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" style='font-family: MJX-STX-ZERO, \"Helvetica Neue\", Helvetica, Roboto, Arial, sans-serif;'><mjx-utext style=\"font-size: 90.6%; padding: 0.828em 0px 0.221em; width: 7px;\" variant=\"-explicitFont\">−</mjx-utext></mjx-mtext><mjx-mrow data-semantic-added=\"true\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"4,7\" data-semantic-content=\"8\" data-semantic- data-semantic-owns=\"4 8 7\" data-semantic-parent=\"10\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-msub data-semantic-children=\"2,3\" data-semantic- data-semantic-owns=\"2 3\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>C</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c>3</mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"9\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\"5,6\" data-semantic- data-semantic-owns=\"5 6\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>N</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c>4</mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow></mjx-mrow></mjx-math></mjx-container> is comparatively elucidated. These results provide alternative structural manipulation degrees of freedom for the inherent properties of two-dimensional van der Waals layered semiconductor materials, and expand ferroelectric candidate materials for next-generation nanoelectronic devices.","PeriodicalId":20082,"journal":{"name":"Physical Review B","volume":"214 1","pages":""},"PeriodicalIF":3.7000,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physical Review B","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1103/physrevb.110.174103","RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Physics and Astronomy","Score":null,"Total":0}
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