محاضرة فيزياء المواد (محاضرة 10) PDF

‫ﺍﻟﺩﻛﺗﻭﺭ ﻣﺣﻣﺩ ﺃﺣﻣﺩ ﺁﻟﺟﻼﻟﻲ – ﻗﺳﻡ ﺍﻟﻔﻳﺯﻳﺎء – ﻛﻠﻳﺔ ﺍﻟﻣﻌﻠﻣﻳﻥ – ﺟﺎﻣﻌﺔ ﺍﻟﻁﺎﺋﻑ‬ ‫ﺳﻠﺳﻠﺔ ﻣﺣﺎﺿﺭﺍﺕ ﻓﻳﺯﻳﺎء ﺍﻟﺟﻭﺍﻣﺩ )ﺍﻟﻣﺣﺎﺿﺭﺓ ﺍﻟﻌﺎﺷﺭﺓ(‬ ‫ﺍﻟﻤﺴﺘﻮﻳﺎﺕ ﺍﻟﺬﺭﻳﺔ ﻳﻨﺸﻄﺮ ﻛﻞ ﻣﻨﻬﺎ ﺇﻟﻰ ﻣﺴﺘﻮﻳﻴﻦ ﺃﻭ ﺃﻥ ﻛﻞ ﺫﺭﺓ ﺗﻘﺪﻡ ﻣﺴﺘﻮﻳﺎﺕ ﻁﺎﻗﺘﻬﺎ ﻟﺘﺸﻜﻞ ﺍﻟﻤﺴﺘﻮﻳﺎﺕ ﺍﻟﺠﺰﻳﺌﻴﺔ ‪،‬ﻭﻓﻲ ﺣﺎﻝ ﺛﻼﺛﺔ ﺫﺭﺍﺕ‬ ‫ﻓﺎﻥ ﺍﻟﻤﺴﺘﻮﻱ ﺍﻟﺬﺭﻱ ﻳﺼﺒﺢ ﺛﻼﺛﺔ ﻣﺴ...

‫ﺍﻟﺩﻛﺗﻭﺭ ﻣﺣﻣﺩ ﺃﺣﻣﺩ ﺁﻟﺟﻼﻟﻲ – ﻗﺳﻡ ﺍﻟﻔﻳﺯﻳﺎء – ﻛﻠﻳﺔ ﺍﻟﻣﻌﻠﻣﻳﻥ – ﺟﺎﻣﻌﺔ ﺍﻟﻁﺎﺋﻑ‬ ‫ﺳﻠﺳﻠﺔ ﻣﺣﺎﺿﺭﺍﺕ ﻓﻳﺯﻳﺎء ﺍﻟﺟﻭﺍﻣﺩ )ﺍﻟﻣﺣﺎﺿﺭﺓ ﺍﻟﻌﺎﺷﺭﺓ(‬ ‫ﺍﻟﻤﺴﺘﻮﻳﺎﺕ ﺍﻟﺬﺭﻳﺔ ﻳﻨﺸﻄﺮ ﻛﻞ ﻣﻨﻬﺎ ﺇﻟﻰ ﻣﺴﺘﻮﻳﻴﻦ ﺃﻭ ﺃﻥ ﻛﻞ ﺫﺭﺓ ﺗﻘﺪﻡ ﻣﺴﺘﻮﻳﺎﺕ ﻁﺎﻗﺘﻬﺎ ﻟﺘﺸﻜﻞ ﺍﻟﻤﺴﺘﻮﻳﺎﺕ ﺍﻟﺠﺰﻳﺌﻴﺔ ‪،‬ﻭﻓﻲ ﺣﺎﻝ ﺛﻼﺛﺔ ﺫﺭﺍﺕ‬ ‫ﻓﺎﻥ ﺍﻟﻤﺴﺘﻮﻱ ﺍﻟﺬﺭﻱ ﻳﺼﺒﺢ ﺛﻼﺛﺔ ﻣﺴﺘﻮﻳﺎﺕ ﻣﻨﻔﺼﻠﺔ ﻗﺮﻳﺒﺔ ﺟﺪﺍ ﻣﻦ ﺑﻌﻀﺎﻫﺎ ﺍﻟﺒﻌﺾ ﻭﻫﻜﺬﺍ ﻧﺠﺪ ﺃﻧﻔﺴﻨﺎ ﺃﻣﺎﻡ ﻋﺪﺓ ﻣﺴﺘﻮﻳﺎﺕ ﻟﻬﺎ ﻋﺮﺽ‬ ‫ﻳﺘﻨﺎﺳﺐ ﻣﻊ ﻋﺪﺩ ﺍﻟﻤﺲﺕﻭﻳﺎﺕ ﺍﻟﻄﺎﻗﻴﺔ ﻧﺴﻤﻲ ﻫﺬﺍ ﺍﻟﻌﺮﺽ ﺑﺸﺮﻳﻂ ﺍﻟﻄﺎﻗﺔ)‪. (Energy band‬‬ ‫ﺃﻱ ﺃﻧﻪ ﻛﻠﻤﺎ ﺍﻧﻀﻤﺖ ﺫﺭﺓ ﺗﻀﺎﻑ ﺳﻮﻳﺔ ﻁﺎﻗﺔ ﻣﺰﺩﻭﺟﺔ ﻟﻜﻞ ﺳﻮﻳﺔ ﻣﻦ ﺍﻟﺴﻮﻳﺎﺕ ‪ 1s,2s,2p…..‬ﻓﻔﻲ ﺣﺎﻝ ‪ N‬ﺫ ﺭﺓ ﻓﺈﻥ ﻛﻞ ﺳﻮﻳﺔ ﻁﺎﻗﺔ ﺫﺭﻳﺔ‬ ‫ﺗﻨﺸﻄﺮ ﺇﻟﻰ ‪ N‬ﺳﻮﻳﺔ ﻓﺮﻋﻴﺔ ﻣﺘﻘﺎﺭﺑﺔ ﻟﺪﺭﺟﺔ ﺃﻧﻪ ﻳﻤﻜﻦ ﺍﻋﺘﺒﺎﺭﻫﺎ ﻣﺴﺘﻤﺮﺓ ﻭﺗﺸﻜﻞ ﺍﻟﺸﺮﻳﻂ ﺍﻟﻄﺎﻗﻲ ﺑﺪﻻ ﻣﻦ ﺍﻟﻤﺴﺘﻮﻱ ﺍﻟﻄﺎﻗﻲ ﻓﻲ ﺍﻟﺬﺭﺓ‬ ‫ﺍﻟﻤﻌﺰﻭﻟﺔ ﻭﺍﻥ ﺷﺮﻳﻂ ﺍﻟﻄﺎﻗﺔ ﺳﻴﺤﻮﻱ )‪ 2N(2Ɩ+1‬ﺳﻮﻳﺔ ﻁﺎﻗﺔ ﻛﻞ ﺳﻮﻳﺔ ﻁﺎﻗﺔ ﻓﻲ ﻫﺬﺍ ﺍﻟﺸﺮﻳﻂ ﺟﺎﻫﺰﺓ ﻻﺳﺘﻴﻌﺎﺏ ﺇﻟﻜﺘﺮﻭﻥ ﻭﺍﺣﺪ ﺣﺴﺐ ﻣﺒﺪﺃ‬ ‫ﺑﺎﻭﻟﻲ ﻓﻲ ﺍﻻﺳﺘﻴﻌﺎﺏ ﺍﻟﻜﺘﺮﻭﻧﻲ ﺍﻟﺸﻜﻞ )‪.(1‬‬ ‫ﺇﻥ ﺍﻛﺒﺮ ﺍﺛﺮ ﻟﻠﺒﻠﻮﺭﺓ ﻋﻠﻰ ﺍﻻﻟﻜﺘﺮﻭﻧﺎﺕ ﻳﻜﻮﻥ ﻋﻠﻰ ﺍﻟﻜﺘﺮﻭﻧﺎﺕ ﺍﻟﺘﻜﺎﻓﺆ ﻭﺗﻜﻮﻥ ﺷﺮﺍﺋﻂ ﺍﻟﻄﺎﻗﺔ ﻟﻬﺬﻩ ﺍﻻﻟﻜﺘﺮﻭﻧﺎﺕ ﺃﻋﺮﺽ ﺍﻟﺸﺮﺍﺋﻂ ﻭﺫﻟﻚ ﺑﻌﻜﺲ‬ ‫ﺍﻻﻟﻜﺘﺮﻭﻧﺎﺕ ﺍﻟﺪﺍﺧﻠﻴﺔ ﺍﻟﺘﻲ ﻻ ﺗﺘﺄﺛﺮ ﻛﺜﻴﺮﺍ ﺑﻮﺟﻮﺩ ﺍﻟﺬﺭﺍﺕ ﺍﻟﻤﺠﺎﻭﺭﺓ ﻭﺗﺒﻘﻰ ﺷﺮﺍﺋﻂ ﻁﺎﻗﺘﻬﺎ ﺿﻴﻘﺔ ‪.‬ﻭﻣﻦ ﺍﻟﻤﻬﻢ ﺟﺪﺍ ﺃﻥ ﻧﻼﺣﻆ ﺃﻧﻪ ﺷﺮﺍﺋﻂ ﺍﻟﻄﺎﻗﺔ ﻫﺬﻩ ﻫﻲ‬ ‫ﺍﻟﺸﺮﺍﺋﻂ ﺍﻟﻤﺴﻤﻮﺣﺔ ﻟﺘﻮﺍﺟﺪ ﺍﻻﻟﻜﺘﺮﻭﻧﺎﺕ ﻓﻴﻬﺎ ﺃﻣﺎ ﺍﻟﻔﺎﺻﻞ ﺑﻴﻦ ﺷﺮﻳﻄﻴﻦ ﻓﺎﻧﻪ ﻳﻤﺜﻞ ﻣﺠﺎﻻﺕ ﺃﻭ ﻓﻮﺍﺻﻞ ﻣﻤﻨﻮﻉ ﻋﻠﻰ ﺍﻹﻟﻜﺘﺮﻭﻥ ﺍﻟﺘﻮﺍﺟﺪ ﺑﻬﺎ ﻭﺗﺴﻤﻰ‬ ‫ﺑﺎﻟﻔﻮﺍﺻﻞ ﺍﻟﻤﻤﻨﻮﻋﺔ)‪ (forbidden bands‬ﺃﻭ ﻓﻮﺍﺻﻞ ﺍﻟﻄﺎﻗﺔ )‪. (Energy Gaps‬‬ ‫ﻭﻫﻜﺬﺍ ﻓﺎﻥ ﻁﻴﻒ ﺍﻟﻄﺎﻗﺔ ﻓﻲ ﺍﻟﺠﻮﺍﻣﺪ ﻳﺘﺄﻟﻒ ﻣﻦ ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﺷﺮﺍﺋﻂ ﺍﻟﻄﺎﻗﺔ ﺍﻟﻤﺴﻤﻮﺡ ﻟﻺﻟﻜﺘﺮﻭﻥ ﺍﻟﺘﻮﺍﺟﺪ ﺑﻬﺎ ﺗﻔﺼﻞ ﺑﻴﻨﻬﺎ ﻓﻮﺍﺻﻞ ﻁﺎﻗﺔ ﻻ ﻳﻤﻜﻦ ﺃﻥ‬ ‫ﺗﺸﻐﻠﻬﺎ ﺍﻻﻟﻜﺘﺮﻭﻧﺎﺕ ﺗﺴﻤﻰ ﻓﻮﺍﺻﻞ ﺍﻟﻄﺎﻗﺔ‪.‬ﺇﻥ ﻣﻘﺪﺍﺭ ﺍﻻﻧﺸﻄﺎﺭ ﻳﻌﺘﻤﺪ ﻋﻠﻰ ﺍﻟﻤﺴﺎﻓﺎﺕ ﺑﻴﻦ ﻧﻮﻯ ﺍﻟﺬﺭﺍﺕ ﻭﻛﻠﻤﺎ ﻛﺎﻥ ﺍﻻﻗﺘﺮﺍﺏ ﻛﺒﻴﺮﺍ ﻛﺎﻥ ﺍﻻﻧﺸﻄﺎﺭ ﺷﺪﻳﺪﺍ‬ ‫ﻭﻳﻜﻮﻥ ﺍﻟﺘﺄﺛﻴﺮ ﺍﻛﺒﺮ ﻋﻠﻰ ﺍﻻﻟﻜﺘﺮﻭﻧﺎﺕ ﺍﻟﺨﺎﺭﺟﻴﺔ ﻣﻨﻪ ﻋﻠﻰ ﺍﻟﺪﺍﺧﻠﻴﺔ ﺑﺴﺒﺐ ﺷﺪﺓ ﺍﺭﺗﺒﺎﻁ ﺍﻻﻟﻜﺘﺮﻭﻧﺎﺕ ﺍﻟﺪﺍﺧﻠﻴﺔ‪ ،‬ﺃﻣﺎ ﺍﻟﺨﺎﺭﺟﻴﺔ ﻓﻴﻤﻜﻦ ﺃﻥ ﺗﺘﺪﺍﺧﻞ ﺍﻟﺸﺮﺍﺋﻂ‬ ‫ﻣﻊ ﺑﻌﻀﻬﺎ ﺍﻟﺒﻌﺾ ﻟﺘﺸﻜﻞ ﺷﺮﻳﻂ ﻭﺍﺣﺪ ﺍﻟﺸﻜﻞ)‪.(2‬‬ ‫‪.3‬ﺩﺍﻟﺔ ﺑﻠﻮﺥ )‪: (The Bloch Function‬‬ ‫)‪ U(r‬ﺩﻭﺭﻳﺔ ﻓﻲ ﺑﻠﻮﺭﺓ ﻣﺜﺎﻟﻴﺔ ﺗﻨﺠﻢ ﻋﻦ ﻣﺤﺼﻠﺔ‬ ‫ﻟﻤﻌﺎﻟﺠﺔ ﻣﺴﺄﻟﺔ ﺷﺮﺍﺋﻂ ﺍﻟﻄﺎﻗﺔ ﻻﺑﺪ ﻣﻦ ﺍﻟﻌﻮﺩﺓ ﺇﻟﻰ ﻣﻌﺎﺩﻟﺔ ﺷﺮﻭﺩ ﻳﻨﺠﺮ ﺑﻮﺟﻮﺩ ﻁﺎﻗﺔ ﺟﻬﺪ‬ ‫ﺟﻬﺪﻳﻦ‪:‬‬ ‫ ﺟﻬﺪ ﺳﺎﻛﻨﺎ ﻳﻨﺘﺞ ﻋﻦ ﺻﻔﻮﻑ ﺍﻻﻳﻮﻧﺎﺕ ) ‪ (z‬ﺍﻟﻤﻮﺟﺒﺔ ﺍﻟﻤﺮﺗﺒﺔ ﺑﺸﻜﻞ ﻣﺜﺎﻟﻲ‪.‬‬ ‫‪+‬‬ ‫‪P‬‬ ‫‪P‬‬ ‫ ﺟﻬﺪ ﻧﺎﺟﻢ ﻋﻦ ﻛﻞ ﺍﻻﻟﻜﺘﺮﻭﻧﺎﺕ ﺍﻟﺨﺎﺭﺟﻴﺔ ﺍﻷﺧﺮﻯ )ﺍﻟﻜﺘﺮﻭﻧﺎﺕ ﺍﻟﺘﻮﺻﻴﻞ(‪ ،‬ﻭﻳﻌﺘﺒﺮ ﺑﻠﻮﺥ ﺃﻥ ﻛﺜﺎﻓﺔ ﺍﻟﺸﺤﻨﺔ ﺍﻟﻮﺳﻄﻴﺔ ﻟﺘﻠﻚ‬ ‫ﺍﻻﻟﻜﺘﺮﻭﻧﺎﺕ ﺛﺎﺑﺘﺔ ﻭﻫﻲ ﻧﻔﺴﻬﺎ ﻓﻲ ﻛﻞ ﺧﻠﻴﺔ ﻣﻦ ﺧﻼﻳﺎ ﺍﻟﺒﻠﻮﺭﺓ ﺍﻟﺸﻜﻞ)‪.(3‬‬  ‫ﺍﻟﺩﻛﺗﻭﺭ ﻣﺣﻣﺩ ﺃﺣﻣﺩ ﺁﻟﺟﻼﻟﻲ – ﻗﺳﻡ ﺍﻟﻔﻳﺯﻳﺎء – ﻛﻠﻳﺔ ﺍﻟﻣﻌﻠﻣﻳﻥ – ﺟﺎﻣﻌﺔ ﺍﻟﻁﺎﺋﻑ‬ ‫ﺳﻠﺳﻠﺔ ﻣﺣﺎﺿﺭﺍﺕ ﻓﻳﺯﻳﺎء ﺍﻟﺟﻭﺍﻣﺩ )ﺍﻟﻣﺣﺎﺿﺭﺓ ﺍﻟﻌﺎﺷﺭﺓ(‬ ‫ﻭﻓﻘﺎ ﻟﻤﺎ ﺳﺒﻖ ﻳﻌﺘﺒﺮ ﺍﻟﺠﻬﺪ ﺩﻭﺭﻱ ‪،‬ﻭﻳﺘﻢ ﺣﻞ ﻣﻌﺎﺩﻟﺔ ﺷﺮﻭﺩ ﻳﻨﺠﺮ ﻟﺠﺴﻴﻢ ﻭﺍﻗﻊ ﻓﻲ ﺑﺌﺮ ﺟﻬﺪ ﺫﻱ ﺍﺭﺗﻔﺎﻉ ﻣﺤﺪﻭﺩ‪،‬ﻭﻣﻨﻪ ﻳﻤﻜﻦ ﺃﻥ ﻧﻜﺘﺐ ﻣﻌﺎﺩﻟﺔ ﺷﺮﻭﺩ ﻳﻨﺠﺮ‬ ‫ﺑﻌﺪ ﺍﻷﺧﺬ ﺑﺎﻟﺤﺴﺒﺎﻥ ﺩﻭﺭﻳﺔ ﺍﻟﺠﻬﺪ ﻭﻓﻘﺎ ﻟﻤﺎ ﻳﻠﻲ‪:‬‬ ‫)‪U(r‬‬ ‫)‪= U(r + T‬‬ ‫‪ˆ = Eψ‬‬ ‫‪Ηψ‬‬ ‫‪2‬‬ ‫‪(−‬‬ ‫)‪+ U(r))ψ(r) =Eψ(r‬‬ ‫‪2m‬‬ ‫‪2m‬‬ ‫‪∇ 2 ψ(r) + 2 (E − U(r))ψ(r) =0‬‬ ‫)‪(1‬‬ ‫‪‬‬ ‫ﻭﻛﻤﺎ ﻧﻌﻠﻢ ﻣﻦ ﻣﻴﻜﺎﻧﻴﻜﺎ ﺍﻟﻜﻢ ﻓﺈﻥ ﺍﻟﻤﻌﺎﺩﻟﺔ ) ‪ (1‬ﺗﺨﻀﻊ ﻟﺤﺎﻟﺘﻴﻦ ﺍﻷﻭﻟﻰ ﻳﻜﻮﻥ ‪E>U‬ﻭﺗﺸﻤﻞ ﻓﻲ ﺣﺎﻟﺘﻨﺎ ﻫﺬﻩ ﺍﻻﻟﻜﺘﺮﻭﻧﺎﺕ ﺍﻟﺤﺮﺓ ﺍﻟﻤﻮﺟﻮﺩﺓ ﻓﻲ ﺍﻟﺸﺮﻳﻂ‬ ‫ﺍﻟﺨﺎﺭﺟﻲ)ﺷﺮﻳﻂ ﺍﻟﺘﻮﺻﻴﻞ ﻓﻲ ﺍﻟﻤﻌﺎﺩﻥ(‪ ،‬ﻭﺍﻟﺜﺎﻧﻴﺔ ﻳﻜﻮﻥ ‪ E