چكيده فارسي :
ﻣﺸﺎﻫﺪﺍﺕ ﺍﺧﯿﺮ ﺩﺭ ﻣﻮﺭﺩ ﻭﺟﻮﺩ ﻓﺎﺯ ﻋﺎﯾﻖ ﻭ ﺍﺑﺮﺭﺳﺎﻧﺎ ﺍﺯ ﻧﻮﻉ ﻫﻤﺒﺴﺘﮥ ﻗﻮﯼ ﺩﺭ ﮔﺮﺍﻓﯿﻦ ﺩﻭﻻﯾﻪ ﭼﺮﺧﯿﺪﻩ TBG، ﯾﮏ ﺑﺴﺘﺮ ﺟﺪﯾﺪ
ﺑﺮﺍﯼ ﻣﻄﺎﻟﻌﮥ ﺍﺑﺮﺭﺳﺎﻧﺎﯾﯽ ﻏﯿﺮ ﻣﺘﻌﺎﺭﻑ ﺍﯾﺠﺎﺩ ﮐﺮﺩﻩ ﺍﺳﺖ. ﻓﺎﺯ ﺍﺑﺮﺭﺳﺎﻧﺎﯾﯽ ﺑﺮﺍﯼ ﺍﯾﻦ ﺳﺎﻣﺎﻧﻪ ﺩﺭ ﺯﺍﻭﯾﮥ ﭼﺮﺧﺸﯽ ﺗﻘﺮﯾﺒﺎ ﺑﺮﺍﺑﺮ ﺑﺎ ◦80٫1
ﺍﺗﻔﺎﻕ ﻣﯽﺍﻓﺘﺪ، ﭼﮕﺎﻟﯽ ﺣﺎﻣﻞﻫﺎ ﺩﺭ ﺍﯾﻦ ﺣﺎﻟﺖ ﺗﻘﺮﯾﺒﺎ ﺑﺮﺍﺑﺮ 2cm− 1101 ﻭ ﺩﻣﺎﯼ ﮔﺬﺍﺭ ﺑﻪ ﺍﺑﺮﺭﺳﺎﻧﺎﯾﯽ ﺑﺮﺍﺑﺮ ◦K7٫1 ﺍﺳﺖ. ﺍﯾﻦ ﺭﻓﺘﺎﺭ ﺍﺑﺮﺭﺳﺎﻧﺎﯾﯽ ﺑﻪ ﺑﺮﻫﻤﮑﻨﺶ ﺍﻟﮑﺘﺮﻭﻥ−ﺍﻟﮑﺘﺮﻭﻥ ﺩﺭ ﺯﻭﺍﯾﺎﯼ ﺧﺎﺹ ﻧﺴﺒﺖ ﺩﺍﺩﻩ ﻣﯽﺷﻮﺩ. ﺯﻣﺎﻧﯽ ﮐﻪ ﺳﺎﻣﺎﻧﻪ ﺍﺯ ﻧﻈﺮ ﺑﺎﺭ ﺍﻟﮑﺘﺮﯾﮑﯽ ﺩﺭ ﺣﺎﻟﺖ ﺧﻨﺜﯽ ﻗﺮﺍﺭ ﺩﺍﺭﺩ ﯾﻌﻨﯽ ﻫﯿﭻ ﺁﻻﯾﺸﯽ ﺩﺭ ﺁﻥ ﺍﺗﻔﺎﻕ ﻧﯿﻔﺘﺎﺩﻩ ﺍﺳﺖ ﻭ ﺑﺮﺍﯼ ﺯﻭﺍﯾﺎﯼ ﭼﺮﺧﺶ ﮐﻮﭼﮏ، ﭼﻬﺎﺭ ﻧﻮﺍﺭ ﺍﻧﺮﮊﯼ
ﺗﺨﺖ ﺩﺭ ﺳﺎﺧﺘﺎﺭ ﻧﻮﺍﺭﯼ ﺳﺎﻣﺎﻧﻪ ﻇﺎﻫﺮ ﻣﯽﺷﻮﺩ ﮐﻪ ﺑﺎﻋﺚ ﺑﺎﺯﺑﻬﻨﺠﺎﺭﺵ ﺳﺮﻋﺖ ﻓﺮﻣﯽ ﺍﻟﮑﺘﺮﻭﻥﻫﺎ ﺩﺭ ﻧﺰﺩﯾﮑﯽ ﻧﻘﺎﻁ ﺩﯾﺮﺍﮎ ﻣﯽﮔﺮﺩﺩ.
ﺩﺭ ﺑﺮﺧﯽ ﺯﺍﻭﯾﻪﻫﺎﯼ ﭼﺮﺧﺶ ﻧﺰﺩﯾﮏ ﺑﻪ ﺻﻔﺮ، ﺳﺮﻋﺖ ﺩﺭ ﻧﻘﻄﮥ ﺩﯾﺮﺍﮎ ﻭﺍﻗﻌﺎ ﺑﻪ ﺻﻔﺮ ﻣﯽﺭﺳﺪ ﮐﻪ ﺍﯾﻦ ﺯﺍﻭﯾﻪﻫﺎ ﺭﺍ ﺯﺍﻭﯾﻪﻫﺎﯼ ﺟﺎﺩﻭﯾﯽ
ﻣﯽﮔﻮﯾﻨﺪ. ﻧﺎﻫﻤﺘﺮﺍﺯﯼ ﺩﻭ ﻻﯾﮥ ﮔﺮﺍﻓﯿﻦ ﺩﺭ ﺯﺍﻭﯾﮥ ﺟﺎﺩﻭﯾﯽ ﺑﺎﻋﺚ ﺍﯾﺠﺎﺩ ﯾﮏ ﺷﺒﮑﮥ ﻣﺘﻨﺎﻭﺏ ﺑﺎ ﺩﻭﺭﮤ ﺗﻨﺎﻭﺏ ﺑﺴﯿﺎﺭ ﺑﺰﺭﮔﺘﺮ ﺍﺯ ﺩﻭﺭﮤ ﺗﻨﺎﻭﺏ ﺷﺒﮑﮥ ﮔﺮﺍﻓﯿﻦ ﻣﯽﺷﻮﺩ. ﺍﯾﻦ ﺷﺒﮑﮥ ﺟﺪﯾﺪ ﺭﺍ ﺷﺒﮑﮥ ﻣﺎﺭﻩ ﻣﯽﮔﻮﯾﻨﺪ ﮐﻪ ﻧﻘﺶ ﺍﺳﺎﺳﯽ ﺩﺭ ﺗﺸﮑﯿﻞ ﻧﻮﺍﺭﻫﺎﯼ ﺗﺨﺖ ﺩﺍﺭﺩ. ﻣﺪﻝ ﻣﻮﺛﺮﯼ ﮐﻪ ﺑﺮﺍﯼ ﺗﻮﺻﯿﻒ ﺍﯾﻦ ﺳﺎﻣﺎﻧﻪ ﺍﺳﺘﻔﺎﺩﻩ ﻣﯽﮐﻨﯿﻢ ﺑﺮ ﺍﺳﺎﺱ ﺍﯾﻦ ﺷﺒﮑﮥ ﻣﺎﺭﻩ ﺳﺎﺧﺘﻪ ﻣﯽﺷﻮﺩ. ﺍﯾﻦ ﻣﺪﻝ ﮐﻪ ﺗﻮﺳﻂ ﮐﺎﻧﮓ ﻭ ﻭﺍﻓﮏ ﭘﯿﺸﻨﻬﺎﺩ ﺷﺪﻩ ﺍﺳﺖ
ﺷﺎﻣﻞ ﭼﻬﺎﺭ ﺍﻭﺭﺑﯿﺘﺎﻝ ﻭﺍﻧﯿﺮ ﺍﺳﺖ ﮐﻪ ﺩﺭ ﺟﺎﯾﮕﺎﻩﻫﺎﯼ ﺷﺒﮑﮥ ﻣﺎﺭﮤ ﻻﻧﻪ ﺯﻧﺒﻮﺭﯼ ﻗﺮﺍﺭ ﮔﺮﻓﺘﻪﺍﻧﺪ. ﻋﻼﻭﻩ ﺑﺮ ﺟﻤﻼﺕ ﭘﺮﺵ ﺑﯿﻦ ﺍﻭﺭﺑﯿﺘﺎﻝﻫﺎ، ﺑﺮﻫﻤﮑﻨﺶﻫﺎﯼ ﺩﺭﻭﻥ−ﺍﻭﺭﺑﯿﺘﺎﻟﯽ ﻭ ﺑﯿﻦ−ﺍﻭﺭﺑﯿﺘﺎﻟﯽ ﺭﺍ ﻫﻢ ﺑﻪ ﻫﺎﻣﯿﻠﺘﻮﻧﯽ ﺳﺎﻣﺎﻧﻪ ﺍﺿﺎﻓﻪ ﮐﺮﺩﻩ ﻭ ﺍﺯ ﯾﮏ ﻣﺪﻝ ﻫﺎﺑﺎﺭﺩ ﺗﻌﻤﯿﻢ ﯾﺎﻓﺘﻪ ﺑﺮﺍﯼ ﺗﻮﺻﯿﻒ ﺁﻥ ﺍﺳﺘﻔﺎﺩﻩ ﻣﯽﮐﻨﯿﻢ. ﻣﺤﺎﺳﺒﺎﺕ ﻣﺎ ﺑﺮ ﺍﺳﺎﺱ ﺭﻭﺵﻫﺎﯼ ﺧﻮﺷﻪﺑﻨﺪﯼ ﮐﻮﺍﻧﺘﻮﻣﯽ ﺍﻧﺠﺎﻡ ﻣﯽﮔﯿﺮﺩ، ﺑﺪﯾﻦ ﺗﺮﺗﯿﺐ ﮐﻪ ﺑﺮﺍﯼ ﺑﺮﺭﺳﯽ ﺍﺑﺮﺭﺳﺎﻧﺎﯾﯽ ﺍﺯ ﺭﻭﺵ ⅭⅮⅯFT ﻭ ﺧﻮﺷﻪﻫﺎﯾﯽ ﻣﺘﺸﮑﻞ ﺍﺯ ﭼﻬﺎﺭ ﺟﺎﯾﮕﺎﻩ ﺷﺒﮑﻪ ﮐﻪ ﺩﺭ ﻣﯿﺎﻥ 6 ﺍﻭﺭﺑﯿﺘﺎﻝ ﺣﻤﺎﻡ ﻗﺮﺍﺭ ﮔﺮﻓﺘﻪﺍﻧﺪ ﺍﺳﺘﻔﺎﺩﻩ ﻣﯽﮐﻨﯿﻢ. ﺩﺭ ﻣﻄﺎﻟﻌﮥ ﻓﺎﺯ ﻋﺎﯾﻖ ﻫﻤﺒﺴﺘﻪ ﺍﺯ ﺭﻭﺵ VⅭA ﺍﺳﺘﻔﺎﺩﻩ ﮐﺮﺩﻩ ﻭ ﺧﻮﺷﻪﻫﺎﯾﯽ ﻣﺘﺸﮑﻞ ﺍﺯ 21 ﺍﻭﺭﺑﯿﺘﺎﻝ ﻭﺍﻧﯿﺮ ﺭﺍ ﺍﻧﺘﺨﺎﺏ ﻣﯽﮐﻨﯿﻢ ﺗﺎ ﺑﺘﻮﺍﻧﯿﻢ ﺑﺮﻫﻤﮑﻨﺶﻫﺎﯼ ﺑﯿﻦ ﺍﻭﺭﺑﯿﺘﺎﻟﯽ ﺩﺭ ﯾﮏ ﺧﻮﺷﻪ ﺭﺍ ﺗﻌﺮﯾﻒ ﻧﻤﺎﯾﯿﻢ. ﺧﻼﺻﻪ ﻧﺘﺎﯾﺠﯽ ﮐﻪ ﺑﻪ ﮐﻤﮏ ﺍﯾﻦ ﻣﺪﻝﺳﺎﺯﯼ ﺑﺪﺳﺖ ﺁﻭﺭﺩﻩﺍﯾﻢ
ﺑﺼﻮﺭﺕ ﺯﯾﺮ ﺍﺳﺖ. ﯾﮏ ﭘﺎﺭﺍﻣﺘﺮ ﻧﻈﻢ ﺍﺑﺮﺭﺳﺎﻧﺎﯾﯽ ﻏﯿﺮ ﺻﻔﺮ ﺑﺎ ﺗﻘﺎﺭﻥ ±ip p ﺩﺭ ﻣﺤﺪﻭﺩﮤ ﮔﺴﺘﺮﺩﻩﺍﯼ ﺍﺯ ﭼﮕﺎﻟﯽ ﺣﺎﻣﻞﻫﺎ ﺑﺪﺳﺖ ﺁﻣﺪﻩ ﺍﺳﺖ. ﺍﯾﻦ ﻣﺤﺪﻭﺩﻩ
ﺍﺯ ﭼﮕﺎﻟﯽ ﮐﻪ ﺍﺑﺮﺭﺳﺎﻧﺎﯾﯽ ﺩﺭ ﺁﻥ ﺍﺗﻔﺎﻕ ﻣﯽﺍﻓﺘﺪ ﺗﻄﺎﺑﻖ ﻗﺎﺑﻞ ﻗﺒﻮﻟﯽ ﺑﺎ ﻣﺸﺎﻫﺪﺍﺕ ﺗﺠﺮﺑﯽ ﺩﺍﺭﺩ. ﺍﻧﺪﺍﺯﻩ ﮔﯿﺮﯼﻫﺎﯼ ﺗﺠﺮﺑﯽ ﻧﺸﺎﻥ ﻣﯽﺩﻫﻨﺪ ﮐﻪ ﺳﺎﻣﺎﻧﻪ ﺩﺭ ﻣﺤﺪﻭﺩﮤ ﭼﮕﺎﻟﯽﻫﺎﯼ 5٫0 = n ﻭ 5٫1 = n ﺭﻓﺘﺎﺭﯼ ﺷﺒﯿﻪ ﺑﻪ ﻋﺎﯾﻖ ﻣﺎﺕ ﺩﺍﺭﺩ، ﺑﻨﺎﺑﺮﺍﯾﻦ ﺍﻧﺘﻈﺎﺭ ﺩﺍﺭﯾﻢ ﮐﻪ ﺍﺑﺮﺭﺳﺎﻧﺎﯾﯽ ﺩﺭ ﺍﯾﻦ ﻣﺤﺪﻭﺩﻩﻫﺎ ﺳﺮﮐﻮﺏ ﻭ ﯾﺎ ﺩﺭ ﺑﻬﺘﺮﯾﻦ ﺣﺎﻟﺖ ﺍﺯ ﺑﯿﻦ ﺑﺮﻭﺩ ﮐﻪ ﺍﯾﻦ ﺍﻧﺘﻈﺎﺭ ﺗﻮﺳﻂ ﻣﺤﺎﺳﺒﺎﺗﯽ ﮐﻪ ﺍﻧﺠﺎﻡ ﺩﺍﺩﻩﯾﻢ ﺗﺎﯾﯿﺪ ﻣﯽﺷﻮﺩ. ﻋﻼﻭﻩ ﺑﺮ ﺍﺑﺮﺭﺳﺎﻧﺎﯾﯽ ﺍﺯ ﻧﻮﻉ ip ± p، ﻓﺎﺯ ﺍﺑﺮﺭﺳﺎﻧﺎﯾﯽ ﺩﯾﮕﺮﯼ ﺍﺯ ﻧﻮﻉ ﺍﺳﭙﯿﻦ ﯾﮏﺗﺎﯾﯽ ﻭ ﺑﺎ ﺗﻘﺎﺭﻥ id ± d ﻧﯿﺰ ﺑﺮﺍﯼ ﺳﺎﻣﺎﻧﻪ ﺑﺪﺳﺖ ﺁﻭﺭﺩﻩﯾﻢ ﮐﻪ ﺍﻧﺪﺍﺯﮤ ﭘﺎﺭﺍﻣﺘﺮ ﻧﻈﻢ ﺁﻥ ﻧﺴﺒﺖ ﺑﻪ ﻓﺎﺯ ip ± p ﮐﻮﭼﮑﺘﺮ ﺍﺳﺖ. ﺑﺎ ﻣﺤﺎﺳﺒﮥ ﺗﺎﺑﻊﻧﻤﺎﯼ ﭘﺎﺗﻮﻑ ﺑﺮﺍﯼ ﺩﻭ ﺍﺑﺮﺭﺳﺎﻧﺎﯾﯽ ﺑﺎ ﺗﻘﺎﺭﻥ ip ± p ﻭ id ± d ﺑﻪ ﺍﯾﻦ ﻧﺘﯿﺠﻪ ﻣﯽﺭﺳﯿﻢ ﮐﻪ ﺍﺑﺮﺭﺳﺎﻧﺎﯾﯽ ﺑﺎ ﺗﻘﺎﺭﻥ ip ± p ﺍﻧﺮﮊﯼ ﺁﺯﺍﺩ ﮐﻮﭼﮑﺘﺮﯼ ﺩﺍﺭﺩ ﻭ ﺩﺭ ﻧﺘﯿﺠﻪ ﻓﺎﺯ ﻏﺎﻟﺐ ﺩﺭ ﺑﯿﻦ ﺍﯾﻦ ﺩﻭ ﻣﻮﺭﺩ ﺍﺳﺖ. ﻋﻼﻭﻩ ﺑﺮ ﻓﺎﺯ ﺍﺑﺮﺭﺳﺎﻧﺎﯾﯽ، ﺑﺮﺭﺳﯽ ﻓﺎﺯ ﻋﺎﯾﻖ ﻫﻤﺒﺴﺘﮥ ﻗﻮﯼ ﮐﻪ ﺩﺭ ﺁﺯﻣﺎﯾﺸﮕﺎﻩ ﻣﺸﺎﻫﺪﻩ ﺷﺪﻩ ﺍﺳﺖ، ﻧﯿﺰ ﺟﺰﻭ ﺍﻫﺪﺍﻑ ﺍﯾﻦ ﭘﺎﯾﺎﻥﻧﺎﻣﻪ ﺑﻮﺩﻩ ﺍﺳﺖ. ﻣﺤﺎﺳﺒﺎﺕ ﻣﺎ ﻭﺟﻮﺩ ﻓﺎﺯ ﻋﺎﯾﻖ ﻫﻤﺒﺴﺘﮥ ﻗﻮﯼ ﺩﺭ ﻣﺤﺪﻭﺩﮤ ﭼﮕﺎﻟﯽﻫﺎﯾﯽ ﮐﻪ ﺩﺭ ﺍﻧﺪﺍﺯﻩ ﮔﯿﺮﯼﻫﺎﯼ ﺗﺠﺮﺑﯽ ﻣﺸﺎﻫﺪﻩ ﺷﺪﻩﺍﻧﺪ ﺭﺍ ﺗﺎﯾﯿﺪ ﻣﯽﮐﻨﺪ. ﻫﻤﭽﻨﯿﻦ ﺑﺮﺭﺳﯽﻫﺎﯾﯽ ﺑﺮﺍﯼ ﺗﺎﯾﯿﺪ ﯾﺎ ﺭﺩ ﻭﺟﻮﺩ ﻓﺎﺯﻫﺎﯼ ﭘﺎﺩﻓﺮﻭﻣﻐﻨﺎﻃﯿﺲ ﻭ ﺁﺭﺍﯾﺶ ﺑﺎﺭ ﻣﻨﻈﻢ ﺩﺭ ﻣﺤﺪﻭﺩﮤ ﭼﮕﺎﻟﯽﻫﺎﯾﯽ ﮐﻪ
ﺳﺎﻣﺎﻧﻪ ﺩﺭ ﻓﺎﺯ ﻋﺎﯾﻖ ﻗﺮﺍﺭ ﺩﺍﺭﺩ ﺍﻧﺠﺎﻡ ﮔﺮﻓﺖ ﮐﻪ ﻭﺟﻮﺩ ﻫﯿﭻ ﮐﺪﺍﻡ ﺍﺯ ﺍﯾﻦ ﻓﺎﺯﻫﺎﯼ ﻣﻨﻈﻢ ﺗﺎﯾﯿﺪ ﻧﺸﺪ.
چكيده انگليسي :
The recent discovery of a correlated insulating state and superconductivity in twisted bilayer Graphene (TBG) has paved a new platform for study- ing unconventional superconductivity. The fact that superconductivity in TBG arises from doping a half filled insulating state makes it even more exciting because of the similarity with the high Tc cuprates. The super- conductivity appears for a twisted bilayer Graphene at an angle 1.08◦ for a very low carrier density of about 1011 cm−2, few orders of magnitude lower than cuprate superconductors, with a Tc of 1.7 K, which is much higher to be explained by the BCS theory given the carrier density.
Such superconducting behaviour is attributed to electron-electron inter- actions in the bilayer Graphene at specific angles. The flattening of bands were observed around the charge neutrality for the bilayer Graphene (charge neutrality means the undoped case) at low twist angles, which lead to the renormalization of the velocity around Dirac points. This indicated the presence of correlation effects in TBG. It was also observed that at specific angles of rotation close to zero, the velocity at the Dirac points vanishes completely, such angles being called the magic angles.
An important experimental observation is the existence of a gap of around 40 meV between the flat energy bands around charge neutrality and the excited bands in both the electron doped and hole doped sides. This gap seems to decouple the physics of the flat bands around charge neutrality from the rest of the bands, so that one can study the physics of magic- angle TBG by only focusing on those bands. It has been identified that the symmetry properties of the low-energy flat bands is crucial to understand the nature of the correlated orbitals and hence to construct a Hubbard model for the system.
The bilayer Graphene is characterized by various features due to the mis- match of the lattice points of both layers, called the moiré patterns. The exact nature of the pattern is related to the relative orientation between the two Graphene layers, characterized by a displacement d between the two layers and an angle θ starting from a given original relative configura- tion of the layers. The moiré patterns strongly modulate the hybridization between the two layers of Graphene. For generic values of the displacement d and the angle θ the moiré pattern is not essentially periodic and does not form a Bravais lattice. Only for specific values of d and θ does the moiré lattice become periodic. It is further found that the moiré pattern at low angles is effectively insensitive to d, which just shifts the pattern in real-space.
The construction of an effective theoretical model to study magic-angle TBG has proved to be a great challenge. First of all, the moiré unit cell at the magic-angle is large, containing thousands of atoms. It has been difficult to construct the effective real-space Wannier orbitals correspond- ing to the low-energy bands, which describe the physics of the magic-angle TBG.
In this thesis, we have used a four-orbital model to describe flat bands. We have defined electron-electron interactions within the orbital as well as long-range interactions in the effective Hamiltonian of the system. With the help of this model and using quantum cluster methods, we have been able to confirm the existence of superconductivity and strongly correlated phases.
